Resonant solitons from the 3×3 operator
aa r X i v : . [ n li n . S I] J un Resonant solitons from the 3 × D. J. Kaup ∗ and Robert A. Van Gorder Department of Mathematics, P.O. Box 161364, University of Central Florida, Orlando, FL 32816-1364 USA ∗ Corresponding author. Email: [email protected]
December 27, 2016
Abstract
We study and detail the features of the resonant soliton of the 3 × ζ -plane, where ζ isthe spectral parameter. When the potential matrix has anti-hermitian symmetry ( Q α,β = − Q ∗ β,α ),each of the two transmission coefficients in the lower half plane become equal to the complexconjugates of one of the two in the upper half plane, leaving only two independent transmissioncoefficients, which we take to be those in the upper half plane. The bound state scattering data forthis operator consists of the zeros of these two transmission coefficients (bound state eigenvalues)and a normalization coefficient associated with each eigenvalue. With two transmission coefficients,there is a wider variety of possible soliton solutions than in the Zakharov-Shabat (ZS) 2 × Q lying next to the diagonal, with all other components of Q vanishing.These soliton solutions will be equivalent to those in the ZS case, up to scaling factors. If we woulddistribute the eigenvalues among both transmission coefficients so that each transmission coefficientwould have at least one eigenvalue, the soliton structure becomes more complex. In this case, anyone eigenvalue will be found to generally make contributions to all non-diagonal components of Q .Of particular interest, inside this class is a special class of solitons which exist only when thesetwo transmission coefficients have exactly equal eigenvalues. Equality of eigenvalues from differenttransmission coefficients give rise to “resonant solitons”. This latter class is the case which wetreat here. We find that there are two different states for the resonant soliton solution, both ofwhich arise from a bifurcation. The bifurcation is shown to arise from the algebraic structure ofthe 3 × sl (3) Soliton Solutions, Resonant Solitons.1 Introduction
The purpose of this paper is to detail the features of the resonant soliton solutions of the nonde-generate 3 × ∂ x V − iζJ · V = − Q · V , J = J J
00 0 J , Q = Q Q Q Q Q Q (1.1)on the interval −∞ < x < + ∞ . We assume J > J > J and that Tr( J ) = 0. Q ( x ) is a potentialmatrix which vanishes like Q ( x → ±∞ ) = o (1 /x ) for large x . The matrix V ( x ) is a 3 × Q and will take the six components of Q to begenerally independent and uniquely different.The Lax pair for the 3WRI nonlinear system was first presented in 1973 [47] along with thesoliton solutions for the explosive and decay cases. Here the components of Q are the envelopes forinteracting waves. Here for the first time, it was demonstrated that in the explosive case, there wasa nonlinear instability whereby a singularity with infinite amplitude would eventually develop in thesolution. In 1975, approximately at the same time, two manuscripts were submitted [24, 48] givingmore details on this system and its solutions. The work by Kaup [24] presented a derivation of theIST for this system, giving a set of Marchenko equations and described the solutions from the pointof view of the Marchenko equations. In the work by Zakharov and Manakov [48], they constructedvarious soliton solutions and described the interactions from the point of view of these solitonsolutions and the quasi-classical approximation. Each work made use of separable initial data forconstructing the general scattering matrix. The final solution and the entire interaction could thenbe described in terms of the initial soliton and radiation (continuous spectra) content of each wave.In 1979, Kaup, Reiman and Bers [25] publish a review of the 3WRI wherein more detailed aspectsof these solutions were given along with comparisons and validations from numerical solutions. Ina related work [35], Reiman extended these results to the case where the medium could containspatial inhomogeneities and studied how they differed from the homogeneous solutions and howthey modified those results. Interestingly, Reiman showed that the inhomogeneous case of the3WRI was also an integrable system solvable by the same IST of the homogeneous 3WRI. Higherorder 3WRI soliton solutions, particularly those corresponding to multiple zeros in the transmissioncoefficients, have recently been given in Ref. [39]. A perturbation treatment of the decay instabilitywas briefly treated in [25]. More recently, a full treatment of the perturbations of (1.1) and theuniversal covering set for the squared eigenfunctions and their adjoints has been given in Ref. [30],including the closure and the orthogonality relations. The general nature of the soliton solutions ofthe n × n generalization of (1.1) has been described in [50]. These soliton solutions have also beenclassified in Ref. [20] according to the group properties of the eigenvalue problem, while a recentoverview of the n × n IST and its soliton structure is found in Ref. [31].For the simple 2 × sl (2) soliton [20]. The soliton solutions that arisein the 3 × sl (3) solitons. The general soliton solutions for three of the sl (3)symmetries have been given in [48, 24, 25].In the 3WRI, the upper triangular components of Q correspond to three different waves whichcan interact resonantly. (The lower triangular components of Q are ± the complex conjugates ofthose in the upper triangle.) There are two general types of sl (3) soliton solutions. First, depending2n the symmetry, any one of the three waves could contain one or several NLS solitons, provided allthe solitons were found in only one wave. This is a trivial example of the sl (3) solitons wherein onehas only one nonzero wave, not three. Thus no interactions occur. These sl (3) solitons are identicalto sl (2) N -soliton solutions. This is expected since sl (2) is a subgroup of sl (3). Thus there wouldbe three different representations of the sl (2) subgoup, one for each wave. The second general typeis the nontrivial case, wherein all three waves are nonzero. For these sl (3) solitons, they interactwith their energy being exchanged back and forth between the three waves in a regular fashion.However the asymptotics of these solutions show that in general, no solitons are exchanged betweenthe three waves.In the general 2 × Q are in-dependent and unrelated, there are two transmission coefficients, one in each complex half-plane,the zeros of which, along with a normalization coefficient associated each zero, determine the sl (2)soliton structure. In the case of the 3 × sl (3) soliton structure. Amongst these structures lies one thatis uniquely different from sl (2) solitons, which is called a “resonant soliton” [30, 31]. A resonantsoliton is one where the two transmission coefficients in each half-plane have a common zero whichgives a bifurcation in the general sl (3) soliton case mentioned above. It is unique among the sl (3)soliton solutions in that not only is energy exchanged between the waves, but also entire solitonscan be exchanged. This feature of the 3WRI, whereby one can transfer packets of energy from awave of one frequency to another wave of a different frequency, and/or back again, has long beensuggested as a basis for possible