New exact multi line soliton and periodic solutions with constant asymptotic values at infinity of the NVN integrable nonlinear evolution equation via dibar-dressing method
aa r X i v : . [ n li n . S I] J a n New exact solutions with constant asymptoticvalues at infinity of the NVN integrable nonlinearevolution equation via ¯ ∂ -dressing method M.Yu. Basalaev, V.G. Dubrovsky and A.V. Topovsky
Novosibirsk State Technical University, Karl Marx prosp. 20, Novosibirsk630092, RussiaE-mail: [email protected]
Abstract.
The classes of exact multi line soliton, periodic solutions andsolutions with functional parameters, with constant asymptotic values at infinity u | ξ + η →∞ → − ǫ , for the hyperbolic and elliptic versions of the Nizhnik-Veselov-Novikov (NVN) equation via ¯ ∂ -dressing method of Zakharov and Manakov wereconstructed.At fixed time these solutions are exactly solvable potentials correspondinglyfor one-dimensional perturbed telegraph and two-dimensional stationarySchr¨odinger equations. Physical meaning of stationary states of quantum particlein exact one line and two line soliton potential valleys was discussed.In the limit ǫ → u (1) , u (2) (line solitons and periodicsolutions) were found which sum u (1) + u (2) (linear superposition) is also exactsolution of NVN equation.PACS numbers: 02.30.Ik, 02.30.Jr, 02.30.Zz, 05.45.Yv
1. Introduction
Exact solutions of differential equations of mathematical physics, linear and nonlinear,are very important for the understanding of various physical phenomena. In the lastthree decades the Inverse Scattering Transform (IST) method has been generalizedand successfully applied to several two-dimensional nonlinear evolution equations suchas Kadomtsev-Petviashvili, Davey-Stewartson, Nizhnik-Veselov-Novikov, Zakharov-Manakov system, Ishimori, two-dimensional integrable Sin-Gordon and others (seebooks [1]-[4] and references therein).The extension of nonlocal Riemann-Hilbert problem by Zakharov and Manakov [5]and ¯ ∂ -problem approach [6] led to the discovery of more general ¯ ∂ -dressing method [7]-[10] which became very powerful method for solving two-dimensional integrablenonlinear evolution equations. In the present paper the ¯ ∂ -dressing method of Zakharovand Manakov was used for the construction of the classes of exact multisoliton andperiodic solutions of the famous (2+1)-dimensional Nizhnik-Veselov-Novikov (NVN)integrable equation u t + κ u ξξξ + κ u ηηη + 3 κ ( u∂ − η u ξ ) ξ + 3 κ ( u∂ − ξ u η ) η = 0 , (1.1)where u ( ξ, η, t ) is scalar function, κ , κ are arbitrary constants, ξ = x + σy, η = x − σy ,and σ = ± ∂ ξ ≡ ∂∂ξ , ∂ η ≡ ∂∂η and ∂ − ξ , ∂ − η are operators inverse to ∂ ξ and ∂ η : ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ∂ − η ∂ η = ∂ − ξ ∂ ξ = 1. Equation (1.1) was first introduced and studied by Nizhnik [11]for hyperbolic version (NVN-II equation) with σ = 1 and independently by Veselovand Novikov [12] for elliptic version (NVN-I equation) with σ = i , κ = κ = κ . TheNVN equation is integrable by the IST due to representation of it as the compatibilitycondition for two linear auxiliary problems [11],[12]: L ψ = (cid:0) ∂ ξη + u (cid:1) ψ = 0 , (1.2) L ψ = (cid:0) ∂ t + κ ∂ ξ + κ ∂ η + 3 κ (cid:0) ∂ − η u ξ (cid:1) + 3 κ (cid:0) ∂ − ξ u η (cid:1)(cid:1) ψ = 0 (1.3)in the form of the Manakov’s triad[ L , L ] = BL , B = 3 (cid:0) κ ∂ − η u ξξ + κ ∂ − ξ u ηη (cid:1) . (1.4)The present paper is the continuation of Dubrovsky et al work and follows thenotations, review of the subject and general considerations presented in the previouspapers [22]-[24]. We apply the ¯ ∂ -dressing method of Zakharov and Manakov for theconstruction of classes of exact solutions with non-zero constant asymptotic values atinfinity: u ( ξ, η, t ) = ˜ u ( ξ, η, t ) + u ∞ = ˜ u ( ξ, η, t ) − ǫ, (1.5)where ˜ u ( ξ, η, t ) → ξ + η → ∞ . In this case the first linear auxiliary problem in(1.2) has the form: (cid:0) ∂ ξη + ˜ u (cid:1) ψ = ǫψ. (1.6)For σ = 1 with real space variables ξ ⇒ t − x, η ⇒ t + y equation (1.6) can beinterpreted as perturbed telegraph equation with potential u = ˜ u − ǫ or perturbedstring equation for ǫ = 0. For σ = i with complex space variables ξ ⇒ x + iy = z, η ⇒ x − iy = z equation (1.6) coincides with the famous two-dimensional stationarySchr¨odinger equation (cid:0) − ∂ z ¯ z + V Schr (cid:1) ψ = Eψ (1.7)with V Schr = − u and E = − ǫ . For this reason the construction via ¯ ∂ -dressingmethod of exact solutions of the NVN equations with constant asymptotic values atinfinity means simultaneous calculation of exact eigenfunctions (wave functions) ψ andexactly solvable potentials u = ˜ u − ǫ and V Schr = − u for above mentioned famouslinear equations.The inverse scattering transform for the first auxiliary linear problem (1.6) (orin particular for 2D Schr¨odinger equation (1.7)) has been developed in a number ofpapers. Detailed review one can find in the book of Konopelchenko [3]. On the basis ofdeveloped for (1.6) IST using time evolution given by second auxiliary problem (1.3)several classes of exact solutions of NVN equation were constructed [3], [4],[11]-[21].Some exact solutions of NVN-II equation with σ = 1 were obtained in the work [11]via the transformation operators. Veselov et al constructed finite zone solutions ofNVN equation [12]. The classes of rational localized solutions of so called N V N − I ± -equation (with E >
E < ψ were presented in the works [14]-[16]. Special care requires thecase of E = 0 for (1.7), i. e. the case of N V N − I equation [17]. The use ofDarbu transformations for the construction of exact solutions of NVN equation wasdemonstrated by Matveev et al [18]. The class of dromion-like solutions of NVNequation via Mottard transformations was constructed by Athorne et al [21]. We havealready constructed classes of exact potentials for perturbed telegraph equation (1.6) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method u = ˜ u − ǫ and perturbed string equation with u = ˜ u, ǫ = 0 via ∂ -dressing method in the paper [22] and obtained some rationally localized solutions ofNVN-equation with simple and multiple pole wave functions ψ via ∂ -dressing method[23],[24].Present work is concentrated on further use of ∂ -dressing method for theconstruction of exact solutions of two-dimensional integrable nonlinear evolutionequations, exact potentials and wave functions of famous linear auxiliary problems(1.6) or (1.7) and the study of their possible applications. While many studies ofthis subject were performed the question of physical interpretation and exploitationof results obtained via ∂ -dressing are still of great interest.The paper is organized as described further. Basic ingredients of the ¯ ∂ -dressingmethod for the NVN equation (1.1) in brief are presented in sections 2,3 and generaldeterminant formula for multi line soliton solutions and useful formulas for theconditions of reality and potentiality of u are obtained. In sections 4 and 5 the classesof exact multi line soliton solutions for hyperbolic version with σ = 1 and for ellipticversion with σ = i of the NVN equation respectively are constructed. The classesof periodic solutions for both versions of NVN equation are constructed in section6. The classes of solutions with functional parameters are constructed in section 7.The simplest examples of exact one, two line soliton solutions with correspondingexact wave functions of auxiliary linear problems, periodic solutions and solutionswith functional parameters are presented in sections 3,4 and 5,6,7 of the paper.
2. Basic ingredients of the ¯ ∂ -dressing method and general determinantformulas for exact solutions As a matter of convenience here we briefly reviewed the basic ingredients of the ¯ ∂ -dressing method [7]-[10] for the NVN equation (1.1) in the case of u ( ξ, η, t ) withgenerically non-zero asymptotic value at infinity (1.6). We followed the treatment ofthe papers [23],[24] without repetition of theirs detailed calculations.At first one postulates the non-local ¯ ∂ -problem: ∂χ ( λ, ¯ λ ) ∂ ¯ λ = ( χ ∗ R )( λ, ¯ λ ) = Z Z C χ ( µ, µ ) R ( µ, µ ; λ, ¯ λ ) dµ ∧ dµ (2.1)where in our case χ and R are the scalar complex-valued functions and χ has canonicalnormalization: χ → λ → ∞ . It should be assumed that the problem (2.1) isunique solvable. Then one introduces the dependence of kernel R of the ¯ ∂ -problem(2.1) on the space and time variables ξ , η , t : ∂R∂ξ = iµR ( µ, µ ; λ, λ ; ξ, η, t ) − R ( µ, µ ; λ, λ ; ξ, η, t ) iλ,∂R∂η = − i ǫµ R ( µ, µ ; λ, λ ; ξ, η, t ) + R ( µ, µ ; λ, λ ; ξ, η, t ) i ǫλ , (2.2) ∂R∂t = i ( κ µ − κ ǫ µ ) R ( µ, µ ; λ, λ ; ξ, η, t ) − R ( µ, µ ; λ, λ ; ξ, η, t ) i ( κ λ − κ ǫ λ ) . Integrating (2.2) one obtains R ( µ, µ ; λ, ¯ λ ; ξ, η, t ) = R ( µ, µ ; λ, ¯ λ ) e F ( µ ; ξ,η,t ) − F ( λ ; ξ,η,t ) (2.3)where F ( λ ; ξ, η, t ) = i (cid:2) λξ − ǫλ η + (cid:0) κ λ − κ ǫ λ (cid:1) t (cid:3) . (2.4) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method D = ∂ ξ + iλ, D = ∂ η − i ǫλ , D = ∂ t + i (cid:16) κ λ − κ ǫ λ (cid:17) (2.5)expressing the dependence (2.2) of kernel R of the ¯ ∂ -problem (2.1) on the space andtime variables ξ , η , t in the following equivalent form[ D , R ] = 0 , [ D , R ] = 0 , [ D , R ] = 0 (2.6)one can construct the operators of auxiliary linear problems˜ L = X l,m,n u lmn ( ξ, η, t ) D l D m D n . (2.7)These operators must satisfy to the conditions h ∂∂λ , ˜ L i χ = 0 , ˜ Lχ ( λ, λ ) | λ →∞ → λ = 0 and λ = ∞ of the complex plane of spectralvariable λ . For such operators ˜ L the function ˜ Lχ obeys the same ¯ ∂ -equation as thefunction χ . There are may be several operators ˜ L i of this type, by virtue of the uniquesolvability of (2.1) one has ˜ L i χ = 0 for each of them. In considered case one constructstwo such operators:˜ L = D D + u D + u D + u, (2.9)˜ L = D + κ D + κ D + V D + V D + V D + V D + V. (2.10)Using the conditions (2.8) and series expansions of wave functions χ near the points λ = 0 and λ = ∞ χ = χ + χ λ + χ λ + . . . , χ = χ ∞ + χ − λ + χ − λ + . . . , (2.11)one obtains the reconstruction formulas for the field variables u , u and V , V , V , V through the coefficients χ and χ ∞ of expansions (2.11) (for calculation details seepapers [23],[24]): u = − χ ∞ η χ ∞ , V = − κ χ ∞ ξ χ ∞ ; (2.12) u = − χ ξ χ , V = − κ χ η χ ; (2.13) V = 3 iκ ǫχ η , V = − iκ χ − ξ . (2.14)According to well known terminology the operator ˜ L in (2.9) is pure potentialoperator when its first derivatives are absent. Due to canonical normalization of wavefunction χ | λ →∞ → χ ∞ = 1): u = − χ ∞ η χ ∞ = 0 , V = − κ χ ∞ ξ χ ∞ = 0 . (2.15)For zero value of the term u ∂ η in ˜ L one must to require χ = const , withoutrestriction we can choose χ = 1, and then due to (2.13) u = − χ ξ χ = 0 , V = − κ χ η χ = 0 . (2.16)Using (2.8),(2.12) - (2.16)(for calculation details see also [23],[24]) one obtains thefollowing expressions for V , V and u : V = 3 iκ ǫχ η = 3 κ ∂ − ξ u η , V = − iκ χ − ξ = 3 κ ∂ − η u ξ , (2.17) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method u = − ǫ − iχ − η = − ǫ + iǫχ ξ . (2.18)The field variable V in (2.10) due to gauge freedom [25] in the present paper is chosento be equal to zero. In terms of the wave function ψ := χe F ( λ ; ξ,η,t ) = χe i (cid:2) λξ − ǫλ η + (cid:0) κ λ − κ ǫ λ (cid:1) t (cid:3) , (2.19)under the reduction u = 0 and u = 0 (the condition of potentiality ˜ L ), one obtainsfrom (2.9),(2.10) due to (2.8) and (2.15)-(2.17) the linear auxiliary system (1.2),(1.3)and NVN integrable nonlinear equation (1.1) as compatibility condition (1.4) of linearauxiliary problems in (1.2), (1.3).The solution of the ¯ ∂ -problem (2.1) with constant normalization χ ∞ = 1 isequivalent to the solution of the following singular integral equation: χ ( λ ) = 1 + Z Z C dλ ′ ∧ d ¯ λ ′ πi ( λ ′ − λ ) Z Z C χ ( µ, ¯ µ ) R ( µ, µ ; λ, ¯ λ ) dµ ∧ d ¯ µ. (2.20)From (2.20) one obtains for the coefficients χ and χ − of the series expansions (2.11)of χ the following expressions: χ = 1 + Z Z C dλ ∧ d ¯ λ πiλ Z Z C χ ( µ, ¯ µ ) R ( µ, µ ; λ, ¯ λ ) e F ( µ ) − F ( λ ) dµ ∧ d ¯ µ (2.21)and χ − = − Z Z C dλ ∧ d ¯ λ πi Z Z C χ ( µ, ¯ µ ) R ( µ, µ ; λ, ¯ λ ) e F ( µ ) − F ( λ ) dµ ∧ d ¯ µ (2.22)where F ( λ ) is short notation for F ( λ ; ξ, η, t ) given by the formula (2.4). The conditionsof reality u and of potentiality of the operator ˜ L give some restrictions for the kernel R of the ¯ ∂ -problem (2.1). In the Nizhnik case ( σ = 1 , ¯ κ = κ , ¯ κ = κ ) of the NVNequations (1.1) with real space variables ξ = x + y , η = x − y the condition of realityof u leads from (2.18) and (2.22) in the limit of ”weak” fields ( χ = 1 in (2.22)) to thefollowing restriction for the kernel R of the ¯ ∂ - problem: R ( µ, µ ; λ, λ ) = R ( − µ, − µ ; − λ, − λ ) . (2.23)For the Veselov-Novikov case ( σ = i, κ = κ = κ = ¯ κ ) of the NVN equations (1.1)with complex space variables ξ = z = x + iy , η = ¯ z = x − iy the condition of realityof u leads from (2.18) and (2.22) in the limit of ”weak” fields to another restrictionon the kernel R of the ¯ ∂ - problem: R ( µ, µ ; λ, λ ) = ǫ | µ | | λ | µλ R ( − ǫλ , − ǫλ , − ǫµ − ǫµ ) . (2.24)The potentiality condition for the operator ˜ L in (2.9) for the choice χ = 1 due to(2.21) has the following form: χ − Z Z C dλ ∧ dλ πiλ Z Z C χ ( µ, µ ) R ( µ, µ ; λ, λ ) e F ( µ ) − F ( λ ) dµ ∧ dµ = 0 . (2.25)Here we obtained general formulas for multisoliton solutions corresponding to thedegenerate delta-kernel R : R ( µ, ¯ µ ; λ, ¯ λ ) = π X k A k δ ( µ − M k ) δ ( λ − Λ k ) . (2.26) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method χ ( λ ) due to (2.20) has the form: χ ( λ ) = 1 + 2 i X k A k Λ k − λ χ ( M k ) e F ( M k ) − F (Λ k ) . (2.27)The coefficient χ − due to (2.22) and (2.26) has the form: χ − = − i X k A k χ ( M k ) e F ( M k ) − F (Λ k ) . (2.28)For the wave functions χ ( M k ) from (2.27) one obtains the following system ofequations: X l ˜ A kl χ ( M l ) = 1 , ˜ A lk = δ lk + 2 iA k M l − Λ k e F ( M k ) − F (Λ k ) . (2.29)Instead of matrix ˜ A in (2.29) it is convenient to introduce matrix A given by expression A lk = δ lk + 2 iA k M l − Λ k e F ( M l ) − F (Λ k ) . (2.30)Both these matrices ˜ A in (2.29) and A (7.19) are connected by the relation A lk = e F ( M l ) ˜ A lk e − F ( M k ) . (2.31)From (2.29) due to (2.31) one derives the expression for the wave function χ at discretevalues of spectral variable: χ ( M l ) = X k ˜ A − lk = X k e F ( M k ) − F ( M l ) A − lk . (2.32)As a matter of convenience hereafter we described some useful formulas for wavefunctions satisfying to linear auxiliary problems (1.2),(1.3). From (2.19) and (2.32)one obtains the wave function ψ ( M l , ξ, η, t ) = χ ( M l ) e F ( M l ) at discrete points λ = M l in the space of spectral variables: ψ ( M l , ξ, η, t ) = χ ( M l ) e F ( M l ) = X k e F ( M k ) A − lk . (2.33)For the wave function (2.19) at arbitrary point λ from (2.27) - (2.32) follows theexpression: ψ ( λ, ξ, η, t ) = χ ( λ ) e F ( λ ) = h i X k A k Λ k − λ e F ( M k ) − F (Λ k ) χ ( M k ) i e F ( λ ) = h i X k,l A k Λ k − λ e − F (Λ k ) A − kl e F ( M l ) i e F ( λ ) . (2.34)Inserting (2.32) into (2.28) one obtains for the coefficient χ − χ − = − i X k,l A k e F ( M k ) − F (Λ k ) e F ( M l ) − F ( M k ) A − kl == − i X k,l A k e F ( M l ) − F (Λ k ) A − kl = i tr (cid:16) ∂A∂ξ A − (cid:17) . (2.35)and due to reconstruction formula u = − ǫ − iχ − η the convenient determinant formulafor the solution u of NVN equation (1.1): u = − ǫ + ∂∂η tr (cid:16) ∂A∂ξ A − (cid:17) = − ǫ + ∂ ∂ξ∂η ln(det A ) . (2.36) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method T r ( ∂A∂ξ A − ) = ∂∂ξ ln(det A ) , trD = det (1 + D ) (2.37)are used; the matrix D from last identity of (7.12) is degenerate with rank 1.Potentiality condition (2.25) by the use of (2.26)-(2.32) can be transformed to theform: χ − − ǫ N X k,l =1 A − kl B lk = 0 (2.38)where degenerate matrix B with rank 1 is defined by the formula B lk = − iǫ Λ k A k e F ( M l ) − F (Λ k ) . (2.39)Due to (7.12)-(7.20) potentiality condition (2.25) takes the form:0 = N X k,m =1 A − km B mk = tr ( A − B ) = det ( BA − + 1) − , (2.40)here matrix BA − is degenerate of rank 1 and in deriving the last equality in (7.16)the second matrix identity of (7.12) is used. Equivalently due to (7.16) the potentialitycondition takes the formdet( A + B ) = det A. (2.41)
3. Fulfilment of potentiality condition. General formulas for one line andtwo line solitons
Formula (2.36) for exact solutions u ( ξ, η, t ) of NVN equations (1.1) is effective if thereality ¯ u = u conditions (2.23),(2.24) and potentiality condition (2.25) of operator L are satisfied. This is the major and the most difficult part of all constructions.Here we demonstrated how one can to fulfil the condition of potentiality (2.25) bydelta-kernel with two terms: R ( µ, ¯ µ ; λ, ¯ λ ) = π (cid:16) Aδ ( µ − µ ) δ ( λ − λ ) + Bδ ( µ − µ ) δ ( λ − λ ) (cid:17) . (3.1)Inserting (3.1) into (2.25) one obtains in the limit of weak fields ( χ = 1 in (2.25)): χ − Z Z C iλ (cid:16) Aδ ( µ − µ ) δ ( λ − λ ) + Bδ ( µ − µ ) δ ( λ − λ ) (cid:17) ×× e F ( µ ) − F ( λ ) dµ ∧ dµ dλ ∧ dλ = 2 i (cid:0) Aλ e F ( µ ) − F ( λ ) + Bλ e F ( µ ) − F ( λ ) (cid:1) = 0 . (3.2)The equality (3.2) is valid if F ( µ ) − F ( λ ) = F ( µ ) − F ( λ ) , Aλ = − Bλ . (3.3)Due to the definition of F ( λ ) = i (cid:2) λξ − ǫλ η + (cid:0) κ λ − κ ǫ λ (cid:1) t (cid:3) from space-dependentpart of (3.3) the system of equations follows: µ − λ = µ − λ , ǫµ − ǫλ = ǫµ − ǫλ . (3.4) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method µ = λ , µ = λ ; 2) µ = − λ , µ = − λ . (3.5)The solution µ = λ , µ = λ corresponds to lump solution and will not be consideredhere, (for more information about lump solutions see [20], [21]). For the secondsolution µ = − λ , µ = − λ taking into account second relation from (3.3) oneobtains: Aλ = − Bλ = Bµ = a, (3.6)where a is some arbitrary complex constant. It is evident that to the potentialitycondition (2.25) the kernel R (which is the sum of expressions of the type (3.1) withparameters defined by (3.4)-(3.6)) R ( µ, ¯ µ ; λ, ¯ λ ) = π N X k =1 h a k λ k δ ( µ − µ k ) δ ( λ − λ k ) + a k µ k δ ( µ + λ k ) δ ( λ + µ k ) i == π N X k =1 A k δ ( M − M k ) δ (Λ − Λ k ) (3.7)with the sets of amplitudes A k and spectral parameters M k , Λ k ( A , .., A N ) := ( a λ , ..., a N λ N ; a µ , ..., a N µ N );( M , ..., M N ) := ( µ , ..., µ N ; − λ , ..., − λ N ) , (Λ , ..., Λ N ) := ( λ , ..., λ N ; − µ , ..., − µ N ) (3.8)satisfies.In order to avoid repetition of similar calculations in the following sections weprepared some useful formulas in general position for calculating one- and two- linesoliton solutions and corresponding wave functions. The determinants of matrix A (7.19) with parameters (3.8) corresponding to the simplest kernels (3.7) with N = 1and N = 2 have the forms:1 . N = 1 : det A = (cid:16) p e ∆ F ( µ ,λ ) (cid:17) ; (3.9)2 . N = 2 : det A = (cid:16) p e ∆ F ( µ ,λ ) + p e ∆ F ( µ ,λ ) + qe ∆ F ( µ ,λ )+∆ F ( µ ,λ ) (cid:17) (3.10)here p k , ∆ F ( µ k , λ k ) ( k = 1 ,
2) and q are given by the expressions p k := ia k µ k + λ k µ k − λ k ; ∆ F ( µ k , λ k ) := F ( µ k ) − F ( λ k ) , (3.11) q := − p p · ( λ − λ )( λ + µ )( µ − µ )( λ + µ )( λ + λ )( λ − µ )( µ + µ )( λ − µ ) . (3.12)The formula for one line soliton solution due to (2.36),(3.9) is: u ( ξ, η, t ) = − ǫ − ǫ p ( µ − λ ) µ λ e ∆ F ( µ ,λ ) (1 + p e ∆ F ( µ ,λ ) ) . (3.13)By using the equations (2.27),(7.19) and (2.32) corresponding to one line solitonsolution (3.13) wave functions one calculates:˜ χ := χ ( µ ) = χ ( − λ ) = 11 + p e ∆ F ( µ ,λ ) ; (3.14) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method χ ( λ ) = 1 − (cid:16) λ λ − λ + µ λ + µ (cid:17) ia e ∆ F ( µ ,λ ) p e ∆ F ( µ ,λ ) . (3.15)Considering (3.14), (3.15) wave functions ψ ( µ ) = χ ( µ ) e F ( µ ) , ψ ( − λ ) = χ ( − λ ) e F ( − λ ) and ψ ( λ ) = χ ( λ ) e F ( λ ) satisfy to linear auxiliary problems (1.2),(1.3) and at the same time to famous linear equations (1.6), (1.7) and have thefollowing forms: ψ ( µ ) = e F ( µ ) p e ∆ F ( µ ,λ ) , ψ ( − λ ) = e − F ( λ ) p e ∆ F ( µ ,λ ) ; (3.16) ψ ( λ ) = e F ( λ ) − (cid:16) λ λ − λ + µ λ + µ (cid:17) ia e ∆ F ( µ ,λ ) e F ( λ ) p e ∆ F ( µ ,λ ) . (3.17)For two line soliton solution one obtains via (2.36),(3.10) after simple calculations theexpression: u ( ξ, η, t ) = − ǫ − ǫ N ( ξ, η, t ) D ( ξ, η, t ) , (3.18)where the nominator N and denominator D are given by the expressions N ( ξ, η, t ) = ( λ − µ ) λ µ e ∆ F ( µ ,λ ) ( qp e F ( µ ,λ ) + p ) ++ ( λ − µ ) λ µ e ∆ F ( µ ,λ ) ( qp e F ( µ ,λ ) + p ) ++ p p ( λ − µ − λ + µ ) (cid:16) λ − µ λ µ − λ − µ λ µ (cid:17) e ∆ F ( µ ,λ )+∆ F ( µ ,λ ) ++ q ( λ − µ + λ − µ ) (cid:16) λ − µ λ µ + λ − µ λ µ (cid:17) e ∆ F ( µ ,λ )+∆ F ( µ ,λ ) , (3.19) D ( ξ, η, t ) = (1 + p e ∆ F ( µ ,λ ) + p e ∆ F ( µ ,λ ) + qe ∆ F ( µ ,λ )+∆ F ( µ ,λ ) ) . (3.20)It is remarkable that for the choice q = p p , i. e. under the condition( λ − λ )( λ + µ )( µ − µ )( λ + µ )( λ + λ )( λ − µ )( µ + µ )( λ − µ ) = − λ µ + λ µ )( λ − µ )( λ − µ ) = 0 (3.22)the formula for two line soliton solution (3.18) with N , D given by (3.19),(3.20) reducesto very simple expression: u ( ξ, η, t ) = − ǫ − ǫ p ( µ − λ ) µ λ e ∆ F ( µ ,λ ) (1 + p e ∆ F ( µ ,λ ) ) −− ǫ p ( µ − λ ) µ λ e ∆ F ( µ ,λ ) (1 + p e ∆ F ( µ ,λ ) ) . (3.23)It should be emphasized that in the present paper multi line soliton solutions areconsidered, for such solutions by construction µ k = λ k , ( k = 1 , q = p p satisfies if λ µ + λ µ = 0 . (3.24) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method p p = q , have thefollowing simple forms: χ ( µ ) = ˜ χ ˜ χ h − i a e ∆ F ( µ ,λ ) ( λ + λ )( λ + µ )( λ + µ )( λ − λ )( λ − µ )( λ − µ ) i , (3.25) χ ( − λ ) = ˜ χ ˜ χ h − i a e ∆ F ( µ ,λ ) ( λ − λ )( λ − µ )( λ + µ )( λ + λ )( λ + µ )( λ − µ ) i , (3.26) χ ( µ ) = ˜ χ ˜ χ h i a e ∆ F ( µ ,λ ) ( λ + λ )( λ − µ )( λ + µ )( λ − λ )( λ + µ )( λ − µ ) i , (3.27) χ ( − λ ) = ˜ χ ˜ χ h i a e ∆ F ( µ ,λ ) ( λ − λ )( λ + µ )( λ + µ )( λ + λ )( λ − µ )( λ − µ ) i , (3.28) χ ( λ ) = 1 + 2 i (cid:16) λ a λ − λ χ ( µ ) e ∆ F ( µ ,λ ) + µ a − µ − λ χ ( − λ ) e ∆ F ( µ ,λ ) ++ λ a λ − λ χ ( µ ) e ∆ F ( µ ,λ ) + µ a − µ − λ χ ( − λ ) e ∆ F ( µ ,λ ) (cid:17) , (3.29)where ˜ χ ˜ χ are the wave functions (see (3.14))˜ χ = χ ( µ ) = χ ( − λ ) = 11 + p e ∆ F ( µ ,λ ) , ˜ χ = χ ( µ ) = χ ( − λ ) = 11 + p e ∆ F ( µ ,λ ) (3.30)corresponding to one line soliton solutions. Two soliton ψ wave functions (2.33),(2.34) satisfying to linear auxiliary problems (1.2), (1.3) and at the same time tofamous linear equations (1.6), (1.7) due to (3.25)-(3.29) have following forms: ψ ( µ ) = e F ( µ ) p e ∆ F ( µ ,λ ) p e ∆ F ( µ ,λ ) ( λ + λ )( λ + µ )( λ − λ )( λ − µ ) p e ∆ F ( µ ,λ ) , (3.31) ψ ( − λ ) = e F ( − λ ) p e ∆ F ( µ ,λ ) p e ∆ F ( µ ,λ ) ( λ − λ )( λ − µ )( λ + λ )( λ + µ ) p e ∆ F ( µ ,λ ) , (3.32) ψ ( µ ) = e F ( µ ) p e ∆ F ( µ ,λ ) − p e ∆ F ( µ ,λ ) ( λ + λ )( λ − µ )( λ − λ )( λ + µ ) p e ∆ F ( µ ,λ ) , (3.33) ψ ( − λ ) = e − F ( λ ) p e ∆ F ( µ ,λ ) − p e ∆ F ( µ ,λ ) ( λ − λ )( λ + µ )( λ + λ )( λ − µ ) p e ∆ F ( µ ,λ ) , (3.34) ψ ( λ ) = e F ( λ ) + 2 i (cid:16) λ a λ − λ ψ ( µ ) e − F ( λ ) + µ a − µ − λ ψ ( − λ ) e F ( µ ) ++ λ a λ − λ ψ ( µ ) e − F ( λ ) + µ a − µ − λ ψ ( − λ ) e F ( µ ) (cid:17) e F ( λ ) . (3.35)All formulas (3.9)-(3.35) derived in the present section will be effective if thereality conditions (2.23), (2.24) are satisfied. The reality condition u = ¯ u imposesadditional restrictions on the parameters a k , λ k , µ k (3.8) of the kernel (3.7).These restrictions and the calculations of exact multi line soliton solutions u withcorresponding wave functions are suitable for hyperbolic and elliptic versions of NVNequation (1.1) separately. ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method
4. Exact multi line soliton solutions of NVN-II equation
