Searching for missing D'Alembert waves in nonlinear system: Nizhnik-Novikov-Veselov equation
aa r X i v : . [ n li n . S I] J u l Searching for missing D’Alembert waves in nonlinear system:Nizhnik-Novikov-Veselov equation
Man JIA and S. Y. LOU
Laboratory of Clean Energy Storage and Conversion,School of Physical Science and Technology, Ningbo University, Ningbo 315211, P. R. China
In linear science, the wave motion equation with general D’Alembert wave solutions is one of thefundamental models. The D’Alembert wave is an arbitrary travelling wave moving along one direc-tion under a fixed model (material) dependent velocity. However, the D’Alembert waves are missedwhen nonlinear effects are introduced to wave motions. In this paper, we study the possible travel-ling wave solutions, multiple soliton solutions and soliton molecules for a special (2+1)-dimensionalKoteweg-de Vries (KdV) equation, the so-called Nizhnik-Novikov-Veselov (NNV) equation. Themissed D’Alembert wave is re-discovered from the NNV equation. By using the velocity resonancemechanism, the soliton molecules are found to be closely related to D’Alembert waves. In fact, thesoliton molecules of the NNV equation can be viewed as special D’Alembert waves. The interactionsolutions among special D’Alembert type waves ( n -soliton molecules and soliton-solitoff molecules)and solitons are also discussed. PACS numbers: 05.45.Yv,02.30.Ik,47.20.Ky,52.35.Mw,52.35.Sb
The wave motion equation u tt − c u xx = 0 (1)is one of the most fundamental equations in linear physics. The general solutions of (1) possess theD’Alembert wave form u = f ( ξ ) + g ( η ) , ξ ≡ x − ct, η ≡ x + ct, (2)where f and g are arbitrary second-order differentiable functions of the traveling wave variables ξ and η ,respectively. The model parameter c , the velocity of the D’Alembert waves, is fixed and related to concretephysical problems. For instance, c is the light velocity in optics and electromagnetics and sound velocity inacoustics and elasticity mechanics.The simplest nonlinear extension of the wave motion equation is the Korteweg-de Vries (KdV) equation[1] u t + u xxx + 6 uu x = 0 , (3)which possesses various applications in physics and other scientific fields. It approximately describes theevolution of long, one-dimensional waves in many physical settings, including shallow-water waves withweakly nonlinear restoring forces, long internal waves in a density-stratified ocean, ion acoustic waves in aplasma, acoustic waves on a crystal lattice [1] and the 2-dimensional quantum gravity [2]. The KdV equationcan be solved by using the inverse scattering transform [3] and other powerful methods such as the Hirota’sbilinear method [4] and the Darboux transformation [5].The KdV equation and other nonlinear extensions lost the D’Alembert type wave solutions. A naturaland important question is can we find possible nonlinear wave extensions such that the missing D’Alemberttype waves are still allowed?In this letter, we study some special types of exact solutions for a special (2+1)-dimensional extension ofthe KdV equation, the so-called Nizhnik-Novikov-Veselov (NNV) equation[6–8] u t + cu x + du y + au xxx + bu yyy + 3 a ( uv ) x + 3 b ( uw ) y = 0 , u x = v y , u y = w x . (4)It is clear that when y = x and v = w = u , the NNV equation (4) is reduced to the KdV equation (3) aftersome scaling and Galileo transformations.The study of soliton molecules (SMs), soliton bound states, is one of the hot topics because some types ofSMs have been observed and applied in several physical areas such as the optics [9–12, 14] and Bose-Einsteincondensates [13]. Some theoretical proposals to form soliton molecules have been established [15, 16]. InRef. [17], the similar soliton molecules were numerically verified to exist in the nonlinear dispersive NLS(n,n)equation. Especially, in Refs. [18, 19], a new mechanism, the velocity resonance, to form soliton moleculesis proposed. It is found that the standard (1+1)-dimensional KdV equation (3) does not possess solitonmolecules. However, in real physics, higher order effects such as the higher order dispersions and higherorder nonlinearities have been neglected when the KdV equation (3) is derived [20]. Whence the higherorder effects are considered to the usual KdV equation, one can really find some types of SMs [18]. By usingthe velocity resonance mechanism, some authors find new types of SMs such as the dromion molecules andhalf periodic kink (HPK) molecules for some other physical systems [21–24].To search for the possible D’Alembert type wave solutions, we study the traveling wave solutions of theNNV equation (4) in the form u = U ( τ ) , v = V ( τ ) , w = W ( τ ) , τ = kx + py + ωt. (5)Substituting (5) into (4) yields( ω + ck + dp ) U τ + ( ak + bp ) U τττ + 3 ak ( U V ) τ + 3 bp ( U W ) τ = 0 , kU τ = pV τ , pU τ = kW τ , (6)say, V = kp U, W = pk U, (7)( ω + ck + dp ) U τ + ( ak + bp ) U τττ + 3 ak + bp pk (cid:0) U (cid:1) τ = 0 . (8)From (8) we know that when setting p = − r ab k, ω = − ck − dp, (9)the travelling wave U becomes an arbitrary D’Alembert type wave in the form u = U ( τ ) , τ = x − r ab y − (cid:18) c − d r ab (cid:19) t, (10)and moves along the direction p ab x + y (perpendicular to x − p ab y ) with the model parameter ( { a, b, c, d } )dependent velocity ~c = { c x , c y } ≡ ( c − d r ab , − (cid:18) c − d r ab (cid:19) r ba ) , | c | = q c x + c y = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d − c r ba (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r (cid:16) ab (cid:17) / . Because of the arbitrariness of the D’Alembert type wave (10), the solitary waves and soliton moleculespossess quite free shapes.Fig. 1 shows some special structures which may be used to describe real nonlinear phenomena. Fig. 1ais a kink molecule which is obtained by taking U ( τ ) = 2 h ln(1 + e k ( τ + τ ) + e k ( τ + τ ) + a e k ( τ + τ )+ k ( τ + τ ) ) i τ (11)with a = b = c = d = k = a = 1 , τ = − τ = 5 and k = 0 . t = 0.Fig. 1b displays the structure of HPK-kink molecule described by U ( τ ) = 2 n ln h − . . τ ))e k ( τ + τ ) + e k ( τ + τ ) + (1 − . . τ ))e k ( τ + τ ) io τ (12)with a = b = c = d = k = 1 , τ = − τ = 6 and k = − . t = 0.Fig.1c is a plot of a periodic kink structure expressed by U ( τ ) = tanh { τ [1 − . . τ )] } [1 − . τ )] (13)with a = b = c = d = 1 at time t = 0.Fig.1d displays a three-soliton molecule presented by U ( τ ) = 6( c − τ ) exp (cid:20) − c − τ ) (cid:21) (14)with a = b = c = d = 1 and c = 1.Fig.1e plots a special molecule constructed by two solitons and a KAK bound state with U given by (14), a = b = c = d = 1 , and c = √ a = b = c = d = 1 , and c = 2 at time t = 0.Now a further interesting question is that can we find some special types of interaction solutions amongsome special D’Alembert type waves and other types of NNV waves? To partially answer this question, westudy the multiple soliton solutions of the NNV equation (4).To find multiple soliton solutions, Hirota’s bilinear method can be used. By taking the transformations u = 2[ln( f )] xy , v = 2[ln( f )] xx , w = 2[ln( f )] yy , (15)the NNV equation becomes( f ∂ x − f x ) D y (2 aD x + 2 cD x + D t ) f · f + ( f ∂ y − f y ) D x (2 bD y + 2 dD y + D t ) f · f = 0 , (16)where the Hirota’s bilinear operator D x i , x i = x, y, z, t is defined by D nx i f · g = ( ∂ x i − ∂ x ′ i ) n f ( x i ) g ( x ′ i ) (cid:12)(cid:12) x ′ i = x i . (17)Eq. (16) can be further bilinearized to D y (2 aD x + 2 cD x + D t + D z ) f · f = 0 , (18) D x (2 bD y + 2 dD y + D t − D z ) f · f = 0 , (19)by introducing an auxiliary variable z .It is straightforward to write down the solutions of (18)–(19) ( ξ i = k i x + p i y + q i z + ω i t + ξ i ) f = X µ exp N X j =1 µ j ξ j + N X ≤ i 2. (f) Soliton-MS-soliton molecule determined by (14) with c = 2. Applying the velocity resonant mechanism [18, 19] to the multiple soliton solution (20) of the NNVequation (4), we have k i k j = p i p j = ak i + bp i + ck i + dp i ak j + bp j + ck j + dp j , k i = ± k j , p i = ± p j , say, p l = − r ab k l , l = i, j. If one considers the velocity resonant conditions for n solitons, one can find the only solution is p l = − r ab k l , ξ l = k l τ + ξ l , l = 1 , , . . . , n, (23)where τ is just the travelling variable (10) of the D’Alembert type waves. In other words, the n -solitonmolecule (15) with (20), (23) and N = n is just a special D’Alembert type wave.Now we consider special interactions among D’Alembert type waves and usual solitons. If a solitonmolecule is constituted by n solitons, then Eq. (20) can be rewritten as f = X µ exp n X j =1 µ j ξ j + N X ≤ i 1, the equation system (42)–(45) possesses many othertypes of special solutions. For instance, F = 1 + e k τ + q z + τ + e k τ + q z + τ + ( k − k ) ( k + k ) e ( k + k ) τ +( q + q ) z + τ + τ , (46) F = e k τ + q z + τ " k − k )( √ bk − k )( k + k )( √ bk + k ) e k τ + q z + τ (47)with q i = 2 abk i + ( d √ a + c √ b ) k i , i = 1 , p = √ ak .Fig.2d shows a special interaction structure between a soliton and a special D’Alembert type wave (amolecule constituted by a soliton and a solitoff) for the field u described by (15) with (41), (46) and (47)under the parameter selections a = b = c = k = 1 , d = 2 , k = 0 . , k = 0 . , τ = − τ = 5 , ξ = z = 0 (48)at time t = 0.In summary, the missing D’Alembert type waves are discovered in a special nonlinear system, the NNVequation. The similar phenomena may be found for other types of nonlinear models [24]. For the NNVequation, the n -soliton molecules are just the special type of D’Alembert waves. The interactions amongsolitons and the soliton molecules can be directly obtained from the multiple soliton solutions. It is foundthat there are other types of interactions among different types of D’Alembert waves and solitons. A specialexample, the interaction solution between a soliton-solitoff molecule and a separated soliton which can notbe obtained from the multiple soliton solutions, is given explicitly. The D’Alembert type waves (includingsoliton molecules and soliton-solitoff molecules) are firstly found in nonlinear systems and deserve furtherinvestigated. Acknowledgements The work was sponsored by the National Natural Science Foundations of China (No.11975131) and K. C.Wong Magna Fund in Ningbo University. [1] D. G. Crighton, Appl. Math. 39 (1995).[2] H. Y. Guo, Z. H. Wang and K. Wu, Phys. Lett. B 277 (1991).[3] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett. 706 (1980).[7] A. P. Veselov, S. P. Novikov, Sov. Math. Dokl. 588 (1984).[8] S. P. Novikov, A. P. Veselov, Physica D 267 (1986). [9] M. Stratmann, T. Pagel and F. Mitschke, Phys. Rev. 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