designs of optical logical elements.Recently this feature has generated further interest for the same purposes in a different setting.Here one utilizes a nonzero background wherein one takes a combination of bright and dark solitons.Degasperis, et al [8, 14, 13] have found exact solutions of this 3WRI when one wave is taken tobe on an asymptotically constant nonzero background, with the other two being localized pulses.Solutions exist where the three interact and move with a common velocity (simultons). Suchsolutions were found to be stable when their velocity was greater than a critical value. Interactionsof simultons with various localized pulses were found to give rise to the excitation (decay) of stable(unstable) simultons by means of the absorption (emission) of the energy carried by a localizedpulse. The speed of these solitons could be continuously varied by means of adjusting the energiesof the two bright pulses.Here we shall only treat the bright soliton case. The major purpose of this paper is to detailthose features of the scattering data which allow this soliton resonance phenomenon to occur.Secondary will be to describe how these features will display themselves in the potential matrix, Q .As is known, physical applications of the IST typically require the potential Q to have certainsymmetries. In the 2 × Q = r = ± q ∗ = ± Q ∗ where ∗ indicates the complex conjugate.If Q ( x → ±∞ ) →
0, then solitons can only occur in r = − q ∗ case [45, 1]. If one allows for anasymptotically constant nonzero background, then in the r = + q ∗ case one can have “dark solitons”[46]. If the solution is to exist only on a semi-infinite or finite interval, virtual solitons [41, 27] foreither symmetry can be found.For the 3 × Q α,β = − Q ∗ β,α , which we shall call the anti-hermitian symmetry. This latter symmetry in the sl (3)case corresponds to the soliton decay case of the 3WRI and in the sl (2) case, to r = − q ∗ .In Section 2, we will briefly outline the results from the direct and the inverse scattering problemsfor Eq. (1.1). For the case of no symmetry, we give the linear dispersion relations (LDRs) from3hich one defines the scattering data. For the anti-hermitian symmetry, the bound state scatteringdata consists of the zeros of two different transmission coefficients, with a normalization coefficientassociated with each zero. From this structure, as discussed earlier, there are two general typesof sl (3) soliton solutions. First, zeros could be in only one transmission coefficient, with the othertransmission coefficient having no zeros. In this case, one has only an sl (2) N -soliton solution whichwould only exist in one of the two conjugate pairs of components in Q adjacent to the diagonal.For the second type, the zeros would be distributed between both transmission coefficients. Thissoliton solution would make contributions to all three conjugate pairs of Q . To specify these sl (3)soliton solutions, we use the notation ( M, N )-soliton solution, where M will be the number of zerosof the first transmission coefficient and N will be the number of zeros in the second transmissioncoefficient. We also give and discuss the general sl (3) (1 , sl (3) soliton solution when an arbitrary number of resonantsolitons are present. We find that there are two distinctly different structures for a resonant soliton,due to a bifurcation in the scattering matrix. These structures are obtained for each branch of thebifurcation, for the case of compact support. Also the LDRs are extended to include resonantsolitons. We then briefly discuss how to handle the non-compact support case and how it will givethe same structures for resonant solitons. In Section 4, under anti-hermitian symmetry, we obtainthe resonant sl (3) (1 , sl (2) subgroup and delineate its scattering data. In Section 5, webriefly discuss how the resonate soliton can also be obtained by taking the limit of the two zerosapproaching each other in the general sl (3) (1 , Here we highlight the direct and inverse scattering problems for the general 3 × × In the direct scattering problem, one addresses the solutions of the eigenvalue problem, what theiranalytical properties are, the adjoint solutions and their properties, what is the scattering matrixand its properties, what are the features of the bound states, if any, and what are the fundamentalanalytical solutions and their adjoints, etc. Each of these topics shall only be defined and discussedto the extent necessary for the construction of soliton solutions. Further details will be found inthe above references, [47, 24, 48, 25, 30].The relevant matrix form of the linear eigenvalue problem associated with the 3WRI is given by(1.1). We assume no symmetry on Q and take the six components of Q to be independent (all thediagonal components are zero). We shall take the diagonal elements of J to be real and to satisfy J > J > J . Q ( x ) is a potential matrix with zero diagonal entries and vanishing as | x | → ∞ .The matrix V ( x ) is a 3 × x → ±∞ . For ζ real, there are twostandard sets which are Φ( x → −∞ ) → e iζJx , Ψ( x → + ∞ ) = e iζJx . (2.1)4ince these two matrices are linearly dependent on the other, we haveΦ = Ψ · S, Ψ = Φ · R, (2.2)where due to the Wronskian relation, det S = 1 , det R = 1 , (2.3)while the inverse relation is R = S S − S S S S − S S S S − S S S S − S S S S − S S S S − S S S S − S S S S − S S S S − S S . (2.4)We will use ± superscripts to indicate the regions of analyticity (+ for the UHP and − for theLHP) where UHP stand for the upper half complex ζ -plane and LHP for the lower half. Subscriptswill used to indicate the components of a matrix quantity. The individual Jost solutions are thecolumns in the Jost solution matrix. For this system, we haveΦ = [ φ − , φ , φ +3 ] , Ψ = [ ψ +1 , ψ , ψ − ] , (2.5)where those components without a ± superscript in general only exist on the real ζ -axis. (Strictlyspeaking, it is not the Jost solution which is analytic in ζ , uniformly for all x , but the variouscolumns in matrix products such as Φ · e − iζJx . With this understood, we shall refer to the ap-propriate Jost solutions as being analytic in ζ if this product is so analytic.) How to generallydetermine the analytical properties of the Jost solutions are detailed in the above references andtextbooks such as Ref. [38, 50].The analytical properties of the scattering matrices S follow from the above and are S = S − S S S S S S S S +33 , R = R +11 R R R R R R R R − . (2.6)In each region of analyticity, there are three linearly independent solutions of (1.1). For the realaxis, those could be either Φ or Ψ, or a suitable mixture. For each half-plane, we already have twoof these Jost functions. The third can be construct from a linear combination of the Jost functionsas detailed by Shabat [38]. Such solutions were constructed in [24]. For inversion about x = + ∞ ,one can take our third independent function to be the meromorphic functions [30] µ +2 = ψ − R R +11 ψ +1 , µ − = ψ − R R − ψ − . (2.7)We define the set of µ solutions, µ + and µ − , as µ + = [ ψ +1 , µ +2 , φ +3 ] , µ − = [ φ − , µ − , ψ − ] , (2.8)From these solutions, one may construct the “fundamental meromorphic solutions” (FMS)which will be designated by Θ. They are defined byΘ ± = µ ± · e − iζJx . (2.9)Θ + provides us with a set of