In the present section the hyperbolic version of NVN equation (1.1) or NVN-IIequation, i. e. the case σ = 1 with real space variables ξ = x + y and η = x − y , willbe covered. In order to satisfy the reality condition (2.23) let us require for each termin the sum (3.7): a k λ k δ ( µ − µ k ) δ ( λ − λ k ) + a k µ k δ ( µ + λ k ) δ ( λ + µ k ) == a k λ k δ ( µ + µ k ) δ ( λ + λ k ) + a k µ k δ ( µ − λ k ) δ ( λ − µ k ) . (4.1)From (4.1) two possibilities follow:1 . a k λ k = a k λ k , a k µ k = a k µ k , µ k = − µ k , λ k = − λ k ; 2 . a k λ k = a k µ k , µ k = λ k . (4.2)In the first case in (4.2) one obtains that the spectral points µ k , λ k and amplitudes a k are pure imaginary: µ k = − µ k := iµ k , λ k = − λ k := iλ k , a k = − a k := − ia k . (4.3)For the second case in (4.2) it is appropiate to introduce the following notations foramplitudes and spectral points a k = a k := a ′ k ; λ ′ k , µ ′ k := λ ′ k . (4.4)So the kernel (2.26), (3.7) satisfying to potentiality (2.25) and reality (2.23) conditionsin considered two cases (4.2) due to (4.3), (4.4) can be chosen in the following form R ( µ, µ, λ, λ ) = π L + N ) X k =1 A k δ ( µ − M k ) δ ( λ − Λ k ) (4.5)of L pairs of the type π (cid:0) a l λ l δ ( µ − iµ l ) δ ( λ − iλ l )+ a l µ l δ ( µ + iλ l ) δ ( λ + iµ l ) (cid:1) , ( l =1 , ..., L ) and N pairs of the type π (cid:0) a ′ n λ ′ n δ ( µ − µ ′ n ) δ ( λ − λ ′ n )+ a ′ n λ ′ n δ ( µ + λ ′ n ) δ ( λ + µ ′ n ) (cid:1) ,with µ ′ n = ¯ λ ′ n ( n = 1 , ..., N ) of corresponding items. In (4.5) for application ofgeneral determinant formulas (7.19), (2.36) and (7.17) due to (4.3)-(4.5) the followingsets of amplitudes A k and spectral parameters M k , Λ k ( A , .., A L + N ) ) == ( a λ , .., a L λ L ; a µ , ..a L µ L ; a ′ λ ′ , .., a ′ N λ ′ N ; a ′ µ ′ , ..a ′ N µ ′ N ) , ( M , .., M L + N ) ) = ( iµ , .., iµ L ; − iλ , .., − iλ L ; µ ′ , .., µ ′ N ; − λ ′ , .., − λ ′ N ) , (Λ , .., Λ L + N ) ) = ( iλ , .., iλ L ; − iµ , .., − iµ L ; λ ′ , .., λ ′ N ; − µ ′ , .., − µ ′ N ) . (4.6)are introduced.General determinant formula (2.36) with matrix A from (7.19) with correspondingparameters (4.6) of kernel R (4.5) of ∂ -problem (2.1) gives exact multi line solitonsolutions u ( ξ, η, t ) with constant asymptotic value − ǫ at infinity of hyperbolic versionof NVN equation. At the same time an application of general scheme of ∂ -dressing method gives exact potentials u and corresponding wave functions χ [ L,N ] ( M l ), ψ [ L,N ] ( M l ) = χ [ L,N ] ( M l ) e F ( M l ) at discrete spectral parameters M l and χ [ L,N ] ( λ ), ψ [ L,N ] ( λ ) = χ [ L,N ] ( λ ) e F ( λ ) at continuous spectral parameter λ of linear auxiliaryproblems (1.2),(1.3) and one-dimensional perturbed telegraph equation (1.6). For theconvenience here and henceforth the symbols χ [ L,N ] , ψ [ L,N ] denote the wave functionsof multi line soliton exact solution corresponding to the general kernel (4.5) with L + N pairs of items.The rest of the present section is devoted to the presentation for considered case(4.2) of the explicit forms of some one line of types [1 , , [0 ,
1] and two line of types ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method , , [0 , , [1 ,
1] soliton solutions of hyperbolic version of NVN equation and exactpotentials with corresponding wave functions of one-dimensional perturbed telegraphequation (1.6). [1 , and [2 , line solitons The kernels of type R (4.5) with values L = 1 , N = 0 (i. e. a l =0 , l = 1 , a ′ n = 0 , n = 1 , ..., N ) in (4.6) are correspond to [1 , ,
0] solitons. Fornonsingular one line [1 ,
0] and two line [2 ,
0] soliton solutions of hyperbolic version ofNVN equation parameters µ k , λ k , a k in general formulas (3.9)-(3.35) of Section 3 mustbe identified due to (4.6) by the following way: µ k = − µ k := iµ k , λ k = − λ k := iλ k , a k = − a k := − ia k , ( k = 1 ,
2) (4.7)and real parameters p k (3.11) p k = a k µ k + λ k µ k − λ k = e φ k > , ( k = 1 ,
2) (4.8)since positive constants must be chosen. The real phases ∆ F ( µ k , λ k ) = F ( µ k ) − F ( λ k ) := ϕ k , ( k = 1 ,
2) (3.9)-(3.35) are given in considered case by the expressions: ϕ k ( ξ, η, t ) = ( λ k − µ k ) ξ + (cid:16) ǫλ k − ǫµ k (cid:17) η − κ (cid:0) λ k − µ k (cid:1) t − κ (cid:16) ǫ λ k − ǫ µ k (cid:17) t. (4.9)One line soliton [1 ,
0] solution generating by simplest kernel R of the type (4.5) with L = 1 , N = 0 and parameters (4.6) due to (3.13) and (4.8), (4.9) is nonsingular linesoliton: u = − ǫ − ǫ ( λ − µ ) λ µ ϕ + φ . (4.10) Figure 1.
One line soliton [1 ,
0] solution ˜ u ( x, y, t = 0) = u ( x, y, t = 0) + ǫ (4.10)(blue) and squared absolute value of corresponding wave function | ψ [1 , ( iµ ) | (green) (4.11) with parameters a = − , ǫ = 1 , λ = 1 , µ = 4. Wave functions ψ [1 , ( iµ ), ψ [1 , ( − iλ ) and ψ [1 , ( λ ) due to formulas (3.16),(3.17) and (4.7)-(4.9) have the following forms: ψ [1 , ( iµ ) = e F ( iµ ) e ϕ + φ , ψ [1 , ( − iλ ) = e − F ( iλ ) e ϕ + φ ; (4.11) ψ [1 , ( λ ) = e F ( λ ) − (cid:16) iλ λ − iλ + iµ λ + iµ (cid:17) a e ϕ + F ( λ ) e ϕ + φ . (4.12)Graphs of one line [1 ,
0] soliton (4.10) and the squared absolute value of wave function ψ [1 , ( iµ ) (4.11) for certain values of parameters are presented in Fig.1. Graph of ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ψ [1 , ( − iλ ) has the similarform but with localization along another one half of potential valleyTwo line soliton [2 ,
0] solution in considered case of kernel R (4.5) withparameters (3.12),(4.6)-(4.8) is given by the formula (3.18). It is remarkable thatunder the condition q = p p (see (3.22)) which is equivalent to the relation:( λ µ + λ µ )( λ − µ )( λ − µ ) = 0 , (4.13)i. e. to relation λ µ + λ µ = 0 (due to λ n = µ n , we do not consider in thepresent paper lumps!), the solution (3.18) radically simplifies and due to (3.23) takesthe form: u ( ξ, η, t ) = − ǫ − ǫ ( λ − µ ) λ µ ϕ ( ξ,η,t )+ φ − ǫ ( λ − µ ) λ µ ϕ ( ξ,η,t )+ φ . (4.14)a) b) Figure 2.
Two line soliton [2 ,
0] solution ˜ u ( x, y, t = 0) = u ( x, y, t = 0) + ǫ (4.14)(a) and squared absolute value of corresponding wave function | ψ [2 , ( iµ ) | (green) (b), with parameters a = 1 , λ = 1 , µ = − a = − , λ = 4 , ǫ = − The corresponding wave functions χ [2 , , ψ [2 , calculated in considered case ofkernel R (4.5) with parameters (4.6)-(4.8) by the formulas (2.27)-(2.34), undercondition p p = q , i. e. under λ µ + λ µ = 0, are given by the simple formulas(3.25)-(3.35). Graphs of two line [2 ,
0] soliton (4.14) and the squared absolute valueof wave function - ψ [2 , ( iµ ) (3.31) for certain values of parameters are presented inFig.2 (the squared absolute values of other wave functions (3.32-3.34) have similarforms but with localization along another three possible halves of two potentialvalleys). ∂ -dressing in present paper is carried out for the fixed nonzero value of parameter ǫ . Nevertheless one can correctly set ǫ = c k µ k , ( k = 1 ,
2) ( c k -arbitrary complexconstant) and consider the limit ǫ → ǫ = 0. Limiting procedure ǫ = c k µ k → , ( k =1 ,
2) can be correctly performed by the following settings in all required formulas: ǫ → µ k → µ µ = − λ λ → c c inaccordance with the relations ǫ = c k µ k and µ λ + µ λ = 0; the last relation isassumed to be valid in considered limit. The two line soliton solution (4.14) in thelimit ǫ → u = − c λ ϕ ( ξ,η,t )+ φ − c λ ϕ ( ξ,η,t )+ φ , (4.15) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ϕ k ( ξ, η, t ) and φ k due to (4.8), (4.9) have in considered limit theforms: ϕ k ( ξ, η, t ) = λ k ξ − c k η − κ λ k t + κ c k t, φ k = ln( − a k ) . (4.16)One can check by direct substitution that NVN-II equation (1.1) with σ = 1 satisfiesby u given by (4.15), it satisfies also by each item u ( k ) = − c k λ k ϕ k ( ξ,η,t )+ φ k , ( k = 1 ,
2) (4.17)of the sum (4.15). So in considered case the linear principle of superposition u = u (1) + u (2) for such special solutions u (1) , u (2) (4.17) is valid. [0 , and [0 , line solitons To [0 , ,
2] solitons the kernels of type R (4.5) with values L = 0; N = 1 , a l = 0 , l = 1 , ..., L ; a ′ n = 0 , n = 1 ,
2) in (4.6) are correspond. For nonsingularone line [0 ,
1] and two line [0 ,
2] soliton solutions of hyperbolic version of NVN equationparameters µ k , λ k , a k in general formulas (3.9)-(3.35) of Section 3 must be identifieddue to (4.6) by the following way: µ k = λ k , a k = a k := a k , ( k = 1 , . (4.18)The parameters p k , ( k = 1 , q in (3.9)-(3.35) due to (4.18) are given by theexpressions: p k = − a k λ kR λ kI := e φ k > , q = p p · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( λ − λ )( λ + ¯ λ )( λ + λ )( λ − ¯ λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.19)where the parameters p k := e φ k > F ( µ k , λ k ) = F ( µ k ) − F ( λ k ) := ϕ k , ( k = 1 ,
2) in (3.9)-(3.35) aregiven due to (2.4) in considered case by the expressions: ϕ k ( ξ, η, t ) = i h ( λ k − λ k ) ξ − ǫ (cid:16) λ k − λ k (cid:17) η + κ ( λ k − λ k ) t − κ ǫ (cid:16) λ k − λ k (cid:17) t i . (4.20) Figure 3.
One line soliton [0 ,
1] solution ˜ u ( x, y, t = 0) = u ( x, y, t = 0) + ǫ (4.21)(blue) and squared absolute value of corresponding wave functions | ψ [0 , ( λ ) | = | ψ [0 , ( − λ ) | (green) (4.22) with parameters a = − , λ R = 0 . , λ I = 2 , ǫ = − ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ,
1] solution generated by simplest kernel R of the type (4.5) with L = 0 , N = 1 and parameters (4.6) due to (3.13) and (4.18)-(4.20) is nonsingular linesoliton: u = − ǫ + 2 ǫλ I | λ | ϕ ( ξ,η,t )+ φ . (4.21) Figure 4.
Two line soliton [0 ,
2] solution ˜ u ( x, y, t = 0) = u ( x, y, t = 0) + ǫ withparameters a = − , λ R = 0 . , λ I = 2; a = − , λ R = 0 . , λ I = 1 , ǫ = − The corresponding wave functions ψ [0 , ( µ ) = χ [0 , ( µ ) e F ( µ ) , ψ [0 , ( − λ ) = χ [0 , ( − λ ) e F ( − λ ) and ψ [0 , ( λ ) = χ [0 , ( λ ) e F ( λ ) of linear auxiliary problems(1.2),(1.3) and exact potential u = ˜ u − ǫ of one-dimensional perturbed telegraphequation (1.6) due to (3.16)-(3.17) and (4.18)-(4.20) have the forms: ψ [0 , ( λ ) = e F ( λ ) e ϕ + φ , ψ [0 , ( − λ ) = e − F ( λ ) e ϕ + φ ; (4.22) ψ [0 , ( λ ) = e F ( λ ) − (cid:16) λ λ − λ + λ λ + λ (cid:17) ia e ϕ + F ( λ ) e ϕ + φ . (4.23)Graphs of one line [0 ,
1] soliton (4.21) and the squared absolute values of wave functions(4.22) for certain values of parameters are shown in Fig.3.Two line soliton solution in considered case of kernel (4.5) with L = 0 , N = 2and parameters (4.6),(4.19) is given by the formula (3.18). It is interesting to notethat the condition q = p p in the considered case of kernel R of the type (4.5)with L = 0 , N = 2 and parameters (4.6),(4.18), (4.19) due to (3.24) takes the form λ µ + λ µ = | λ | + | λ | = 0 and can not be satisfied for λ k = 0, by this reasonsplitting of two line soliton solution (3.18)-(3.20) into the simple form (3.23) in thepresent case is impossible. Graph of two line [0 ,
2] soliton given by (3.18)-(3.20) forcertain values of corresponding parameters is shown in Fig.4. [1 , line soliton To [1 ,
1] soliton corresponds the kernel of type R (4.5) with values L = 1; N = 1(i. e. a = 0 , a ′ = 0) in (4.6). For nonsingular two line [1 ,
1] soliton solution ofhyperbolic version of NVN equation parameters µ k , λ k , a k in general formulas (3.9)-(3.35) of Section 3 must be identified due to (4.6) by the following way: µ = − µ := iµ , λ = − λ := iλ , a = − a := − ia ,µ = µ ′ , λ = λ ′ = µ ′ , a = a ′ = a ′ := a ′ , (4.24) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method a , λ , µ in formulas (3.18)-(3.35) due (4.24) must be identified with a ′ , λ ′ , µ ′ in(4.5). The parameters p k , ( k = 1 , q in (3.9)-(3.35) due to (3.11) and (4.24) aregiven by expressions: p = a µ + λ µ − λ := e φ > , p = − a λ R λ I := e φ > . (4.25)Two line soliton [1 ,
1] solution in considered case with parameters (3.12), (4.24) isgiven by the formula (3.18). It is remarkable that under the condition q = p p (see(3.22)) which is equivalent to the relation:( − λ µ + | λ | )( iλ − iµ )( λ − λ ) = 0 , (4.26)i. e. to relation − λ µ + | λ | = 0 (due to λ n = µ n , we do not consider in thepresent paper lumps!), the solution (3.19) radically simplifies and due to (3.23) takesthe form: u ( ξ, η, t ) = − ǫ − ǫ ( λ − µ ) λ µ ϕ ( ξ,η,t )+ φ + 2 ǫλ I | λ | ϕ ( ξ,η,t )+ φ , (4.27)where phases ϕ ( ξ, η, t ), ϕ ( ξ, η, t ) are given by the formulas (4.9),(4.20). Figure 5.