three linearly independent, meromorphic solutions in the UHP andexisting on the real ζ -axis, while Θ − provides us with another similar set on the real ζ -axis and inthe LHP. 5he remaining part of the direct scattering problem is to detail the asymptotics of the Jostsolutions as one approaches any essential singularity on the boundary of the region of analyticity.This provides a means for obtaining the potentials, given the Jost solutions. There is only oneessential singularity at | ζ | = ∞ in this problem. Taking ζ to be real, then Φ and Ψ have a commonasymptotic expansion which isΦ , Ψ = ( I + i B (1) /ζ + B (2) /ζ + . . . ) · e − iζJx as | ζ | → ∞ . (2.10)In terms of Θ ± , this leads to Θ ± ( | ζ | → ∞ ) = I + i B (1) /ζ + O (1 /ζ ) , (2.11)in the appropriate half-plane. One finds that when no pairs of the components of J are equal, then B (1) can be given by [ B (1) , J ] = Q , whose solution is B (1) = X Q J − J Q J − J Q J − J X Q J − J Q J − J Q J − J X , (2.12)where the X ’s are generally integrals over quadratic products of the components of the potentialmatrix, Q . On the real axis, which is the boundary between the two regions of analyticity, it follows thatΘ + and Θ − will be related linearly. Thus one may construct what are called “linear dispersionrelations” (LDRs), whereby one obtains a set of nonhomogeneous, linear, algebro-singular integralequations relating Θ + in the UHP (and on the real axis) to Θ − in the LHP (and on the real axis),and vice versa. As shown in [30], one can reduce these six equations to a set of three linearlyindependent equations, from which one can define the scattering data and which also provides asolution for the inverse scattering problem, if it exists. (As is well known, sometimes potentialcomponents in Q have singularities which are physical, as in the explosive case of the 3WRI [47].)An alternate derivation of the LDRs has been given in Ref. [31], whereby one directly obtains aset of 3 linearly independent LDRs for the 3 × n linearly independent LDRs for the general n × n case of (1.1).Taking the results from [30], we have the four LDRs (of which only three are linearly independenton the real axis)Θ +1 ( ζ ) = − πi Z R dζ ′ ζ ′ − ζ (cid:20) S S − (cid:18) Θ − ( ζ ′ ) E ( ζ ′ ) + R R − Θ − ( ζ ′ ) E ( ζ ′ ) (cid:19) + S S − Θ − ( ζ ′ ) E ( ζ ′ ) (cid:21) (2.13) − N − X k =1 ζ − ,k − ζ ) S ,k S −′ ,k Θ − ( ζ − ,k ) E ( ζ − ,k ) , (2.14)6 +2 ( ζ ) = − πi Z R dζ ′ ζ ′ − ζ (cid:20) R R +11 Θ +1 ( ζ ′ ) E ( ζ ′ ) − R R − Θ − ( ζ ′ ) E ( ζ ′ ) (cid:21) (2.15)+ N +11 X k =1 ζ +11 ,k − ζ R ,k R + ′ ,k Θ +1 ( ζ +11 ,k ) E ( ζ +11 ,k ) + N − X k =1 ζ − ,k − ζ ) R ,k R −′ ,k Θ − ( ζ − ,k ) E ( ζ − ,k ) , (2.16)and for ℑ ( ζ ) ≤ − ( ζ ) = − πi Z R dζ ′ ζ ′ − ζ (cid:20) R R +11 Θ +1 ( ζ ′ ) E ( ζ ′ ) − R R − Θ − ( ζ ′ ) E ( ζ ′ ) (cid:21) + N +11 X k =1 ζ +11 ,k − ζ R ,k R + ′ ,k Θ +1 ( ζ +11 ,k ) E ( ζ +11 ,k ) + N − X k =1 ζ − ,k − ζ ) R ,k R −′ ,k Θ − ( ζ − ,k ) E ( ζ − ,k ) , (2.17)Θ − = + 12 πi Z R dζ ′ ζ ′ − ζ (cid:20) S S +33 Θ +1 ( ζ ′ ) E ( ζ ′ ) + S S +33 (cid:18) Θ +2 ( ζ ′ ) E ( ζ ′ ) + R R +11 Θ +1 ( ζ ′ ) E ( ζ ′ ) (cid:19)(cid:21) (2.18) − N +33 X k =1 ζ +33 ,k − ζ ) S ,k S + ′ ,k Θ +2 ( ζ +33 ,k ) E ( ζ +33 ,k ) . (2.19)As to the notation and quantities given above, R indicates that the path of the integral is along thereal axis, N +11 (cid:0) N +33 (cid:1) is the number of zeros of R +11 ( ζ ) (cid:0) S +33 ( ζ ) (cid:1) in the UHP (assumed finite), ζ +11 ,k (cid:16) ζ +33 ,k (cid:17) is the k th zero of R +11 ( ζ ) (cid:0) S +33 ( ζ ) (cid:1) in the UHP, N − (cid:0) N − (cid:1) is the number of zeros of S − ( ζ ) (cid:0) R − ( ζ ) (cid:1) in the LHP (assumed finite), ζ − ,k (cid:16) ζ − ,k (cid:17) is the k th zero of S − ( ζ ) (cid:0) R − ( ζ ) (cid:1) , R + ′ ,k (cid:16) S + ′ ,k , S −′ ,k , R −′ ,k (cid:17) is the value of dR +11 ( ζ ) /dζ (cid:0) dS +33 ( ζ ) /dζ , dS − ( ζ ) /dζ , dR − ( ζ ) /dζ (cid:1) at its k th zero, and R ,k , etc., in the case of compact support, are just the values of R , etc. at the appropriate zeros(and in the case of non-compact support, would just be some coefficients). Finally, E pq ( ζ ) = exp [ iζ ( J p − J q ) x ] . (2.20)We observe that the integrands of the integrals in (2.16) and (2.17) are equal and oppose in signwhile the discrete contributions in each are identical. From their difference, taking the limit of theimaginary part of ζ vanishing, we have the relationshipΘ +2 − Θ − = R R − Θ − E ( ζ ) − R R +11 Θ +1 E ( ζ ) , ℑ ( ζ ) = 0 , (2.21)which according to (2.7), is an identity. Whence (2.16) and (2.17), for ℑ ( ζ ) = 0, are linearlydependent. However, since the other two equations require the values of Θ ± at the various poles,we need to retain both of these in the set of LDRs.This total set of nonhomogeneous, linear, algebro-singular integral equations is a minimal setof LDRs, from which one may reconstruct the Jost solutions, given the scattering data.7 .3 The scattering data Once we have the LDRs in the above form, then it becomes possible to define the scattering datafor this problem. In order to solve these equations, we first must specify the quantities: • the reflection coefficients σ j = S j S − ( j = 2 , , σ j = S j S +33 ( j = 1 , , ρ = R R +11 and ρ = R R − on the real axis, • the zeros of R +11 ( ζ ) in the UHP ( ζ +11 ,k ; k = 1 , , ..., N +11 ) and the values of C ,k = R ,k R + ′ ,k ateach such zero, • the zeros of S +33 ( ζ ) in the UHP ( ζ +33 ,k ; k = 1 , , ..., N +33 ) and the values of C ,k = S ,k S + ′ ,k at eachsuch zero, • the zeros of R − ( ζ ) in the LHP ( ζ − ,k ; k = 1 , , ..., N − ) and the values of C ,k = R ,k R −′ ,k ateach such zero, • the zeros of S − ( ζ ) in the LHP ( ζ − ,k ; k = 1 , , ..., N − ) and the values of C ,k = S ,k S −′ ,k at eachsuch zero.Note that we have six reflection coefficients but only four sets of eigenvalues and normalizationcoefficients. Under compact support, the four normalization coefficients would be the residues offour of the reflection coefficients at the appropriate eigenvalues. Thus there are two reflectioncoefficients, σ and σ , which will not be associated with any normalization coefficients. In thelinear limit, each reflection coefficient can be matched to the Fourier transform of one of the sixcomponents of Q . With only four normalization coefficients, we may specify the position and phaseof solitons in only four of the potential components. If solitons are to occur in the remaining twocomponents, then their positions and phases cannot be given independently. Instead the positionsand phases of solitons in these components must be determined by the positions and phases of theother four solitons. It is this point that we want to further understand. We shall return to thispoint later.Under the anti-hermitian symmetry, the adjoint solutions of (1.1) are linear combinations of thehermitian conjugate of the Jost solutions, V . As a consequence of this symmetry, R α,β ( ζ ) = S ∗ β,α ( ζ ∗ )which, on the real ζ -axis, leads to σ = ρ ∗ and ρ = σ ∗ . For the bound state data, N − = N +11 , N − = N +33 , ζ − ,k = ζ + ∗ ,k , ζ − ,k = ζ + ∗ ,k , C ,k = C ∗ ,k and C ,k = C ∗ ,k . Thus each quantity inthe LHP is the Hermitian conjugate of another quantity in the UHP. This symmetry gives rise tononsingular sl (3) soliton structures which most closely match that of the familiar sl (2) solitonswhere r = − q ∗ . In the following subsection, we shall obtain the (1 , sl (3)-soliton solution. Using that, we will describe theresonant nature of this soliton solution. The solution method for the (
M, N )-soliton solution when there is no symmetry, is given in Ap-pendix A. We use those results to obtain the (1,1)-soliton solution for the anti-hermitian symmetry,provided that the two eigenvalues are distinct. To that end, take N +11 = N − = N +33 = N − = 1.Since there is only one zero of each transmission coefficient, we can simplify the notation. We elim-inate the triple subscripts on the eigenvalues, replacing them with only one subscript, replacing8 +11 , with ζ +1 , ζ − , with ζ − , etc. We also drop the last subscript on the C ’s since there is only oneeigenvalue arising from each transmission coefficient.Turning to (2.11) and (2.12), we may solve for the components of Q . First we impose thesymmetry condition Q αβ = − Q ∗ βα which translates into the symmetry R αβ ( ζ ) = S ∗ βα ( ζ ∗ ) for thescattering matrices. There are four fundamental quantities in the LDRs, on which the solutionsdepend. These are given below and due to the assumed symmetry, can be reduced to only two x -dependent complex quantities. R = C η E , +1 = e iγ e iζ +1 ( J − J )( x − x ) (2.22) S = C η E , − = R ∗ (2.23) R = C η E , − = e − iγ e − iζ − ( J − J )( x − x ) (2.24) S = C η E , +3 = R ∗ (2.25)where η is the imaginary part of ζ +1 , η is the imaginary part of ζ +3 , γ and γ are real phasesand x and x are real spatial positions. Then from the results at the end of Appendix A andequations (2.11) and (2.12), we have Q = ( J − J ) 2 iη R D (cid:18) ζ − − ζ +1 ζ +3 − ζ +1 |R | (cid:19) , (2.26) Q = ( J − J ) 4 iη η R ∗ R D (cid:0) ζ +3 − ζ +1 (cid:1) , (2.27) Q = − ( J − J ) 2 iη R ∗ D (cid:18) ζ +3 − ζ − ζ +3 − ζ +1 |R | (cid:19) , (2.28)where D = (cid:16) |R | (cid:17) (cid:16) |R | (cid:17) + 4 η η (cid:12)(cid:12) ζ +3 − ζ +1 (cid:12)(cid:12) |R | |R | , (2.29)and the other three components of Q follow from the assumed symmetry. This solution wasoriginally given in Refs. [47, 24], although in a different form.Contained in these solutions are the two simple (1 , , , S +33 without a zero and the residue R would be absent from(2.26)-(2.29). In this case only Q would be nonzero and it would have the form Q = ( J − J ) 2 iη R |R | , (2.30)which is proportional to a simple sl (2) soliton located at x = x . Similarly for the (0 , R +11 without a zero and the residue R would be absent from (2.26)-(2.29).Now only Q would be nonzero and it would be given by Q = − ( J − J ) 2 iη R ∗ |R | , (2.31)which is also proportional to a simple sl (2) soliton located at x = x .Note that there are only two positions and two phases in (2.22) - (2.25) while there are threewaves. Thus one wave can be considered to be determined or driven by the other two. If we take Q to be driven, then its phase and position is determined by Q and Q . As to the variety of9he solutions that one could obtain here, one finds both analytical and numerical solutions in theliterature (see, for instance, [8], [32], [13], [33], [12], [15]).Before leaving this solution, there are features of it that we should point out, particularly itsasymptotics. To do this, we shall take | ζ +3 − ζ +1 | to be generally nonzero and on the order of theimaginary parts or even larger, and the differences J − J and J − J to be roughly the same. Let ustranslate our x coordinate so that one of the two positions will be at zero. Then the general natureof this solution can be obtained by letting the other position vary from −∞ to + ∞ . The phasesare relatively unimportant in determining this nature so they will be ignored. Taking x = 0,then for x large, either positive or negative, Q is an NLS soliton, generally localized around x = 0. Meanwhile Q is also an NLS soliton, but generally localized around x = x , while Q is generally exponentially small. As | x | decreases (the two NLS solitons approaching each other)and approaches zero, the two NLS solitons interact and Q grows. However its growth is limitedby two main factors. First, if the real parts, or even the imaginary parts, of the eigenvalues arewidely divergent, then the maximum growth in Q is obviously limited by the | ζ +3 − ζ +1 | term in thedenominator. Second, the mathematical structure of the solution for Q is such that the amplitudeof Q will generally be bounded by about a third of that of an NLS soliton with a similar width.In terms of the 3WRI, the variation of x from −∞ to + ∞ models two NLS solitons (in differentwaves) coming together, colliding, interacting, and then during the interaction some amount of Q is produced. As the two NLS solitons pass through each other and then recede, what energy wasin Q is returned to the two NLS solitons, which have suffered no consequences for this collisionexcept for shifts in positions and phases. This regime of the (1 , R and R to be fixed and take the limit where this difference vanishes, then all components of Q vanish.However this ignores the dependence on x . As x increases, both R and R will smoothly passtoward zero. One concludes then that as the difference in the eigenvalues approach zero, for fixed C and C , the dominate change to the soliton structure will be that the entire structure willsimply shift to larger values of x . Alternately, one could achieve the same effect by allowing theamplitudes of C and C to become smaller and smaller, and at such a rate that the generalposition of the soliton structure would not shift to larger x values. However there is a problem ifwe allow C and C to vanish: we lose the soliton phase and position information. So there ismore involved here than simply a scaling.To understand this better, for the sake of argument, let us consider the case of compact supportwhere all components of the scattering matrices can be extended into the entire complex plane.Then from (2.4), we have S +33 = R +11 R − R R . (2.32)Now, R +11 ( ζ +1 ) = 0 and S +33 ( ζ +3 ) = 0. So if ζ +3 → ζ +1 , then the left hand side vanishes as well as thefirst term on the right. Thus it follows that if ζ +3 = ζ +1 , then the product R R must also vanish(whenever the potential is on compact support). Now, compact support is crucial to this argumentand pure soliton solutions are never on compact support. Nevertheless this still indicates that carewill have to be used whenever these two eigenvalues approach each other. We will resolve this inmore detail in the next section. 10 The resonant soliton case