Two line soliton [1 ,
1] solution ˜ u ( x, y, t = 0) = u ( x, y, t = 0) + ǫ (4.27)with parameters a = − . , λ = 2 , ǫ = − a = − . , λ R = 0 . , λ I = 1. a b Figure 6.
Nonbounded | ψ [1 , ( iµ ) | (a) and bounded | ψ [1 , ( λ ) | = | ψ [1 , ( − λ ) | (b) squared absolute values of wave functions (green) correspondingto solution in the Fig.5. The corresponding wave functions χ [1 , , ψ [1 , calculated in considered case ofkernel R (4.5) with parameters (4.6),(4.24) and by the formulas (2.27)-(2.34), under ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method p p = q , i. e. under − λ µ + | λ | = 0, are given by the simple formulas(3.25)-(3.35). Graphs of two line [1 ,
1] soliton (4.27) and squared absolute values ofsome wave functions given by (3.31)-(3.33) for certain values of parameters are shownin Fig.5-Fig.6 (graphs of | ψ [1 , ( iµ ) | and | ψ [1 , ( − iλ ) | are similar to each otherbut with localization along two different halves of corresponding potential valley).In all considered cases for NVN-II equation (hyperbolic version) multi linesolitons are finite but corresponding wave functions can take infinite values in someareas of the plane ( x, y ), (Fig.1, 2, 6a). Only in two considered cases, for soliton[0 ,
1] and soliton [1 ,
1] the squared absolute value of corresponding wave functions | ψ [0 , ( λ ) | = | ψ [0 , ( − λ ) | (Fig.3) and | ψ [1 , ( λ ) | = | ψ [1 , ( − λ ) | (Fig.6b) arefinite.We have to mention that exact potentials (of types [0,1] and [1,0]) of (1.6) withcorresponding wave functions (4.11), (4.22) in the paper [22] have been calculated andused for the construction of exact solutions of two-dimensional generalized integrablesine-Gordon equation (2DGSG). In the present paper time evolution (2.2) is takeninto account and corresponding multi line soliton solutions of NVN-II equation arecalculated.
5. Exact multi line soliton solutions of NVN-I equation
For elliptic version of NVN equation (1.1), or NVN-I equation, with σ = − ξ := z = x + iy , η := ¯ z = x − iy an application of realitycondition (2.24) to each term of the sum (3.7) for R gives the following relation: a k λ k δ ( µ − µ k ) δ ( λ − λ k ) + a k µ k δ ( µ + λ k ) δ ( λ + µ k ) == ǫ | λ | | µ | λµ h a k λ k δ (cid:16) − ǫλ − µ k (cid:17) δ (cid:16) − ǫµ − λ k (cid:17) + a k µ k δ (cid:16) − ǫλ + λ k (cid:17) δ (cid:16) − ǫµ + µ k (cid:17)i == ǫa k µ k δ (cid:16) λ + ǫµ k (cid:17) δ (cid:16) µ + ǫλ k (cid:17) + ǫa k λ k δ (cid:16) λ − ǫλ k (cid:17) δ (cid:16) µ − ǫµ k (cid:17) . (5.1)We should underline that in the present paper complex delta functions (with complexarguments) are used. The last equality in (5.1) by the well known property of complexdelta functions δ ( ϕ ( z )) = P k δ ( z − z k ) / | ϕ ′ ( z k ) | is obtained; z k in last formula aresimple roots of equation ϕ ( z k ) = 0.From (5.1) two possibilities are follow:1 . a k λ k = ǫa k µ k , λ k = − ǫµ k , µ k = − ǫλ k ; 2 . a k λ k = ǫa k λ k , λ k = ǫλ k , µ k = ǫµ k . (5.2)For the first case in (5.2) taking into account the reality of ǫ one obtains a k = − a k := ia k , ǫ = − µ k λ k = − µ k λ k ; arg( µ k ) = arg( λ k ) + mπ, (5.3)i. e. pure imaginary amplitudes a k ( a k = a k ) and the relation between argumentsof discrete spectral points µ k and λ k with m arbitrary integer. From the secondpossibility in (5.2) for satisfying the reality condition (2.24) the following relations a k = a k := a ′ k , ǫ = | µ ′ k | = | λ ′ k | ; arg( µ ′ k ) = arg( λ ′ k ) + δ k (5.4)with real amplitudes a k = a k := a ′ k and arbitrary constants δ k are follow. ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method R ( µ, µ, λ, λ ) = π L + N ) X k =1 A k δ ( µ − M k ) δ ( λ − Λ k ) (5.5)of L pairs of the type iπ (cid:0) a l λ l δ ( µ − µ l ) δ ( λ − λ l ) + a l µ l δ ( µ + λ l ) δ ( λ + µ l ) (cid:1) (here ǫ = − µ l λ l = − µ l λ l , ( l = 1 , .., L )); and N pairs of the type π (cid:0) a ′ n λ ′ n δ ( µ − µ ′ n ) δ ( λ − λ ′ n ) + a ′ n µ n δ ( µ + λ ′ n ) δ ( λ + µ ′ n ) (cid:1) (here ǫ = | λ ′ n | = | µ ′ n | , ( n = 1 , .., N )) ofcorresponding items. Here in (5.5) for application of general determinant formulas(7.19), (2.36) and (7.17) due to (5.2)-(5.4) the following sets of amplitudes A k andspectral parameters M k , Λ k ( A , .., A L + N ) ) == ( ia λ , .., ia L λ L ; ia µ , .., ia L µ L ; a ′ λ ′ , .., a ′ N λ ′ N ; a ′ µ ′ , .., a ′ N µ ′ N ) , ( M , .., M L + N ) ) = ( µ , .., µ L ; − λ , .., − λ L ; µ ′ , .., µ ′ N ; − λ ′ , .., − λ ′ N ) , (Λ , .., Λ L + N ) ) = ( λ , .., λ N ; − µ , .., − µ N ; λ ′ , .., λ ′ N ; − µ ′ , .., − µ ′ N ) (5.6)are introduced.General determinant formula (2.36) with matrix A from (7.19) with correspondingparameters (5.6) of kernels R (5.5) of ∂ -problem (2.1) gives exact multi linesoliton solutions u ( z, z, t ) with constant asymptotic value − ǫ at infinity of ellipticversion of NVN equation. Simultaneously an application of general scheme of ∂ -dressing method gives exact potentials u and corresponding wave functions χ [ L,N ] ( M l ), ψ [ L,N ] ( M l ) = χ [ L,N ] ( M l ) e F ( M l ) at discrete spectral parameters M l and χ [ L,N ] ( λ ), ψ [ L,N ] ( λ ) = χ [ L,N ] ( λ ) e F ( λ ) at continuous spectral parameter λ of linear auxiliaryproblems (1.2),(1.3) and two-dimensional stationary Schr¨odinger equation (1.7). Hereand below the symbols χ [ L,N ] , ψ [ L,N ] denote the wave functions of multi line solitonexact solution corresponding to the general kernel (5.5) with L + N pairs of items.The rest of the section is devoted to the presentation for considered two cases (5.2)of the explicit forms of some one line of types [1 , , [0 ,
1] and two line soliton solutionsof types [2 , , [0 , , [1 ,
1] of elliptic version of NVN equation and exact potentialswith corresponding wave functions of two-dimensional stationary Schr¨odinger equation(1.7). [1 , , [2 , line solitons To [1 , ,
0] line solitons the kernels of type R (5.5) with values L = 1 , N =0 (i. e. a l = 0 , l = 1 , a ′ n = 0 , n = 1 , ..., N ) in (5.6) are correspond.For nonsingular one line [1 ,
0] and two line [2 ,
0] soliton solutions of elliptic versionof NVN equation parameters µ k , λ k , a k in general formulas (3.9)-(3.35) of Section 3must be identified due to (5.6) by the following way: a k = − a k := ia k , µ k = − ǫλ k ( k = 1 , , (5.7)and real parameters p k (3.11) p k = a k λ k + µ k λ k − µ k = e φ k > , ( k = 1 ,
2) (5.8)as positive constants must be chosen. The real phases ∆ F ( µ k , λ k ) = F ( µ k ) − F ( λ k ) := ϕ k , ( k = 1 ,
2) in (3.9)-(3.35) are given in considered case by the expressions: ϕ k ( z, ¯ z, t ) = i (cid:2) ( µ k − λ k ) z − ( µ k − λ k ) z + κ ( µ k − λ k ) t − κ ( µ k − λ k ) t (cid:3) . (5.9) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ,
0] solution corresponding to simplest kernel R of the type (5.5)with parameters (5.6) due to (5.7)-(5.9) is nonsingular line soliton: u = − ǫ − ǫ ( λ − µ ) λ µ ϕ + φ = − ǫ + | λ − µ | ϕ + φ ; ǫ = − λ ¯ µ . (5.10)a bc Figure 7.
Potential V Shr (5.13) (blue) with the energy level E (yellow)and corresponding squared absolute values of wave functions | ψ [1 , ( µ ) | = | ψ [1 , ( − λ ) | (5.11) (green) with parameters: a) a = − . , λ = e i π , µ =4 e i π , E = − ǫ = −
8; b) a = − . , λ = e i π , µ = 4 e i π , E = − ǫ = 8; c) a = 0 . , λ = e i π , µ = 0 , E = − ǫ = 0. The corresponding wave functions ψ [1 , ( µ ) = χ [1 , ( µ ) e F ( µ ) , ψ [1 , ( − λ ) = χ [1 , ( − λ ) e F ( − λ ) and ψ [1 , ( λ ) = χ ( λ ) e F ( λ ) of linear auxiliary problems (1.2),(1.3)and exact potential V Shr of 2D stationary Schr¨odinger equation (1.7) with energy level E := − ǫ due to (2.34), (3.14)-(3.17) have the forms: ψ ( µ ) = e F ( µ ) e ϕ + φ , ψ ( − λ ) = e − F ( λ ) e ϕ + φ , (5.11) ψ ( λ ) = e F ( λ ) + (cid:16) λ λ − λ + µ λ + µ (cid:17) a e ϕ + F ( λ ) e ϕ + φ ; (5.12) V Schr = − E ( λ − µ ) λ µ ϕ + φ = − | λ − µ | cosh ϕ + φ ; E = − ǫ = 2 λ ¯ µ . (5.13) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method
20a bc
Figure 8.
Potential V Shr corresponding two line soliton [2 ,
0] solution (5.14)(blue) with the energy level E (yellow) with parameters: a) a = − , λ = e i π , µ = 1 . e i π ; a = − , τ = 1 , E = − ǫ = − .
1; b) a = − . , λ = e i π , µ = 4 e i π ; a = − . , τ = 1 , E = − ǫ = 8; c) a = 0 . , λ = e i π , µ =0; a = 0 . , τ = 1 , E = − ǫ = 0. Graphs of Schr¨odinger potentials (5.13) (connected with one line [1 ,
0] solitons V Schr = − u (5.10)) and squared absolute values of wave functions (5.11) forstationary states with energies E < E > E = 0 (equation (1.7) forparticle with mass m = 1) for certain values of corresponding parameters are shown inFig.7. One can prove that two wave functions (5.11) for all signs of energy correspondto stationary states of a particle with opposite to each other conserved projections(on direction of valley) of momentum. In all above mentioned stationary states withwave functions (5.11) particle is bounded in transverse direction to potential valleyand moves freely along the direction of potential valley.Two line soliton [2 ,
0] solution in considered case of kernel R of the type (5.5)with parameters (5.6) is given by the formula (3.18), it is remarkable that under thecondition q = p p this solution radically simplifies. Indeed, due to (3.24) condition q = p p is satisfied if λ µ + λ µ = 0 and in this case two line soliton solution (3.18)takes the form (3.23): u ( z, ¯ z, t ) = − ǫ − ǫ ( λ − µ ) λ µ ϕ ( z, ¯ z,t )+ φ − ǫ ( λ − µ ) λ µ ϕ ( z, ¯ z,t )+ φ == − ǫ + | λ − µ | ϕ ( z, ¯ z,t )+ φ + | λ − µ | ϕ ( z, ¯ z,t )+ φ . (5.14)From the relation λ µ + λ µ = 0 taking into account the first condition (5.2) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method λ µ = λ µ = λ µ = λ µ = − ǫ ) follows µ /µ = − µ /µ = λ /λ and from thelast relation one obtains µ = iτ µ , λ = iτ − λ , τ = ¯ τ (5.15)with arbitrary real constant τ .Wave functions corresponding to two line soliton [2 ,
0] solution (5.14) in consideredcase of kernel R of the type (5.5) with parameters (5.6) and (5.7)-(5.9), undercondition p p = q , are given by very simple expressions (3.25)-(3.35).a bc Figure 9.