With the above in mind, let us study this particular singular case where these two transmissioncoefficients have exactly equal zeros. As one may easily verify, when there is a zero of R +11 ( ζ ) inthe UHP, ζ +11 ,k , which exactly matches one of the zeros of S +33 ( ζ ) in the UHP, ζ +33 ,ℓ , or when thereis a zero of R − ( ζ ) in the LHP, ζ − ,k , which exactly matches one of the zeros of S − ( ζ ) in the LHP, ζ − ,ℓ , then one will find a singularity in the LDRs, (2.13) - (2.18). For example, let us take ζ +11 ,k tobe exactly equal to ζ +33 ,ℓ for some value of k and ℓ . Then to solve these LDRs, we need the value ofΘ +2 ( ζ +33 ,ℓ ) to insert into (2.19). This value will follow from (2.16). However, when we evaluate thisequation at ζ = ζ +33 ,ℓ , we see that there will be a term of the form1 ζ +11 ,k − ζ +33 ,ℓ R ,k R + ′ ,k Θ +1 ( ζ +11 ,k ) E ( ζ +11 ,k ) , which has a denominator that, if R ,k Θ +1 ( ζ +11 ,k ) is nonzero, becomes singular if ζ +11 ,k is exactlyequal to ζ +33 ,ℓ . Thus if a solution is to exist, R ,k Θ +1 ( ζ +11 ,k ) must vanish. That is essentially whathappens, which we shall now detail. For simplicity, we shall assume compact support so that R and S , and the Jost solutions Φ e − iζJx and Ψ e − iζJx are entire functions. Later on in this section,we shall briefly discuss how to handle the noncompact support case.First, we will collect the consequences of having such paired eigenvalues and introduce ournotation. Since R and S are not transmission coefficients, we have the subscript ”2” free to use.So let us designate this common zero instead by ζ +2 ,k where the range of k will be over all such zerosin the UHP. We shall assume this zero is simple in both transmission coefficients. Consider the(1 ,
1) component of (2.4) at ζ = ζ +2 ,k . Then it follows that the product S S must also have a zeroat ζ = ζ +2 ,k . In order to avoid double zeros in this product, we shall also assume that S S + ′ = R + ′ at ζ +2 ,k . Due to compact support, either S or S must contain this simple zero. If we take S to have this zero, it follows from (2.4) that R must also have this same zero. On the other hand,if we take S to have this zero, then it similarly follows that R must also have the same zero.Thus we find that we have a bifurcation with two different possible options.Similarly in the LHP, assuming no symmetry, when ζ = ζ − ,k is a simple common zero of both R − and S − in the LHP, then it follows that S S must also have the same simple common zero.Taking S to be non-zero at ζ − ,k , we find that S = 0 = R at ζ − ,k . If we take S to be non-zero,then S = 0 = R at ζ − ,k .These conditions follow directly from the matrix structure of the sl (3) group and compactsupport. All that has to happen is for a zero of R +11 ( ζ ) to match some zero of S +33 ( ζ ) in the UHP,with both zeros being simple. Then the other zeros follow, and similarly in the LHP. In the following, we shall assume no symmetry on the matrix Q , which we shall take to be oncompact support. Let us turn to the structure of the Jost solutions when R +11 and S +33 havea common zero. In the following, we shall freely make use of (2.4) and the two orthogonalityrelations (in tensor notation) R α, S ,β + R α, S ,β + R α, S ,β = δ αβ = S α, R ,β + S α, R ,β + S α, R ,β , (3.1)where δ αβ is the Kronecker delta. 11irst, from our definitions of µ ± in (2.8), we have that µ ± , R +11 µ +2 = χ +2 , R − µ − = χ − and µ ± ,are entire functions of ζ , where χ ± can be taken to be defined by the above relations. In general µ ± is a meromorphic function while χ ± is an analytic function [30]. Consider the T -matrix and itsinverse given in Ref. [30] µ + = µ − · T , µ − = µ + · T − , (3.2)where T = S − − R S − R +11 S S − − S S − R S S − R +11 − R S − R R − − S R +11 R − R − , T − = R +11 − S R +11 R − R R +11 − R S +33 R S S +33 R − − S S +33 S S +33 − R R − S +33 S +33 . (3.3)We have that the lhs of (3.2) are meromorphic functions. It then follows that the matrix producton the rhs of these equations must also be meromorphic and must match any poles and residuesfound on the lhs. Due to the poles in µ ± , we can have double poles as well as single poles in (3.2).From the first column in the second equation in (3.2), at ζ = ζ +2 ,k , we obtain (cid:2) R χ +2 (cid:3) ( ζ +2 ,k ) = 0 and (cid:20) µ +1 − S + ′ (cid:0) R ′ χ +2 + R χ + ′ (cid:1) + R + ′ S S + ′ µ +3 (cid:21) ( ζ +2 ,k ) = 0 . (3.4)From the second column and (3.4), assuming that S − does not vanish at ζ +2 ,k , we obtain the singleequation (cid:2) χ +2 + S µ +3 (cid:3) ( ζ +2 ,k ) = 0 , (3.5)and from the third column, we obtain (cid:2) S χ +2 (cid:3) ( ζ +2 ,k ) = 0 and (cid:20) R S + ′ R + ′ µ +1 − R + ′ (cid:0) S ′ χ +2 + S χ + ′ (cid:1) + µ +3 (cid:21) ( ζ +2 ,k ) = 0 . (3.6)From the above conditions, there are two different options in which these conditions may be satisfied.Thus we have a bifurcation where two different solutions are possible at each such common zero,which we now describe. For the first solution, let us take the option to have S = 0 = R at ζ +2 ,k , which we will call OptionUA. Then the first equations in (3.4) and (3.6) are trivially satisfied while the second ones, alongwith (3.5), reduce to (cid:2) R µ +1 + χ +2 (cid:3) ( ζ +2 ,k ) = 0 , (cid:2) S µ +1 − µ +3 (cid:3) ( ζ +2 ,k ) = 0 , (3.7)where µ +2 has a simple pole at ζ = ζ +2 ,k , whose residue is χ +2 ( ζ +2 ,k ) / R + ′ ,k . For this option, there isonly one independent Jost solution at ζ +2 ,k . (For the adjoint Jost functions, there are two.)12 .3 UHP - Option UB For the other solution, the remaining option is to take S = 0 = R at ζ +2 ,k . Then (3.5) and thefirst equations in (3.4) and (3.6) give that χ +2 must vanish at ζ +2 ,k . Thus χ +2 ( ζ +2 ,k ) = 0 , (3.8)while the second equations in (3.4) and (3.6) reduce to only one condition R ,k R + ′ ,k µ +1 ( ζ +2 ,k ) − S ,k S + ′ ,k µ +2 ( ζ +2 ,k ) + 1 S + ′ ,k µ +3 ( ζ +2 ,k ) = 0 , (3.9)where by l’Hopital’s rule we have made use of µ +2 ( ζ +2 ,k ) = χ + ′ ( ζ +2 ,k ) / R + ′ ,k . For this option, we havetwo linearly independent Jost solutions at ζ +2 ,k (while the adjoint only has one). Now let us turn to the LHP and the structure of the Jost solutions when S − and R − have acommon zero. Continuing as before but with the first equation in (3.2) instead, from the firstcolumn we obtain (cid:2) S χ − (cid:3) ( ζ − ,k ) = 0 , (cid:20) µ − − R −′ (cid:0) S ′ χ − + S χ −′ (cid:1) + S −′ R R −′ µ − (cid:21) ( ζ − ,k ) = 0 , (3.10)from the second column and (3.10), assuming that R +11 does not vanish at ζ − ,k , we obtain (cid:2) χ − + R µ − (cid:3) ( ζ − ,k ) = 0 , (3.11)and from the third column, we obtain (cid:2) R χ − (cid:3) ( ζ − ,k ) = 0 , (cid:20) S R −′ S −′ µ − − S −′ (cid:0) R ′ χ − + R χ −′ (cid:1) + µ − (cid:21) ( ζ − ,k ) = 0 . (3.12) Continuing as before, under the option S ,k = 0 = R ,k at ζ − ,k , we have that χ − ( ζ − ,k ) must vanish.From l’Hopital’s rule it follows that µ − ( ζ − ,k ) = χ −′ ( ζ − ,k ) / R −′ ,k giving µ − to be analytic at ζ − ,k .The other equations give only the one condition1 S −′ ,k µ − ( ζ − ,k ) − S ,k S −′ ,k µ − ( ζ − ,k ) + R ,k R −′ ,k µ − ( ζ − ,k ) = 0 . (3.13)and there are two linearly independent Jost solutions for this option in the LHP (with the adjointhaving only one such solution). Under the option S ,k = 0 = R ,k at ζ − ,k , we find that there is only one independent solution forthis option. The other two Jost solutions are given by χ − ( ζ − ,k ) + R ,k µ − ( ζ − ,k ) = 0 , µ − ( ζ − ,k ) − S ,k µ − ( ζ − ,k ) = 0 , (3.14)Here we have again that µ − has a simple pole at ζ − ,k , whose residue is χ − ( ζ − ,k ) / R −′ ,k .13 .7 LDRs with Resonant Solitons Given the above, one may extend the LDRs given in Ref. [30] to include resonant solitons. To dothis, one would start with Eqs. (3.08)-(3.11) from that reference, which are still correct, even inthe case of common zeros between the transmission coefficients. Starting from there, in order toreduce those Eqs. (3.08)-(3.11) to a minimal set of linear independent relations, it is necessary toremove all bound state Φ-type Jost solutions in favor of the Ψ-type Jost solutions. As was donefor non-common zeros in Ref. [30], one can extend those equations by using the above optionsto eliminate the bound state Φ-type Jost solutions. The only change to the previous form of theLDRs, will be additional sums over any common zeros. (As we shall point out later, by treating thecommon zero case as a limit of different zeros approaching each other in Eqs. (3.15)-(3.19) of Ref.[30], and by including the four possible options, one can obtain the same results as given below.)A priori, in the common zero case and without any symmetry on Q , one must allow all fouroptions to coexist. To express this, we need to extend the notation in Ref. [30]. We shall use ζ ± a,k for the common zeros for option UA and LA with k = 1 , , . . . , N ± a where N ± a is the numberof these common zeros. Similarly ζ ± b,k will designate the common zeros of options UB and LBwhere k = 1 , , . . . , N ± b with N ± b being the number of these common zeros. Meanwhile the currentnotation used in (2.14)-(2.19) will continue to be used for those zeros which are not common withany in the other transmission coefficient.Using the above options, it is straight forward to give these additional sums. One obtains for ℑ ( ζ ) ≥ +1 ( ζ ) = − πi Z R dζ ′ ζ ′ − ζ (cid:20) S S − (cid:18) Θ − ( ζ ′ ) E ( ζ ′ ) + R R − Θ − ( ζ ′ ) E ( ζ ′ ) (cid:19) + S S − Θ − ( ζ ′ ) E ( ζ ′ ) (cid:21) − N − X k =1 ζ − ,k − ζ ) S ,k S −′ ,k Θ − ( ζ − ,k ) E ( ζ − ,k ) − N − a X k =1 ζ − a,k − ζ ) " S ,k S −′ ,k Θ − ( ζ − a,k ) E ( ζ − a,k ) − R ,k R −′ ,k Θ − ( ζ − a,k ) E ( ζ − a,k ) − N − b X k =1 ζ − b,k − ζ ) S ,k S −′ ,k Θ − ( ζ − b,k ) E ( ζ − b,k ) , (3.15)Θ +2 ( ζ ) = − πi Z R dζ ′ ζ ′ − ζ (cid:20) R R +11 Θ +1 E ( ζ ′ ) − R R − Θ − E ( ζ ′ ) (cid:21) + N +11 X k =1 ζ +11 ,k − ζ R ,k R + ′ ,k Θ +1 ( ζ +11 ,k ) E ( ζ +11 ,k ) + N − X k =1 ζ − ,k − ζ ) R ,k R −′ ,k Θ − ( ζ − ,k ) E ( ζ − ,k )+ N +2 a X k =1 ζ +2 a,k − ζ R ,k R + ′ ,k Θ +1 ( ζ +2 a,k ) E ( ζ +2 a,k ) + N − b X k =1 ζ − b,k − ζ ) R ,k R −′ ,k Θ − ( ζ − ,k ) E ( ζ − b,k ) , (3.16)14nd for ℑ ( ζ ) ≤ − ( ζ ) = − πi Z R dζ ′ ζ ′ − ζ (cid:20) R R +11 Θ +1 E ( ζ ′ ) − R R − Θ − E ( ζ ′ ) (cid:21) + N +11 X k =1 ζ +11 ,k − ζ R ,k R + ′ ,k Θ +1 ( ζ +11 ,k ) E ( ζ +11 ,k ) + N − X k =1 ζ − ,k − ζ ) R ,k R −′ ,k Θ − ( ζ − ,k ) E ( ζ − ,k )+ N +2 a X k =1 ζ +2 a,k − ζ R ,k R + ′ ,k Θ +1 ( ζ +2 a,k ) E ( ζ +2 a,k ) + N − b X k =1 ζ − b,k − ζ ) R ,k R −′ ,k Θ − ( ζ − ,k ) E ( ζ − b,k ) , (3.17)Θ − = + 12 πi Z R dζ ′ ζ ′ − ζ (cid:20) S S +33 Θ +1 ( ζ ′ ) E ( ζ ′ ) + S S +33 (cid:18) Θ +2 ( ζ ′ ) E ( ζ ′ ) + R R +11 Θ +1 ( ζ ′ ) E ( ζ ′ ) (cid:19)(cid:21) − N +33 X k =1 ζ +33 ,k − ζ ) S ,k S + ′ ,k Θ +2 ( ζ +33 ,k ) E ( ζ +33 ,k ) − N +2 a X k =1 ζ +2 a,k − ζ ) S ,k S + ′ ,k Θ +1 ( ζ +2 a,k ) E ( ζ +2 a,k )+ N +2 b X k =1 ζ +2 b,k − ζ ) " R ,k R + ′ ,k Θ +1 ( ζ +2 b,k ) E ( ζ +2 b,k ) − S ,k S + ′ ,k Θ +2 ( ζ +2 b,k ) E ( ζ +2 b,k ) . (3.18)These equations will be referred to as the “LDRs with resonances” or “extended LDRs” while thoseof [30] will be referred to as the “regular LDRs” or just plain “LDRs”. The same relations as above could also be obtained in the case of non-compact support. We shallbriefly outline here the approach that one would use. Starting from the relation (2.31) in Ref. [30]and the definition of the χ states (the Fundamental Analytical Solutions, or FAS) in (2.15) - (2.19)of the same reference, one can obtain χ + · χ A + = Diagonal (cid:2) R +11 , R +11 S +33 , S +33 (cid:3) where χ A ± is theadjoint FAS. (A similar expression also exists in the LHP.) Now at a non-common zero of either R +11 or S +33 in the UHP, one has that there will be two linearly independent components of χ + ,as described in [30]. However, if we have a common zero of these two transmission coefficients,then we have that χ + · χ A + becomes a zero matrix, as well as the middle diagonal componentbecoming a double zero. From these conditions, one can deduce that at a common zero, χ + musthave either only one or only two linearly independent columns. (At the same time, the adjointsolutions, χ A ± , correspondingly must have only two or only one linearly independent solutions.)These two conditions can then be seen to give rise to the above two options. The only differencewould be that the coefficients in (3.7) - (3.9) will become arbitrary coefficients, generally unrelatedto the components of R and S (since the off-diagonal components of these matrices do not generallyexist off the real axis, in the non-compact support case). Once these options are established basedon the required linear dependences and order of the zeros, then the same LDRs would result.15 The Resonant (1 , -Soliton Solution from the Extended LDRs Under the anti-hermitian symmetry, one can obtain a (1 , R and S . First, take the scattering data to be in the form of Option A.Here we have N +2 a = 1 = N − a and whence k = 1 only. Thus the sum can be omitted. Furthermorethe k subscript can also be deleted since it is no longer needed. Then Eqs. (3.15)-(3.18) become,for ℑ ( ζ ) ≥ +1 ( ζ ) = − ζ − a − ζ ) (cid:2) C Θ − ( ζ − a ) E ( ζ − a ) − C Θ − ( ζ − a ) E ( ζ − a ) (cid:3) , (4.1)Θ +2 ( ζ ) = + C ζ +2 a − ζ Θ +1 ( ζ +2 a ) E ( ζ +2 a ) , (4.2)and for ℑ ( ζ ) ≤
0, Θ − ( ζ ) = + C ζ +2 a − ζ Θ +1 ( ζ +2 a ) E ( ζ +2 a ) , (4.3)Θ − = − C ζ +2 a − ζ Θ +1 ( ζ +2 a ) E ( ζ +2 a ) . (4.4)Let us define R = C η E ( ζ +2 a ) = e iγ e iζ +2 a ( J − J )( x − x ) , (4.5) S = C η E ( ζ − a ) = R ∗ , (4.6) R = C η E ( ζ − a ) = e − iγ e − iζ − a ( J − J )( x − x ) , (4.7) S = C η E ( ζ +2 a ) = R ∗ , (4.8)From the above, we find the solution of (4.1) - (4.4) to beΘ +1 ( ζ ) = + 2 η ( ζ − a − ζ ) D a i ( D − −S R , (4.9)Θ ± ( ζ ) = + 2 η R ( ζ +2 a − ζ ) D a − i S i R , (4.10)Θ − ( ζ ) = + 2 η S ( ζ +2 a − ζ ) D a − i S − i R . (4.11)where η is the imaginary part of ζ +2 a and D a = 1 + |R | + |R | . (4.12)16rom (2.12) we then obtain the components of the potential matrix Q = i ( J − J )2 η R D a , (4.13) Q = − i ( J − J )2 η S D a , (4.14) Q = − i ( J − J )2 η S S D a , (4.15)with the other three components following from the symmetry relation on the potential matrix.Again, we have only two positions and phases to freely specify. The position and phase of Q canbe considered to be driven by the other two.In the case of Option B we get essentially the same structure but with some indices interchanged.Taking R = C η E ( ζ +2 b ) = e iγ e iζ +2 b ( J − J )( x − x ) , (4.16) S = C η E ( ζ − b ) = R ∗ , (4.17) R = C η E ( ζ − b ) = e − iγ e − iζ − b ( J − J )( x − x ) , (4.18) S = C η E ( ζ +2 b ) = R ∗ , (4.19)one obtains for the potential components Q = i ( J − J )2 η R R D b , (4.20) Q = i ( J − J )2 η R D b , (4.21) Q = − i ( J − J )2 η S D b , (4.22)where D b = 1 + |R | + |R | . (4.23)Again we have one component being driven by the other two. In this case, it is Q which can beconsidered to be driven.The asymptotics of these equations are much easier to handle than those of (2.22) - (2.29).In Option A, for x << x , we have two NLS solitons present, with the one in Q far to theleft and the one in Q far to the right, with Q vanishingly small. As we reverse this and take x << x , then we find that the original two NLS solitons have vanished and a single NLS