Squared absolute values of wave functions | ψ [2 , ( µ ) | = | ψ [2 , ( − λ ) | (green) corresponding to different values of energy E in theFig.8(a,b,c). Graphs of Schr¨odinger potentials (connected with two line [2 ,
0] soliton V Schr = − u solutions (5.14)) and squared absolute values | ψ [2 , ( µ ) | = | ψ [2 , ( − λ ) | ofsome wave functions from (3.31)-(3.34) for certain values of parameters are shownin Fig.8 and Fig.9 (graphs of | ψ [2 , ( µ ) | = | ψ [2 , ( − λ ) | are similar to graphs of | ψ [2 , ( µ ) | = | ψ [2 , ( − λ ) | but with localization along another soliton valley).Calculated via ∂ -dressing method wave functions (3.31)-(3.34) at discrete valuesof spectral parameters correspond to possible physical basis states of particle localizedin the field of two potential valleys. ∂ -dressing in present paper is carried out for thefixed nonzero value of parameter ǫ or, in context of present section, for nonzero energy E = 0. Nevertheless one can correctly consider the limit ǫ → E = − ǫ = 0.Limiting procedure E = − ǫ = µ k ¯ λ k + ¯ µ k λ k → , ( k = 1 ,
2) can be correctly performedby the following settings in all required formulas: ǫ → µ k → ǫµ k → − ¯ λ k in accordance with the relation ǫ = − µ k ¯ λ k ; inaddition the formula λ = iτ − λ (5.15) (followed from the relations ¯ µ k λ k = µ k ¯ λ k ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method µ λ + µ λ = 0) with arbitrary real constant τ is assumed to be valid. The twoline soliton solution due to (5.14) in considered limit has the form: u = | λ | ϕ ( z, ¯ z )+ φ + | λ | ϕ ( z, ¯ z )+ φ , (5.16)the phases ϕ k ( z, ¯ z ) and φ k due to (2.4),(5.9),(5.8) have in considered limit the forms: ϕ k ( z, ¯ z, t ) = − i (cid:0) λ k z − λ k z + κλ k t − κλ k t (cid:1) , φ k = ln a k . (5.17)One can check by direct substitution that NVN-I equation (1.1) with σ = i satisfiesby u = ˜ u = − V Schr / u ( k ) = | λ k | ϕ k ( z, ¯ z )+ φ k , ( k = 1 ,
2) (5.18)of the sum (5.16). Thus, in considered case the linear principle of superposition u = u (1) + u (2) for such special solutions u (1) , u (2) (5.18) is valid. One can showusing (5.15),(5.17) that line solitons u (1) and u (2) are propagate in the plane ( x, y ) inperpendicular to each other directions. Schr¨odinger potentials V Schr (of the types [1,0]and [2,0]) with corresponding squared absolute value wave functions of zero energylimit E = 0 are also pictured by graphs of Fig.7, Fig.8 and Fig.9. [0 , , [0 , line solitons The kernels of type R (5.5) with values L = 0; N = 1 , a l =0 , l = 1 , .., L ; a ′ n = 0 , n = 1 ,
2) in (5.6) correspond to [0 , ,
2] line solitons. Fornonsingular one line [0 ,
1] and two line [0 ,
2] soliton solutions of elliptic version of NVNequation parameters a k , µ k , λ k in general formulas (3.9)-(3.35) of Section 3 must beidentified due to (5.6) by the following way: a k = a k := a k , ǫ = | µ k | = | λ k | , ( k = 1 , . (5.19)Real parameters p k due to (3.11), (5.6) and (5.19) p k = ia k µ k + λ k µ k − λ k = a k cot δ k e φ k > , µ k := λ k e iδ k , ( k = 1 ,
2) (5.20)appear as positive constants. The real phases ∆ F ( µ k , λ k ) = F ( µ k ) − F ( λ k ) := ϕ k , ( k =1 ,
2) are given in considered case by the expressions: ϕ k ( z, ¯ z, t ) = i [( µ k − λ k ) z − ( µ k − λ k ) z + κ ( µ k − λ k ) t − κ ( µ k − λ k ) t ] . (5.21)One line soliton [0 ,
1] solution corresponding to simplest kernel R of the type (5.5)with parameters (5.6) due to (3.13) and (5.19)-(5.20) and (5.21) is nonsingular linesoliton: u = − ǫ + | λ − µ | ϕ + φ = − ǫ + 2 ǫ sin δ cosh ϕ + φ . (5.22)The corresponding wave functions ψ [0 , ( µ ) = χ [0 , ( µ ) e F ( µ ) , ψ [0 , ( − λ ) = χ [0 , ( − λ ) e F ( − λ ) and ψ [0 , ( λ ) = χ [0 , ( λ ) e F ( λ ) of linear auxiliary problems(1.2),(1.3) and exact potential V Shr of 2D stationary Schr¨odinger equation (1.7) withenergy level E := − ǫ have forms: ψ ( µ ) = e F ( µ ) e ϕ + φ , ψ ( − λ ) = e − F ( λ ) e ϕ + φ , (5.23) ψ ( λ ) = e F ( λ ) − (cid:16) λ λ − λ + µ λ + µ (cid:17) ia e ϕ + F ( λ ) e ϕ + φ ; (5.24) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method
23a b
Figure 10.
Potential V Shr (5.22) (blue) with the energy level E (yellow) andcorresponding squared absolute value of wave function | ψ [0 , ( µ ) | (5.23) (green)with parameters: a) a = − , λ = 2 − i, δ = π , E = − ǫ = −
10; b) a = − , λ = 2 − i, δ = π , E = − ǫ = − V Schr = − | λ − µ | cosh ϕ + φ = − ǫ sin ( δ )cosh ϕ + φ ; E = − ǫ = − | λ | = − | µ | . (5.25)Graphs of Schr¨odinger potential V Schr (5.25) (connected with one line [0 ,
1] soliton V Schr = − u solution (5.22)) and the squared absolute value of wave function ψ [0 , ( µ ) from (5.23) for certain values of parameters are shown in Fig.10: a)( V Shr ) min < E <
0, b) ( V Shr ) min = E < | ψ [0 , ( − λ ) | has the similar form but with localization along another one half of potential valley).Two line soliton [0 ,
2] solution in considered case of kernel kernel R of the type(5.5) with parameters (5.6) and (5.19),(5.20) and (5.21) is given by the formula (3.18).It is remarkable that under the condition q = p p this solution radically simplifies.Indeed, due to (3.24) condition q = p p is satisfied if λ µ + λ µ = 0, in this casetwo line soliton solution (3.18) takes the form (3.23): u ( z, ¯ z, t ) = − ǫ + | λ − µ | ϕ ( z, ¯ z,t )+ φ + | λ − µ | ϕ ( z, ¯ z,t ) , + φ == − ǫ + 2 ǫ sin δ cosh ϕ ( z, ¯ z,t )+ φ + 2 ǫ sin δ cosh ϕ ( z, ¯ z,t )+ φ , µ k = λ k e iδ k , ǫ = | λ k | = | µ k | . (5.26)a b ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method Figure 11.
Potential V Shr corresponding two line soliton [0 ,
2] solution(5.26)(blue) with the energy level E (yellow) with parameters: (a) a = − , λ =2 − i, δ = π ; a = 1 , δ = π , E = − ǫ = −
10; (b) a = 1 , λ = 2 − i, δ = π ; a = 1 , δ = π , E = − ǫ = − The corresponding to two line soliton solution (5.26) wave functions in consideredcase of kernel R of the type (5.5) with parameters (5.6) and (5.19),(5.20) and (5.21),under condition p p = q , are given by very simple expressions (3.25)-(3.35).a b Figure 12.
Squared absolute value of wave function | ψ [0 , ( µ ) | (green) for thedifferent types of crossings of potentials valleys by energy planes in the Fig.11(a,b). Graphs of Schr¨odinger potentials (connected with two line [0 ,
2] solitons V Schr = − u (5.26)) and squared absolute value | ψ [0 , ( µ ) | of one wave function from fourlinear independent partners (3.31)-(3.34) for certain values of parameters are shownin Fig.11 and Fig.12 (the squared absolute values of other wave functions have thesimilar forms but with localization along another three possible halves of two potentialvalleys). In all considered in the present section cases of one line [0,1] and two line[0,2] solitons u = ˜ u − ǫ and Schr¨odinger potentials V Schr = − u corresponding wavefunctions (Fig.10, Fig.12) are not bounded. [1 , line soliton The kernel of type R (5.5) with values L = 1; N = 1 (i. e. a = 1; a ′ = 1) in(5.6) correspond to [1 ,
1] line soliton. For this soliton solution parameters a k , µ k , λ k in general formulas (3.9)-(3.35) of Section 3 must be identified due to (5.6) by thefollowing way: a = − a := ia , ǫ = − µ λ a = a ′ = a ′ := a ′ , µ = µ ′ , λ = λ ′ , ǫ = | µ ′ | = | λ ′ | . (5.27) a , λ , µ in formulas (3.18)-(3.35) due (5.27) must be identified with a ′ , λ ′ , µ ′ in(5.5). Real parameters p , p due to (3.11), (5.6) and (5.27) p = − a µ + λ µ − λ := e φ > , p = ia µ + λ µ − λ = a cot δ e φ > , (5.28)appear as positive constants.Two line soliton [1 ,
1] solution in considered case of kernel kernel R of the type(5.5) with parameters (3.12) and (5.27),(5.28) is given by the formula (3.18). It is ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method q = p p this solution radically simplifies. Indeed,due to (3.24) condition q = p p is satisfied if λ µ + λ µ = 0, in this case two linesoliton solution (3.18) takes the form (3.23: u ( z, ¯ z, t ) = − ǫ + | λ − µ | ϕ ( z, ¯ z,t )+ φ + | λ − µ | ϕ ( z, ¯ z,t )+ φ (5.29)where | λ | = | µ | = − µ λ = − µ λ = ǫ and the phases ϕ , ϕ are given byformulas (5.9),(5.21). Graphs of Schr¨odinger potentials (connected with two line [1 , V Schr = − u (5.29)) and squared absolute values of some wave functions from(3.31)-(3.34) for certain values of parameters are shown in Fig.13 and Fig.14 (graphsof | ψ [1 , ( − λ ) | and | ψ [1 , ( µ ) | are similar to each other but with localization alongtwo different halves of corresponding potential valley). Figure 13.
Potential V Shr corresponding two line soliton [1 ,
1] solution(3.18)(blue) and energy level E (yellow) with parameters a = − , λ = 1 e π , µ =1 . e π ; a = − , λ = 1 . e π , µ = 1 . e π , E = − ǫ = − . a b Figure 14.
Bounded | ψ [1 , ( µ ) | = | ψ [1 , ( − λ ) | (a) and nonbounded | ψ [1 , ( µ ) | (b) squared absolute values of wave functions (green) given by (3.31)-(3.33) corresponding to potential and energy in the Fig.13. In considered in present section case of two line [1,1] soliton u = ˜ u − ǫ (5.29)with corresponding Schr¨odinger potential V Schr = − u squared absolute values of ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method | ψ [1 , ( µ ) | = | ψ [1 , ( − λ ) | are bounded (Fig.14a), but the squaredabsolute values of other basis wave functions | ψ [1 , ( µ ) | and | ψ [1 , ( − λ ) | are notbounded (Fig.14b).In conclusion of Section 5 let us mention that all constructed in subsection 5.1solitons and corresponding wave functions are finite and have appropriate physicalinterpretation. For example, the wave function (5.12) of continuous spectral parameter λ for discrete values of this parameter λ = µ or λ = − λ coincides with wavefunctions (5.11); for positive values of energy E = − ǫ > λ = µ , λ = − λ ,under condition | λ | = − ǫ = E/ >
0, the wave function (5.12) corresponds tostationary states of nonlocalized on the plane ( x, y ) particle which do not reflectsfrom the constructed potential (5.13). In considered in subsections 5.2 and 5.3 casesmulti line solitons are finite but corresponding wave functions can take infinite valuesin some areas of the plane ( x, y ), (Fig.10, 12, 14b); only for two line soliton [1 , | ψ [1 , ( µ ) | = | ψ [1 , ( − λ ) | (Fig.14a) arefinite. The question of more detailed physical interpretation and applications of exactpotentials and corresponding wave functions of 2D stationary Schr¨odinger equationwill be considered elsewhere.
6. Periodic solutions of the NVN equation
The restrictions (2.23) and (2.24) on the kernel R of the ¯ ∂ -problem (2.1) which leadto real solutions u = ¯ u of the NVN equations (1.1) are obtained in section 2 by theuse of reconstruction formula (2.18) u = − ǫ − iχ − η = − ǫ + iχ − η (6.1)in the limit of ”weak” fields , i.e. χ − in (6.1) is calculated from its exact expression(2.22) with approximation χ ≃
1. It is shown in section 4 and 5 that reality conditions(2.23) and (2.24) work and lead to multi line soliton solutions of the NVN equation.Such use of reality condition was considered in all previous papers (see for example[22]-[24]) devoted to constructions of classes of exact solutions of integrable nonlinearevolution equations via ∂ -dressing method. But there is existing possibility of nonuse the limit of weak fields and imposing the reality condition u = u directly to exactsolutions (3.13) of NVN equation calculated in sections 2, 3 and satisfying only topotentiality condition.Thus one starts from the general kernel R (3.7) of ∂ -dressing problem (withparameters (3.8)) which satisfies to potentiality condition χ − R (3.7) with N = 1 the requirements of reality (6.1), i.e. χ − η = − χ − η , leads due to (2.22) and (3.9)-(3.13) to the conclusion: ǫ a ( λ − µ ) λ µ h e − ϕ − ia λ + µ λ − µ e ϕ i = − ǫ a ( λ − µ ) λ µ h e − ϕ + ia λ + µ λ − µ e ϕ i (6.2)with the phase ϕ given due to (2.4) by expressions: ϕ ( ξ, η, t ) = F ( µ ) − F ( λ ) = i h ( µ − λ ) ξ − (cid:16) ǫµ − ǫλ (cid:17) η + κ ( µ − λ ) t − κ (cid:16) ǫ µ − ǫ λ (cid:17) t i (6.3)in hyperbolic case and ϕ ( z, z, t ) = F ( µ ) − F ( λ ) = i h ( µ − λ ) z − (cid:16) ǫµ − ǫλ (cid:17) z + κ ( µ − λ ) t − κ (cid:16) ǫ µ − ǫ λ (cid:17) t i (6.4) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ϕ = ϕ (this case leads to multi line soliton solutions considered insections 4,5) as long as for imaginary phase ϕ = − ϕ . The last case leads to periodicsolutions of the NVN equation. Hereafter we described separately the cases of thehyperbolic and elliptic NVN equations. The hyperbolic case . The condition of imaginary phase ϕ = − ϕ due to (6.3)leads to relation: i h ( µ − λ ) ξ − (cid:16) ǫµ − ǫλ (cid:17) η + κ ( µ − λ ) t − κ (cid:16) ǫ µ − ǫ λ (cid:17) t i == i h ( µ − λ ) ξ − (cid:16) ǫµ − ǫλ (cid:17) η + κ ( µ − λ ) t − κ (cid:16) ǫ µ − ǫ λ (cid:17) t i . (6.5)From space-dependent part of (6.5) one obtains the following system of equations: µ − λ = µ − λ , ǫµ − ǫλ = ǫµ − ǫλ . (6.6)Supposing that µ = λ (the solution µ = λ of (6.6) leads to lump solutions, whichare not considered here, see the papers [23], [24]) one obtains from (6.6) the equivalentsystem of equations µ − λ = µ − λ , µ λ = µ λ . (6.7)The system (6.7) has two solutions:1) µ = − λ , λ = λ , µ = µ (6.8)where λ and µ are real constants. One can show that time-dependent part of(6.5) doesn’t lead to new equations and satisfies due to the system (6.7). For solution µ = − λ of the system (6.7) the phase ϕ given by (6.3) is pure imaginary and hasform: ϕ ( ξ, η, t ) = − i h ( λ + λ ) ξ − (cid:16) ǫλ + ǫλ (cid:17) η + κ ( λ + λ ) t − κ (cid:16) ǫ λ + ǫ λ (cid:17) t i := − i ˜ ϕ . (6.9)Inserting µ = − λ and (6.9) into (6.2) one obtains the relation: (cid:16) λ µ − µ λ (cid:17)(cid:2) a e i ˜ ϕ − a e − i ˜ ϕ (cid:3)h | a | (cid:16) λ + µ λ − µ (cid:17) i = 0 , (6.10)which nontrivially satisfies under the condition: | a | = ± i λ − µ λ + µ = ± λ R λ I . (6.11)The solution of the NVN equation (1.1) due to (2.18) and (6.11) for the choice | a | = λ R λ I has the form: u = − ǫ − iǫ | a | ( λ − λ ) | λ | e i arg a h e i ˜ ϕ + e i arg a e − i ˜ ϕ i = − ǫ +2 ǫ λ R | λ | ( ˜ ϕ − arg a ) . (6.12)The solution of the NVN equation (1.1) for | a | = − λ R λ I due to (2.18) and (6.11) hasthe form: u = − ǫ − iǫ | a | ( λ − λ ) | λ | e i arg a h e i ˜ ϕ − e i arg a e − i ˜ ϕ i = − ǫ +2 ǫ λ R | λ | ( ˜ ϕ − arg a ) . (6.13) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method λ = λ , µ = µ of the system (6.8) pure imaginary phase ϕ given by (6.3) has the form: ϕ ( ξ, η, t ) = i h ( µ − λ ) ξ − (cid:16) ǫµ − ǫλ (cid:17) η + κ ( µ − λ ) t − κ (cid:16) ǫ µ − ǫ λ (cid:17) t i := i ˜ ϕ . (6.14)Inserting λ = λ , µ = µ and ϕ = i ˜ ϕ from (6.14) into (6.2) one obtains the therelation: (cid:16) λ µ − µ λ (cid:17)(cid:2) a e i ˜ ϕ + a e − i ˜ ϕ (cid:3)h − | a | (cid:16) λ + µ λ − µ (cid:17) i = 0 , (6.15)which nontrivially satisfies for | a | = ± λ − µ λ + µ . (6.16)The solution u ( ξ, η, t ) of the NVN equation (1.1) due to (2.18), (6.2), (6.14) and (6.16)is given by expression: u = − ǫ − ǫ ( λ − µ ) λ µ ( ˜ ϕ +arg a ∓ π ) , (6.17)where ∓ π/ ± signs in (6.16). ∂ -dressing in present paper is carried out for the fixed nonzero value of parameter ǫ . Nevertheless as in subsections 4.1 and 5.1 one can correctly consider the limit ǫ →
0, for this one can set ǫ = c k µ k , ( k = 1 ,
2) ( c k -arbitrary real constant) and takethe limit ǫ = c k µ k → , ( k = 1 ,
2) in all derived formulas. Limiting procedure canbe correctly performed by the following settings in all required formulas: ǫ → µ k → µ µ = − λ λ → c c in accordance withthe relations ǫ = c k µ k and µ λ + µ λ = 0 (3.24); the last relation is assumedto be valid in considered limit. The periodic solution (3.23) in the limit ǫ → u = − c λ ( ˜ ϕ +arg a − π ) − c λ ( ˜ ϕ +arg a − π ) , (6.18)where the phases ˜ ϕ k ( ξ, η, t ) due to (6.14) are given in considered limit by theexpressions: ˜ ϕ k ( ξ, η, t ) = ( − λ k ξ − c k η − κ λ k t − κ c k t ) . (6.19)One can check by direct substitution that NVN-II equation (1.1) with σ = 1 satisfiesby u given by (6.18), it satisfies also by each item u ( k ) = − c k λ k ( ˜ ϕ +arg a k − π ) , ( k = 1 ,
2) (6.20)of the sum (6.18). So in considered case the linear principle of superposition u = u (1) + u (2) for such special solutions u (1) , u (2) (6.20) is valid. The elliptic case . For elliptic version of NVN equation (1.1) the condition ofimaginary phase ϕ = − ϕ given by (6.4) leads to the relation: ϕ = i h ( µ − λ ) z − (cid:16) ǫµ − ǫλ (cid:17) z + κ ( µ − λ ) t − κ (cid:16) ǫ µ − ǫ λ (cid:17) t i == i h ( µ − λ ) z − (cid:16) ǫµ − ǫλ (cid:17) z + κ ( µ − λ ) t − κ (cid:16) ǫ µ − ǫ λ (cid:17) t i . (6.21) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method µ − λ = − ǫµ + ǫλ , µ − λ = − ǫµ + ǫλ . (6.22)The solution µ = λ of (6.22) leads to lumps solutions u ( ξ, η, t ) of NVN equation(1.1), which are not considered here (see the papers [23], [24]). Excluding parameter ǫ from (6.22) one obtains the relations: ǫ = µ λ µ − λ µ − λ = µ λ µ − λ µ − λ , (6.23)and their consequence:( | µ | − | λ | )( µ λ − µ λ ) = 0 . (6.24)Due to (6.23) and (6.24) the system (6.22) has the solutions:1 . ǫ = −| µ | = −| λ | , . ǫ = λ µ = λ µ . (6.25)One can show that time-dependent part of (6.21) satisfies by solutions (6.25) of thesystem (6.22). For both solutions of the system (6.22) the pure imaginary ϕ givenby (6.21) takes the form: ϕ ( z, ¯ z, t ) = i [( µ − λ ) z + ( µ − λ )¯ z + κ ( µ − λ ) t + κ ( µ − λ ) t ] := i ˜ ϕ ( z, ¯ z, t )(6.26)The condition (6.2) of reality of u for the first case in (6.25) gives the relation: (cid:16) λ µ − µ λ (cid:17)(cid:2) a e i ˜ ϕ − a e − i ˜ ϕ (cid:3)h | a | (cid:16) λ + µ λ − µ (cid:17) i = 0 , (6.27)which nontrivially satisfies for the following choice of amplitude a | a | = ± λ − µ λ + µ = ± tan δ δ := arg( µ ) − arg( λ ) . (6.28)For | a | = tan δ due to (2.18) and (6.2), (6.25) - (6.28) one obtains the periodicsolution u with constant asymptotic values − ǫ at infinity of elliptic NVN equation: u ( z, ¯ z, t ) = − ǫ − | λ − µ | (cid:16) ˜ ϕ +arg( a )2 (cid:17) = − ǫ + 2 ǫ sin δ cos (cid:16) ˜ ϕ +arg( a )2 (cid:17) , (6.29)and for | a | = − tan δ another periodic solution u ( z, ¯ z, t ) = − ǫ − | λ − µ | (cid:16) ˜ ϕ +arg( a )2 (cid:17) = − ǫ + 2 ǫ sin δ sin (cid:16) ˜ ϕ +arg( a )2 (cid:17) . (6.30)The condition (6.2) of reality of u for the second case in (6.25) gives the relation: (cid:16) λ µ − µ λ (cid:17)(cid:2) a e i ˜ ϕ + a e − i ˜ ϕ (cid:3)h − | a | (cid:16) λ + µ λ − µ (cid:17) i = 0 (6.31)which satisfies for | a | = ± λ − µ λ + µ . (6.32)For the second case in (6.25) periodic solution u ( ξ, η, t ) for the NVN equation (1.1)due to (2.18), (6.26), (6.32) has the form: u = − ǫ − | λ − µ | ( ˜ ϕ +arg a ∓ π ) , ǫ = λ ¯ µ = ¯ λ µ , (6.33) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ∓ π/ ± signs in (6.32). ∂ -dressing in present paper is carried out for the fixed nonzero value of parameter ǫ or, in context of present section, for nonzero energy E = 0. Nevertheless as insubsections 4.1 and 5.1 one can correctly consider the limit ǫ → E = − ǫ = 0. Limiting procedure E = − ǫ = − µ k ¯ λ k − ¯ µ k λ k → , ( k = 1 ,
2) can becorrectly performed by the following settings in all required formulas: ǫ → µ k → ǫµ k → ¯ λ k in accordance with therelation ǫ = µ k ¯ λ k ; in addition the formula λ = iτ − λ (followed from the relations¯ µ k λ k = µ k ¯ λ k and µ λ + µ λ = 0) with arbitrary real constant τ is assumed to bevalid. The periodic solution due to (3.23) in considered limit has the form: u = − | λ | ( ˜ ϕ +arg a − π ) − | λ | ( ˜ ϕ +arg a − π ) , (6.34)the phases ˜ ϕ k ( z, ¯ z, t ) due to (6.26) have in considered limit the forms:˜ ϕ k ( z, ¯ z, t ) = (cid:0) − λ k z − λ k z − κλ k t − κλ k t (cid:1) . (6.35)One can check by direct substitution that NVN-I equation (1.1) with σ = i satisfiesby u = ˜ u = − V Schr / u ( k ) = − | λ k | ( ˜ ϕ k +arg a k − π ) , ( k = 1 ,
2) (6.36)of the sum (6.34). Thus, in considered case the linear principle of superposition u = u (1) + u (2) for such special periodic solutions u (1) , u (2) (6.36) is valid. One canshow using relation λ = iτ − λ , (6.35) that periodic solutions u (1) and u (2) arepropagate in the plane ( x, y ) in perpendicular to each other directions.a b Figure 15. a)Periodic solution u ( x, y, t = 0) (6.29) (blue) and the squaredabsolute value of corresponding wave functions | ψ ( µ ) | = | ψ ( − λ ) | (3.16)(green) with parameters arg( a ) = π , δ = π , λ = 1 − . i, ǫ = 1 .
25, b)Two-periodic solution u ( x, y, t = 0) (3.23) with parameters arg( a ) = π , δ = π , λ =1 − . i ; arg( a ) = π , δ = π , λ = 0 . − . i, ǫ = 1 . Last two figures, Fig.15 a) and Fig.15 b), demonstrate the simplest one - (N=1 inkernel R (3.7)) and two-periodic (N=2 in kernel R (3.7)) solutions of NVN equation(1.1) calculated by the formulas (6.29) and (3.23) under certain values of correspondingparameters. It is assumed also that for two-periodic solution the condition (3.24) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method ∂ -dressing method will be continuedelsewhere.
7. Solutions of NVN equation with functional parameters
Constructed in the previous sections multi line soliton and periodic solutions can beembedded into more general class of exact solutions with functional parameters. Suchsolutions correspond to degenerate kernel R ( µ, µ ; λ, λ ) of ∂ -problem (2.1) R ( µ, µ, λ, λ ) = π N X k =1 f k ( µ, µ ) g k ( λ, λ ) . (7.1)As in section 2 one can easily derive general determinant formula for the class ofexact solutions u ( ξ, η, t ) with constant asymptotic value − ǫ at infinity with functionalparameters of the NVN equation (1.1). Indeed, inserting (7.1) into (2.20) andintegrating one obtains χ ( λ ) = 1 + π N X k =1 h k ( ξ, η, t ) Z Z C dλ ′ ∧ dλ ′ πi ( λ ′ − λ ) g k ( λ ′ , λ ′ ) e − F ( λ ′ ) (7.2)where h k ( ξ, η, t ) := Z Z C χ ( µ, µ ) e F ( µ ) f k ( µ, µ ) dµ ∧ dµ. (7.3)From (7.2), (7.3) follows the system of linear algebraic equations for the quantities h k : N X k =1 A lk h k = α l , ( l = 1 , · · · , N ) (7.4)with α l ( ξ, η, t ) := Z Z C f l ( µ, µ ) e F ( µ ) dµ ∧ dµ (7.5)and matrix A is given by expression: A lk := δ lk + π Z Z C dλ ∧ dλ Z Z C dλ ′ ∧ dλ ′ πi e F ( λ ) − F ( λ ′ ) λ − λ ′ f l ( λ, λ ) g k ( λ ′ , λ ′ ) . (7.6)Introducing the quantities β l ( ξ, η, t ) := Z Z C g l ( λ, λ ) e − F ( λ ) dλ ∧ dλ (7.7)one can rewrite the matrix A lk (7.6) in the following form: A lk = δ lk + 12 ∂ − ξ α l β k . (7.8)The functions α k ( ξ, η, t ), β k ( ξ, η, t ) given by (7.5) and (7.7) are known as functionalparameters. By the definitions (2.4) and (7.5), (7.7) the functional parameters α n and β n to the following linear equations are satisfy: α nξη = ǫα n , α nt + κ α nξξξ + κ α nηηη = 0 , (7.9) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method β nξη = ǫβ n , β nt + κ β nξξξ + κ β nηηη = 0 . (7.10)From (2.22) and (7.4)-(7.7) follows compact formula for the coefficient χ − of theexpansion (2.11) χ − = − i N X k =1 h k β k = − i N X l,k =1 A − kl α l β k = i N X k,l =1 A − kl ∂A lk ∂ξ == iT r ( A − ∂A∂ξ ) = i ∂ ξ (ln det A ) . (7.11)Here and below useful determinant identities T r ( ∂A∂ξ A − ) = ∂∂ξ ln(det A ) , trB = det (1 + B ) (7.12)are used. The matrix B in the last identity of (7.12) is degenerate with rank 1.Using reconstruction formula (2.18) and the expression (7.11) one obtains generaldeterminant formula for the solution u with constant asymptotic values − ǫ at infinitywith functional parameters α k ( ξ, η, t ), β k ( ξ, η, t ) (given by (7.5),(7.7)) of the NVNequation (1.1): u ( ξ, η, t ) = − ǫ − iχ − η = − ǫ + ∂ ∂ξ∂η ln det A. (7.13)Potentiality condition (2.25) due to (7.1), (7.3)-(7.7) also can be expressed interms of functional parameters χ − − ǫ N X k =1 h k β kη = − ǫ N X k,m =1 A − km α m β kη = − ǫ N X k,m =1 A − km B mk = 0 (7.14)where degenerate matrix B with rank 1 is defined by the formula B mk = α m β kη . (7.15)Due to (2.25) and (7.15) potentiality condition (7.14) takes the form0 = N X k,m =1 A − km B mk = tr ( A − B ) = det ( BA − + 1) − , (7.16)here matrix BA − is degenerate of rank 1 and in deriving the last equality in (7.16)second matrix identity (7.12) is used. So due to (7.16) the potentiality condition takesthe following convenient form:det( A + B ) = det A. (7.17)Important class of exact multi line soliton solutions of the NVN equation (1.1)can be obtained from solutions with functional parameters by the following choice ofthe functions f k ( µ, µ ), g k ( λ, λ ) in the kernel R (7.1): f k ( µ, µ ) = δ ( µ − M k ) , g k ( λ, λ ) = A k δ ( λ − Λ k ) . (7.18)Inserting (7.18) into (7.6) one obtains A lk = δ lk + 2 i A k M l − Λ k e F ( M l ) − F (Λ k ) . (7.19)For the matrix B due to (7.1), (7.7) and (7.15), (7.18) one derives the expression: B lk = α l β kη = − iǫ Λ k A k e F ( M l ) − F (Λ k ) . (7.20) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method L in (1.2). In order to satisfy to the condition of potentiality (2.25) theterms in the sum (7.1) for the kernel R can be grouped by pairs. Indeed, insertingthe expression R = πp ( µ, µ ) q ( λ, λ ) + πp ( µ, µ ) q ( λ, λ ) into (2.25) and performingthe change of variables µ ↔ − λ in the second term one obtains in the limit of weakfields ( χ = 1 in the equality (2.25)): Z Z C Z Z C h p ( µ, µ ) q ( λ, λ ) λ − p ( − λ, − λ ) q ( − µ, − µ ) µ i e F ( µ ) − F ( λ ) dµ ∧ dµ dλ ∧ dλ = 0 . (7.21)The relation (7.21) will be satisfied if λ p ( µ, µ ) q ( λ, λ ) = µ p ( − λ, − λ ) q ( − µ, − µ ), orseparating variables, if q ( λ, λ ) λp ( − λ, − λ ) = q ( − µ, − µ ) µp ( µ, µ ) = c (7.22)where c is some constant. Due to (7.22) p and q through q and q are expressed p ( λ, λ ) = − cλ q ( − λ, − λ ) , q ( µ, µ ) = − cµp ( − µ, − µ ) . (7.23)So to the potentiality condition (2.25) due to (7.23) is satisfied the following kernel R ( µ, µ, λ, λ ) = π N X k =1 (cid:16) p k ( µ, µ ) q k ( λ, λ ) + q k ( − µ, − µ ) µ λp k ( − λ, − λ ) (cid:17) (7.24) R of the ∂ -problem (2.1) with N pairs of correlated with each other terms.The conditions (2.23) and (2.24) of reality u = u give further restrictions on thefunctions p k and q k in the sum (7.24). It is convenient to perform the calculations ofthese restrictions and exact solutions u ( ξ, η, t ) separately for Nizhnik σ = 1, ξ = x + y , η = x − y and Veselov-Novikov σ = − ξ = z = x + iy , η = z = x − iy versions ofthe NVN equation (1.1).