solitonhas appeared in Q .For Option B, if we take x << x , then we have two NLS solitons present, with the one in Q far to the left and the one in Q far to the right. As we reverse this and take x << x , thenwe find that the original two NLS solitons have vanished and a single NLS soliton has appearedin Q . From these asymptotics, one sees clearly that these two options are distinctly differentsolutions since the spatial orientation of Q and Q become reversed.There is one case where the solutions of these two options do overlap. This is when R = 0 = S , giving that Q = 0 = Q with only Q remaining nonzero, for each option. This case is also17he third case wherein a single sl (2) soliton can occur in this system. As was mentioned earlier,the (0 , , Q and the other with one in Q . Each one of these NLS solitons only require one zero in one transmission coefficient. Here wesee that Q can also contain a single NLS soliton, but to obtain this sl (2) soliton, one has to insistthat there be a common zero in two different transmission coefficients, a more difficult objective toachieve practically.From the above, one can also argue the instability of the resonant soliton. The NLS solitonsin the (0 , , Q NLS soliton is predicated on more than one coefficient being exactly zero.Let either one of these coefficients become shifted, then one has the case where the two commoneigenvalues are no longer exactly equal. As discussed in Section 2.4 and as seen from (2.22) - (2.29),the asymptotics will be distinctly different from those found above for the resonant soliton.
Here we will briefly discuss how the resonate soliton can be obtained as a limit of a zero of R +11 ( ζ )and a zero of S +33 ( ζ ) approaching each another. As we have already seen, one has two choices asto how this can be done. Once that is realized, then one must choose one of two options andit becomes straight forward to proceed in that manner and obtain the extended LDRs from theregular LDRs. We will leave that as an exercise to the reader.The same thing can be done with the general (1,1)-soliton solution, (2.26) - (2.29). Looking at itsstructure, one sees if one chooses C to be proportional to ( ζ +1 − ζ +3 ) in (2.22) and also renormalizes x to take this into account, then as the eigenvalues approach each other, the solution degeneratesinto the Option A case. Similarly by taking C to be proportional to ( ζ +1 − ζ +3 ) in (2.25), andrenormalizing x , then one obtains the solution for Option B. There were two reasons for this study of the soliton solutions of the 3 × n × n eigenvalue problem. As has been shown here, the resonant soliton can be obtained from either theLDRs or the ( M, N )-soliton solution, by taking the limit of two eigenvalues in different transmissioncoefficients approaching each other. The general (
M, N )-soliton solution for the 3 × n × n case, upon being given the LRDs for the n × n case. (For the case of the n × n operator, one would include more algebraic equations of the form given in (A.3)-(A.4), whichcould then be solved in the same manner outlined in Appendix A.) Of course, as n increases [49],the number of transmission coefficients will also increase, resulting in even more complex forms ofresonant soliton solutions, as well as increased options for the number of independent Jost functionsat any given resonant eigenvalue.The simpler soliton solutions of the 3 × sl (3) solitons and generallywill express themselves as multiple sl (2) solitons in the appropriate components of the potential18atrix Q . However whenever any of these solitons are near to one in the other wave, the cornercomponents of Q could also be excited.There are several symmetries which could be applied to the 3 × sl (3) soliton solutions by ( M, N )where M and N are the number of zeros in the two transmission coefficients in the UHP. Thesimple case of a (1 , , Q . Moving up to the next case of a (1 , , , , sl (3) (1 , sl (3) soliton is found to have a presence in all off-diagonal components of Q . As thepositions of the two NLS solitons come closer together, the amplitude of the sl (3) soliton in the Q wave increases while the components in the other waves Q and Q correspondingly decrease.However the component of the soliton in the Q wave never fully forms into a NLS soliton. As thedifference between the eigenvalues becomes smaller and smaller, the Q component comes closerand closer to forming a complete NLS soliton. This is the regime of a near nonlinear resonancewherein the soliton in Q never does fully form.When the eigenvalues become exactly equal, one has the fully nonlinear resonant case. Herethe (1 , Q and Q with essentially no component of the sl (3) soliton in Q , to the NLS solitons in Q and Q asymptotically being completely absorbedinto a fully formed NLS soliton in Q . The asymptotic limit of this latter solution provides uswith the third sl (2) soliton. This asymptotic limit for (4.12)-(4.15) follows upon taking C = 0and the same can be obtained from (4.20)-(4.23) upon taking C = 0.As shown earlier, there are two different structures for resonant solitons, depending on whetherthere is one (Option A) or two (Option B) independent Jost solutions at the eigenvalue. Thegeneral solution for Option A of the resonant (1 , C and C both vanish. When that happens, the solution degeneratesinto the form of a single NLS soliton in Q (and Q ) with all other components of Q zero. Herewe have a single NLS 1-soliton solution resulting from two zeros, each in a different transmissioncoefficient. As mentioned earlier, in this resonant case, viewed from the evolution of the 3WRI, thesolution for one option is just the time-reversed solution of the other.The reason this resonance is a “nonlinear” resonance is that it requires an exact equality betweentwo eigenvalues, each of which is a zero of a different transmission coefficient, and each of whichis independent of the other. From the direct scattering problem, the positions of these zerosare nonlinear functions of the potential matrix Q . From the inverse scattering problem, it is theeigenvalue which determines the spatial shape of the amplitude and phase of a soliton solution. (Thenormalization coefficient determines only the position and overall phase of the soliton solution.)When these two eigenvalues are equal, then the shape of the solitons in Q and Q are such that,19n the language of the 3WRI, as these two waves collide, they will coherently and constructivelyinterfere with each other, transferring all their energy into the third wave, Q . Recall that forplane waves in the 3WRI, the resonance conditions are that the sums of two of the wave-vectorsand the sums of two of the frequencies must equal those of the third. However for plane waves, onenever achieves a complete conversion since the inverse processes will limit and prevent completeconversion. Thus the resonant soliton is a solution to the 3WRI problem of how to shape and phasethe envelopes of two colliding beams, Q and Q , such that one could achieve a total conversioninto the third wave, Q . The condition for this nonlinear resonance and complete conversion isthe equality of the eigenvalues. Similarly, any fully resonant (every eigenvalue for one wave beingpaired with another in the other wave) ( N, N )-soliton solution would also do the same.