8. Exact solutions with functional parameters of NVN-II equation
Let us consider at first the case σ = 1 of real space variables ξ = x + y , η = x − y or hyperbolic version of the NVN equation (1.1). To the condition (2.23) of reality u = u one can satisfy imposing on each pair of terms in the sum (7.24) the followingrestriction: p n ( µ, µ ) q n ( λ, λ ) + 1 µ q n ( − µ, − µ ) λp n ( − λ, − λ ) = (8.1)= p n ( − µ, − µ ) q n ( − λ, − λ ) + 1 µ q n ( µ, µ ) λp n ( λ, λ ) . Due to (8.1) two cases are possible8 .A. p n ( µ, µ ) q n ( λ, λ ) = p n ( − µ, − µ ) q n ( − λ, − λ ) , (8.2)8 .B. p n ( µ, µ ) q n ( λ, λ ) = 1 µ q n ( µ, µ ) λp n ( λ, λ ) . (8.3) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method .A by separating variables p n ( µ, µ ) p n ( − µ, − µ ) = q n ( − λ, − λ ) q n ( λ, λ ) = c n (8.4)one obtains the following restrictions on the functions p n ( µ, µ ) and q n ( λ, λ ): p n ( µ, µ ) = c n p n ( − µ, − µ ) , q n ( λ, λ ) = 1 c n q n ( − λ, − λ ) . (8.5)Constants c n in (8.5) without restriction of generality can be chosen equal to unity.In the case 8 .B by separating variables µp n ( µ, µ ) q n ( µ, µ ) = λp n ( λ, λ ) q n ( λ, λ ) = c − n (8.6)one obtains the another restrictions on the functions p n ( µ, µ ) and q n ( λ, λ ): q n ( λ, λ ) = λc n p n ( λ, λ ) . (8.7)The constants c n in (8.7) due to (8.6) are real.In applying general determinant formula (7.13) for exact solutions u one mustto identify the corresponding kernels (7.1) and (7.24). For the case 8 .A taking intoaccount (7.24) and (8.5) one has: R ( µ, µ, λ, λ ) = π N X n =1 f n ( µ, µ ) g n ( λ, λ ) == π N X n =1 (cid:16) p n ( µ, µ ) q n ( λ, λ ) + 1 µ q n ( µ, µ ) λp n ( λ, λ ) (cid:17) (8.8)and from (8.8) one can choose the following convenient sets f and g of functions f n , g n : f := ( f , . . . , f N ) = ( p ( µ, µ ) , . . . , p N ( µ, µ ); 1 µ q ( µ, µ ) , . . . , µ q N ( µ, µ )) , (8.9) g := ( g , . . . , g N ) = ( q ( λ, λ ) , . . . , q N ( λ, λ ); λp ( λ, λ ) , . . . , λp N ( λ, λ )) . (8.10)Due to definitions (7.5), (7.7) and (8.9), (8.10) taking into account (8.5) one can derivethe following interrelations between different functional parameters: α n := Z Z C p n ( µ, µ ) e F ( µ ) dµ ∧ dµ = α n , β n := Z Z C q n ( λ, λ ) e − F ( λ ) dλ ∧ dλ = β n , (8.11) α N + n := Z Z C µ q n ( µ, µ ) e F ( µ ) dµ ∧ dµ = iǫ β nη , (8.12) β N + n := Z Z C λp n ( λ, λ ) e − F ( λ ) dλ ∧ dλ = i α nξ , ( n = 1 , . . . , N ) . (8.13)So due to (8.11)-(8.13) the sets of functional parameters have the following structure:( α , . . . , α N ) := ( α , . . . , α N ; iǫ β η , . . . , iǫ β Nη ) (8.14)( β , . . . , β N ) := ( β , . . . , β N ; i α ξ , . . . , i α Nξ ) (8.15) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method N independent real functional parameters ( α , . . . , α N )and ( β , . . . , β N ).General determinant formula (7.13) with matrix A (7.8) corresponding to thekernel R (8.8) of the ∂ -problem (2.1) gives the class of exact solutions u withconstant asymptotic value − ǫ at infinity of hyperbolic version of the NVN equation(1.1). By construction these solutions depend on 2 N real functional parameters( α , . . . , α N ) and ( β , . . . , β N ) given by (8.14),(8.15). In the simplest case N = 1( α , α ) := ( α , iǫ β η ) , ( β , β ) := ( β , i α ξ ) the determinant of A due to (7.8) isgiven by expressiondet A = (cid:0) ∂ − ξ α β (cid:1)(cid:0) − ǫ ∂ − ξ α ξ β η (cid:1) + 18 ǫ α ∂ − ξ β β η == (cid:16) ∂ − ξ α β − α β η ǫ (cid:17) = ∆ . (8.16)The corresponding solution u due to (7.13) and (8.16) has the form: u ( ξ, η, t ) = − ǫ + 12∆ ( α η β − ǫ α ξ β ηη ) − ǫ ( α β − ǫ α ξ β η )( α η β η − α β ηη ) . (8.17)For the delta-functional kernel R (7.24) of the type (8.8) with p n ( µ, µ ) = δ ( µ − iµ n ) , q n ( λ, λ ) = a n λ n δ ( λ − iλ n ) , n = 1 , . . . , N (8.18)the general determinant formula (7.13) leads to corresponding exact multisolitonsolutions. In the simplest case of N = 1 from (8.11) one obtains the functionalparameters α = − ie F ( iµ ) , β = − ia λ e − F ( iλ ) and from (8.17), under thecondition a ( λ + µ ) λ − µ = − e ϕ <
0, the exact nonsingular line soliton solution of thehyperbolic NVN equation: u ( ξ, η, t ) = − ǫ − ǫ ( λ − µ ) λ µ ϕ ( ξ,η,t )+ ϕ (8.19)where the phase ϕ has the form ϕ ( ξ, η, t ) := F ( iµ ) − F ( iλ ) == ( λ − µ ) ξ + (cid:16) ǫλ − ǫµ (cid:17) η − κ (cid:0) λ − µ (cid:1) t − κ (cid:16) ǫ λ − ǫ µ (cid:17) t. (8.20)For the case 8 .B taking into account (8.7) and identifying expressions for R givenby (7.1) and (7.24) one obtains R ( µ, µ, λ, λ ) = π N X n =1 f n ( µ, µ ) g n ( λ, λ ) == π N X n =1 (cid:16) c n p n ( µ, µ ) λp n ( λ, λ ) − c n p n ( − µ, − µ ) λp n ( − λ, − λ ) (cid:17) . (8.21)From (8.21) one can choose the following convenient sets f , g of functions f n , g n : f := ( f , . . . , f N ) = ( p ( µ, µ ) , . . . , p N ( µ, µ ); p ( − ¯ µ, − µ ) , . . . , p N ( − µ, − µ )) , (8.22) g := ( g , . . . , g N ) =( c λp ( λ, λ ) , . . . , c N λp N ( λ, λ ); − c λp ( − λ, − λ ) , . . . , − c N λp N ( − λ, − λ )) . (8.23) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method α n := Z Z C p n ( µ, µ ) e F ( µ ) dµ ∧ dµ, β n := Z Z C c n λp n ( λ, λ ) e − F ( λ ) dλ ∧ dλ = ic n α nξ , (8.24) α N + n = Z Z C p n ( − µ, − µ ) e F ( µ ) dµ ∧ dµ = α n , (8.25) β N + n = − Z Z C c n λp n ( − λ, − λ ) e − F ( λ ) dλ ∧ dλ = β n , ( n = 1 , · · · , N ) . (8.26)So due to (8.24) and (8.25), (8.26) the sets α , β of functional parameters α := ( α , α , . . . , α N ) = ( α , . . . , α N ; α , . . . , α N ) , (8.27) β := ( β , β , . . . , β N ) = ( ic α ξ , . . . , ic N α Nξ ; − ic α ξ , . . . , ic N α Nξ ) (8.28)express through the N independent complex parameters ( α , . . . , α N ).General determinant formula (7.13) with matrix A given by (7.8) with kernel R (8.23) of the ∂ -problem (2.1) gives another class of exact solutions with constantasymptotic value at infinity of the hyperbolic version of the NVN equation (1.1).By construction these solutions depend on N independent complex parameters( α , . . . , α N ) given by (8.27), (8.28). In the simplest case N = 1 ( α , α ) :=( α , α ) , ( β , β ) := ( ic α ξ , − ic α ξ ) the determinant of A due to (7.8) is givenby expression det A = (1 + ic ∂ − ξ α α ξ )(1 − ic ∂ − ξ α ξ α ) − c | α |
16 = (8.29)= (1 + ic ∂ − ξ ( α α ξ − α ξ α )) = ∆ . The corresponding solution u due to (7.13) and (8.29) has the form: u ( ξ, η, t ) = − ǫ + ic
2∆ ( α η α ξ − α η α ξ ) + c ( α α ξ − α α ξ )( α η α − α η α ) . (8.30)For the delta-functional kernel of the type (8.21) with p n ( µ, µ ) = δ ( µ − iλ n ) , n = 1 , . . . , N (8.31)general determinant formula (7.13) taking into account (8.22)-(8.28) leads tocorresponding exact multi line soliton solutions. In the simplest case of N = 1from (8.24) one obtains the functional parameter α = − ie F ( λ ) and due to (8.30)corresponding exact solution u , under the condition c λ R λ I = e ϕ >
0, is the one linenonsingular soliton: u ( ξ, η, t ) = − ǫ + 8 ǫc λ R λ I e ϕ ( ξ,η,t ) | λ | (1 + c λ R λ I e ϕ ( ξ,η,t ) ) = − ǫ + 2 ǫλ I | λ | ϕ ( ξ,η,t )+ ϕ (8.32)where the phase ϕ has the form ϕ ( ξ, η, t ) = i h ( λ − λ ) ξ − (cid:16) ǫλ − ǫλ (cid:17) η + κ (cid:0) λ − λ (cid:1) t − κ (cid:16) ǫ λ − ǫ λ (cid:17) t i . (8.33) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method
9. Exact solutions with functional parameters of NVN-I equation
Let us consider also the case σ = − ξ = z = x + iy , η = ¯ z = x − iy or elliptic version of the NVN equation (1.1). To the condition (2.24)of reality u = u one can satisfy imposing on each pair of terms in the sum (7.24) thefollowing restriction: p n ( µ, µ ) q n ( λ, λ ) + 1 µ q n ( − µ, − µ ) λp n ( − λ, − ¯ λ ) == ǫ | λ | | µ | λµ p n (cid:16) − ǫλ , − ǫλ (cid:17) q n (cid:16) − ǫµ , − ǫµ (cid:17) + ǫ | λ | | µ | λµ λq n (cid:16) ǫλ , ǫλ (cid:17) µ p n (cid:16) ǫµ , ǫµ (cid:17) . (9.1)Due to (9.1) two cases are possible9 .A. p n ( µ, µ ) q n ( λ, λ ) = ǫ | λ | | µ | λµ p n (cid:16) − ǫλ , − ǫλ (cid:17) q n (cid:16) − ǫµ , − ǫµ (cid:17) , (9.2)9 .B. p n ( µ, µ ) q n ( λ, λ ) = ǫ λ | λ | | µ | λ q n (cid:16) ǫλ , ǫλ (cid:17) p n (cid:16) ǫµ , ǫµ (cid:17) . (9.3)In the case 9 .A separating in (9.2) the variables p n ( µ, µ ) | µ | µq n ( − ǫµ , − ǫµ ) = ǫ | λ | λ p n ( − ǫλ , − ǫλ ) q n ( λ, λ ) = c − n (9.4)one obtains the following relations on the functions q n and p n : p n ( µ, µ ) = 1 c n | µ | µ q n (cid:16) − ǫµ , − ǫµ (cid:17) , q n ( λ, λ ) = ǫ c n | λ | λ p n (cid:16) − ǫλ , − ǫλ (cid:17) . (9.5)Comparing two relations in (9.5) one concludes that constant c n are pure imaginary: c n = i a n . In applying general determinant formula (7.13) for exact solutions u onemust to identify the corresponding expressions (7.1) and (7.24) for the kernel R , dueto relations (9.5) one obtains R ( µ, µ, λ, λ ) = π N X n =1 f n ( µ, µ ) g n ( λ, λ ) == π N X n =1 (cid:16) p n ( µ, µ ) i a n ǫ | λ | λ p n (cid:16) − ǫλ , − ǫλ (cid:17) − i a n ǫ | µ | p n (cid:16) ǫµ , ǫµ (cid:17) λp n ( − λ, − λ ) (cid:17) . (9.6)From (9.6) one can choose the following convenient sets f , g of functions f n , g n : f := ( f , . . . , f N ) = (cid:16) p ( µ, µ ) , . . . , p N ( µ, µ ); ǫ | µ | p (cid:16) ǫµ , ǫµ (cid:17) , . . . , ǫ | µ | p N (cid:16) ǫµ , ǫµ (cid:17)(cid:17) , (9.7) g := ( g , . . . , g N ) = (cid:16) i ǫ a | λ | λ p (cid:16) − ǫλ , − ǫλ (cid:17) , . . . , i ǫ a N | λ | λ p N (cid:16) − ǫλ , − ǫλ (cid:17) ; − ia λp ( − λ, − λ ) , . . . , − ia N λp N ( − λ, − λ ) (cid:17) . (9.8)Due to definitions (7.5), (7.7) and (9.7), (9.8) taking into account (9.5) one canderive the interrelations between different functional parameters: α n := Z Z C p n ( µ, µ ) e F ( µ ) dµ ∧ dµ, (9.9) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method β n := i a n ǫ Z Z C | λ | λ p n ( − ǫλ , − ǫλ ) e − F ( λ ) dλ ∧ dλ = (9.10)= − a n ǫ∂ z Z Z C p n ( λ, λ ) e F ( λ ) dλ ∧ dλ = − ǫa n α nz ,α N + n := Z Z C ǫ | µ | p n ( ǫµ , ǫµ ) e F ( µ ) dµ ∧ dµ = ǫ Z Z C p n ( µ, µ ) e F ( ǫµ ) dµ ∧ dµ = ǫα n , (9.11) β N + n := − i Z Z C λa n p n ( − λ, − λ ) e − F ( λ ) dλ ∧ dλ = a n α nz , ( n = 1 , . . . , N ) . (9.12)So due to (9.9)-(9.12) the sets of functional parameters( α , . . . , α N ) := ( α , . . . , α N , ǫα , . . . , ǫα N ) (9.13)( β , β , . . . , β N ) = ( − ǫa α z , . . . , − ǫa N α Nz ; a α z , . . . , a N α Nz ) . (9.14)are express through N independent complex functional parameters ( α , . . . , α N ).General determinant formula (7.13) with matrix A (7.8) corresponding to thekernel R (9.6) of the ∂ -problem (2.1) gives the class of exact solutions u with constantasymptotic value − ǫ at infinity of the elliptic version of the NVN equation (1.1). Byconstruction these solutions depends on N complex functional parameters α , . . . , α N .In the simplest case N = 1 ( α , α ) := ( α , ǫα ) , ( β , β ) := ( − ǫa α z , a α z ) anddue to (7.8) the determinant of A is given by expression:det A = (1 − a ǫ ∂ − z ( α α z ))(1 + a ǫ ∂ − z ( α α z )) + a ǫ | α | == (1 − a ǫ ∂ − z ( α α z ) + a ǫ | α | ) = ∆ . (9.15)The corresponding solution u due to (7.13) and (9.15) has the form: u ( z, ¯ z, t ) = − ǫ + a ǫ
2∆ ( | α z | − | α z | ) − a ǫ | α α z − α α z | . (9.16)For the delta-functional kernel of the type (9.6) with p n ( µ, µ ) = δ ( µ − µ n ) , n = 1 , . . . , N (9.17)and λ n µ n = µ n λ n = − ǫ , general determinant formula (7.13) taking into account (9.7)-(9.14) leads to corresponding exact multi line soliton solutions. In the simplest caseof N = 1 from (9.9) one obtains the functional parameter α = − ie F ( µ ) and due to(9.16) corresponding exact solution u , under the condition ǫa µ + λ λ − µ = − e ϕ <
0, isthe nonsingular one line soliton: u ( z, z, t ) = − ǫ + | λ − µ | ϕ ( z,z,t )+ ϕ (9.18)where the phase ϕ has the form ϕ ( z, z, t ) = i [( µ − λ ) z − ( µ − λ ) z + κ ( µ − λ ) t − κ ( µ − λ ) t ] . (9.19)In the case 9 .B separating in (9.3) the variables p n ( µ, µ ) | µ | ǫ p n ( ǫµ ) = ǫλ q n ( ǫλ , ǫλ ) q n ( λ, λ ) = c n (9.20) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method q n ( λ, λ ) and p n ( µ, µ ): p n ( µ, µ ) = c n ǫ | µ | p n (cid:16) ǫµ , ǫµ (cid:17) , q n ( λ, λ ) = ǫc n λ q n (cid:16) ǫλ , ǫλ (cid:17) . (9.21)The constants c n in (9.20), (9.21) without loss of generality can be choosen equal tounity. In applying general determinant formula (7.13) for exact solutions u one mustto identify the corresponding expressions (7.1) and (7.24) for the kernel R ∂ -problem(2.1). In the considered 9 .B case taking into account (9.21) one obtains from (7.1)and (7.24): R ( µ, µ, λ, λ ) = π N X n =1 f n ( µ, µ ) g n ( λ, λ ) == π N X n =1 (cid:16) p n ( µ, µ ) q n ( λ, λ ) + ǫµµ q n (cid:16) − ǫµ , − ǫµ (cid:17) λ ǫ | λ | p n (cid:16) − ǫλ , − ǫλ (cid:17)(cid:17) . (9.22)From (9.22) one can choose the following convenient sets f , g of functions f n , g n : f := ( f , . . . , f N ) = (cid:16) p ( µ, µ ) , . . . , p N ( µ, µ ); ǫµµ q (cid:16) − ǫµ , − ǫµ (cid:17) , . . . , ǫµµ q N (cid:16) − ǫµ , − ǫµ (cid:17)(cid:17) , (9.23) g := ( g , . . . , g N ) = (cid:16) q ( λ, λ ) , . . . , q N ( λ, λ ); ǫ λ λ p (cid:16) − ǫλ , − ǫλ (cid:17) , . . . , ǫ λ λ p N (cid:16) − ǫλ , − ǫλ (cid:17)(cid:17) . (9.24)Due to definitions (7.5), (7.7) and (9.23), (9.24) taking into account (9.21) onecan derive the interrelations between different functional parameters: α n := Z Z C ǫ | µ | p n (cid:16) ǫµ , ǫµ (cid:17) e F ( µ ) dµ ∧ dµ = Z Z C p n ( µ, µ ) e F ( µ ) dµ ∧ dµ = α n , (9.25) β n := Z Z C ǫλ q n (cid:16) ǫλ , ǫλ (cid:17) e − F ( λ ) dλ ∧ dλ = Z Z C ǫλ q n ( λ, λ ) e − F ( ǫλ ) dλ ∧ dλ = − ǫ β nzz , (9.26) α N + n := Z Z C ǫµµ q n (cid:16) − ǫµ , − ǫµ (cid:17) e F ( µ ) dµ ∧ dµ = − iǫ β nz , (9.27) β N + n := Z Z C ǫ λ λ p n (cid:16) − ǫλ , − ǫλ (cid:17) e − F ( λ ) dλ ∧ dλ = iα nz , ( n = 1 , . . . , N ) . (9.28)From (9.26) it follows that β nz = − β nz = − β nz , ( n = 1 , . . . , N ) . (9.29)So due to (9.25)-(9.29) the sets of functional parameters( α , . . . , α N ) := ( α , . . . , α N ; iǫ β z , . . . , iǫ β Nz ) (9.30)( β , β , . . . , β N ) = ( β , . . . , β N ; i α z , . . . , i α Nz ) . (9.31) ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method N independent functional parameters ( α , . . . , α N ) and( β , . . . , β N ) given by (9.25)-(9.28).General determinant formula (7.13) with matrix A (7.8) corresponding to thekernel R (9.22) of the ∂ -problem (2.1) gives the class of exact solutions u with constantasymptotic value − ǫ at infinity of the elliptic version of the NVN equation (1.1).By construction these solutions depend in fact due to (9.29) on N real functionalparameters α , . . . , α N and N real functional parameters iβ z , . . . , iβ Nz . In thesimplest case N = 1 ( α , α ) := ( α , iǫ β z ) , ( β , β ) := ( β , i α z ) the determinantof A due to (7.8) and (9.30), (9.31) is given by expressiondet A = (1 + 12 ∂ − z α β )(1 − ǫ ∂ − z α z β z ) + α β z ǫ == (1 + 12 ∂ − z ( α β ) − ǫ α β z ) = ∆ . (9.32)Using identity ∂ − z ( α β ) − ∂ − z ( α β ) = ǫ α β z (which is valid due to the relations(9.25)-(9.29)) one obtains explicitly real expression for det A :det A = (1 + 14 ∂ − z ( α β ) + 14 ∂ − z ( α β )) = ∆ . (9.33)Using (9.33) one calculates by (7.13) the corresponding exact solution u = − ǫ + 12∆ (cid:16) ( α β ) z + ( α β ) z (cid:17) − (cid:16) α β + 1 ǫ α z β z (cid:17)(cid:16) α β + 1 ǫ α z β z (cid:17) . (9.34)For the delta-functional kernel of the type (9.22) with p n ( µ, µ ) = iδ ( µ − µ n ) , q n ( λ, λ ) = − ia n λ n δ ( λ − λ n ) , ( n = 1 , . . . , N ) (9.35)with | µ n | = | λ n | = ǫ and real constants a n = a n general determinant formula(7.13) taking into account (9.23)-(9.31) leads to corresponding exact multi line solitonsolutions. In the simplest case of N = 1 from (9.25)-(9.26) one obtains the functionalparameters α = 2 e F ( µ ) , β = − a λ e − F ( λ ) and due to (9.34) corresponding exactsolution u , under the condition i a µ + λ µ − λ = − e ϕ <
0, is nonsingular line soliton: u ( z, z, t ) = − ǫ + ǫ ( δ )cosh ( ϕ ( z,z,t )+ ϕ ) (9.36)where δ = arg µ − arg λ and the phase ϕ has the form ϕ ( z, z, t ) = i [( µ − λ ) z − ( µ − λ ) z + κ ( µ − λ ) t − κ ( µ − λ ) t ] . (9.37)
10. Conclusions and Acknowledgments
The powerful ¯ ∂ -dressing method of Zakharov and Manakov, discovered a quarter ofcentury ago, continues to develop and successfully apply for construction of exactsolutions of multidimensional integrable nonlinear equations. The realization ofthe method goes due to basic idea of IST through the careful study of auxiliarylinear problems by the methods of modern theory of functions of complex variables.Following this way one constructs exact complex wave functions (with rich analyticalstructure) of linear auxiliary problems and by using the wave functions, viareconstruction formulas, exact (or solvable) potentials - exact solutions of integrablenonlinear equations.Constructed in the paper exact solutions of hyperbolic and elliptic versions ofNVN equation (1.1) as exact potentials for one-dimensional perturbed telegraph (or ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method References [1] Novikov S.P., Zakharov V.E., Manakov S.V., Pitaevsky L.V., Soliton Theory: the inversescattering method, New York, Plenum Press, 1984.[2] Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering,London Mathematical Society Lecture Notes Series, vol. 149, Cambridge, Cambridge Univ.Press, 1991.[3] Konopelchenko B.G., Introduction to multidimensional integrable equations: the inverse spectraltransform in 2+1-dimensions, New York - London, Plenum Press, 1992.[4] Konopelchenko B.G., Solitons in multidimensions: inverse spectral transform method,Singapore, World Scientific, 1993.[5] Manakov S.V., The inverse scattering transform for the time-dependent Schr¨odinger equationand Kadomtsev-Petviashvili equation, Physica D, 1981, V.3, 420-427.[6] Beals R., Coifman R.R., The ¯ ∂ approach to inverse scattering and nonlinear equations, PhysicaD, 1986, V.18, 242-249.[7] Zakharov V.E., Manakov S.V., Construction of multidimensional nonlinear integrable systemand their solutions, Funct. Anal. Pril., 1985, V.19, 2, 11-25 (in Russian).[8] Zakharov V.E., Commutating operators and nonlocal ¯ ∂ -problem, in Proc. of Int. Workshop”Plasma Theory and Nonlinear and Turbulent Processes in Physics”, Naukova Dumka, Kiev,1988 Vol.I, p.152.[9] Bogdanov L.V., Manakov S.V., The nonlocal ¯ ∂ -problem and (2+1)-dimensional solitonequations, J.Phys.A: Math.Gen., 1988, V. 21, L537-L544.[10] Fokas A.S. and Zakharov V.E., The dressing method for nonlocal Riemann-Hilbert problems,Journal of Nonlinear Sciences, 1992, V.2, 1, 109-134.[11] Nizhnik L. P. DAN SSSR, 1980, V.254, 332.[12] Veselov A. P. and Novikov S. P., Finite-gap two-dimensional potential Schr¨odinger operators,DAN SSSR, 1984, V.279, 1, 20-24 (in Russian).[13] Grinevich P.G., Manakov S.V., Inverse problem of scattering theory for the two-dimensionalSchr¨odinger operator, the ¯ ∂ -method and nonlinear equations, Funct. Anal. Pril., 1986, V.20,7, 14-24 (in Russian).[14] P.G. Grinevich, Rational solitons of the Veselov-Novikov equation - reflectionless at fixed energytwo-dimensional potentials, Teor. Mat. Fiz., Vol. 69, 307 (1986).[15] Grinevich P.G., Manakov S.V., Inverse scattering problem for the two-dimensional Schr¨odingeroperator at fixed negative energy and generalized analytic functions, in Proc. of Int. Workshop”Plasma Theory and Nonlinear and Turbulent Processes in Physics” (April 1987, Kyiv),Editors Bar’yakhtar, Chernousenko V.M., Erokhin N.S., Sitenko A.G., and Zakharov V.E.,Singapore World Scietific,1988, V.1, 58-85.[16] Grinevich P.G., Novikov R.G., A two-dimensional ”inverse scattering problem” for negativeenergies, and generalized-analytic functions. I. Energies lower than ground state,Funct.Anal.Pril., 1988, V. 22, 1, 23-33 (in Russian).[17] Boiti M., Leon J.J.P., Manna M., Pempinelli F., On a spectral transform of a KdV-like equationrelated to the Schr¨odinger operator in the plane, Inverse Problems, Vol. 3, 25 (1987).[18] Matveev V.B., Salle M.A., Darbu Transformations and solitons, in Springer series in NonlinearDynamics Springer, Berlin, Heidelberg, 1991.[19] Grinevich P.G.,Transformation of dispersion for the two-dimensional Schr¨odinger operator forone energy and connected with it integrable equations of mathematical physics, DoctorateThesis, Chernogolovka, 1999 (in Russian).[20] Grinevich P.G. , Scattering tranformation at fixed non-zero energy for the two-dimensionalSchr¨odinger operator with potential decaying at infinity, Russian Math. Surveys, 2000, V.55,6, 1015-1083. ew exact solutions of the NVN nonlinear equation via ¯ ∂ -dressing method [21] Athorne C., Nimmo J.J.C., On the Moutard transformation for the integrable partial differentialequations, Inverse Problems 7, 1991, 809-826.[22] V.G. Dubrovsky, B.G. Konopelchenko, The 2+1-dimensional generalization of the sine-Gordonequation. II. Localized solutions, Inverse Problems, 9, 391-416(1993).[23] Dubrovsky V.G., Formusatik I.B., The construction of exact rational solutions with constantasymptotic values at infinity of two-dimensional NVN integrable nonlinear evolution equationsvia ¯ ∂ -dressing method, J.Phys.A.: Math. and Gen., v.34A, 1837-1851 (2001).[24] Dubrovsky V.G., Formusatik I.B., New lumps of Veselov-Novikov equation and new exactrational potentials of two-dimensional Schr¨odinger equation via ¯ ∂∂