A Appendix A: Derivation of the general soliton solution; non-common zeros
We obtain the (
M, N )-soliton solution in the absence of any symmetry. We assume no commonzeros between the M zeros and the N zeros and that each M zero and each N zero is simple. Onecan substitute the expressions for the Θ ’s from (2.15) and (2.17) into the remaining expressions(2.13) and (2.18) to getΘ +1 ( ζ ) = − N − X k =1 ζ − ,k − ζ ) S ,k S −′ ,k E ( ζ − ,k ) − N − X k =1 ζ − ,k − ζ ) S ,k S −′ ,k N +11 X ℓ =1 ζ +11 ,ℓ − ζ − ,k R ,ℓ R + ′ ,ℓ Θ +1 ( ζ +11 ,ℓ ) E ( ζ +11 ,ℓ ) E ( ζ − ,k ) − N − X k =1 ζ − ,k − ζ ) S ,k S −′ ,k N − X ℓ =1 ζ − ,ℓ − ζ − ,k ) R ,ℓ R −′ ,ℓ Θ − ( ζ − ,ℓ ) E ( ζ − ,ℓ ) E ( ζ − ,k ) , (A.1)Θ − = − N +33 X k =1 ζ +33 ,k − ζ ) S ,k S + ′ ,k E ( ζ +33 ,k ) − N +33 X k =1 ζ +33 ,k − ζ ) S ,k S + ′ ,k N +11 X ℓ =1 ζ +11 ,ℓ − ζ +33 ,k R ,ℓ R + ′ ,ℓ Θ +1 ( ζ +11 ,ℓ ) E ( ζ +11 ,ℓ ) E ( ζ +33 ,k ) − N +33 X k =1 ζ +33 ,k − ζ ) S ,k S + ′ ,k N − X ℓ =1 ζ − ,ℓ − ζ +33 ,k ) R ,ℓ R −′ ,ℓ Θ − ( ζ − ,ℓ ) E ( ζ − ,ℓ ) E ( ζ +33 ,k ) , (A.2)or, in a more compact notation,Θ +1 ( ζ ) = −F ( ζ )0 − N +11 X ℓ =1 F ,ℓ ( ζ )Θ +1 ( ζ +11 ,ℓ ) − N − X ℓ =1 F ,ℓ ( ζ )Θ − ( ζ − ,ℓ ) , (A.3)Θ − ( ζ ) = −G ( ζ )1 − N +11 X ℓ =1 G ,ℓ ( ζ )Θ +1 ( ζ +11 ,ℓ ) − N − X ℓ =1 G ,ℓ ( ζ )Θ − ( ζ − ,ℓ ) , (A.4)20here the F ’s and G ’s are defined in the obvious manner. Now, using these expressions andevaluating Θ +1 ( ζ ) at ζ = ζ +11 ,k ,Θ +1 ( ζ +11 ,k ) = −F ( ζ +11 ,k )0 − N +11 X ℓ =1 F ,ℓ ( ζ +11 ,k )Θ +1 ( ζ +11 ,ℓ ) − N − X ℓ =1 F ,ℓ ( ζ +11 ,k )Θ − ( ζ − ,ℓ ) , (A.5)for k = 1 , , . . . , N +11 , and evaluating Θ − ( ζ ) at ζ = ζ − ,k ,Θ − ( ζ − ,k ) = −G ( ζ − ,k )1 − N +11 X ℓ =1 G ,ℓ ( ζ − ,k )Θ +1 ( ζ +11 ,ℓ ) − N − X ℓ =1 G ,ℓ ( ζ − ,k )Θ − ( ζ − ,ℓ ) , (A.6) k = 1 , , . . . , N − , which together are a system of N +11 + N − vector equations for the Θ +1 ( ζ +11 ,k )’sand Θ − ( ζ − ,k )’s. Solving this system, and plugging them back into the original expressions (A.3)and (A.4), we recover the closed form solutions for Θ +1 ( ζ ) and Θ − ( ζ ), as well as Θ ± ( ζ ). Explicitly,we may write the system in matrix form, to wit: M ϑ = σ , (A.7)where ϑ T = h Θ +1 ( ζ +1 ) , . . . , Θ +1 ( ζ +11 ,N +11 ) , Θ − ( ζ − , ) , . . . , Θ − ( ζ − ,N − ) i , (A.8) σ T = h σ +1 , . . . , σ + N +11 , σ − , . . . σ − N − , i , (A.9) σ + k = −F ( ζ +11 ,k )0 , for k = 1 , , . . . , N +11 , σ − k = −G ( ζ − ,k )1 , for k = 1 , , . . . , N − , (A.10) M = F , ( ζ +1 ) + 1 · · · F ,N +11 ( ζ +1 ) F , ( ζ +1 ) · · · F ,N − ( ζ +1 ) F , ( ζ +11 , ) · · · F ,N +11 ( ζ +11 , ) F , ( ζ +11 , ) · · · F ,N − ( ζ +11 , )... · · · ... ... · · · ... F , ( ζ +11 ,N +11 ) · · · F ,N +11 ( ζ +11 ,N +11 ) + 1 F , ( ζ +11 ,N +11 ) · · · F ,N − ( ζ +11 ,N +11 ) G , ( ζ − , ) · · · G ,N +11 ( ζ − , ) G , ( ζ − , ) + 1 · · · G ,N − ( ζ − , ) G , ( ζ − , ) · · · G ,N +11 ( ζ − , ) G , ( ζ − , ) · · · G ,N − ( ζ − , )... · · · ... ... · · · ... G , ( ζ − ,N − ) · · · G ,N +11 ( ζ − ,N − ) G , ( ζ − ,N − ) · · · G ,N − ( ζ − ,N − ) + 1 , (A.11)where the +1’s occur with the diagonal terms only. Then, (assuming M is invertible) the generalsolutions for Θ +1 ( ζ +11 ,k ) and Θ +3 ( ζ − ,k ) are (by Cramer’s rule)Θ +1 ( ζ +11 ,k ) = det( M k )det( M ) , Θ +3 ( ζ − ,k ) = det( M N +11 + k )det( M ) , (A.12)where M k is the matrix formed by replacing the k th column of M with σ and M N +11 + k is thematrix formed by replacing the ( N +11 + k )th column of M with σ . Arranging all terms properly, weobtain Θ +1 ( ζ ) = −F ( ζ )0 − N +11 X ℓ =1 F ,ℓ ( ζ ) det( M ℓ )det( M ) − N − X ℓ =1 F ,ℓ ( ζ ) det( M N +11 + ℓ )det( M ) , (A.13)21 +2 ( ζ ) = + N +11 X k =1 C ,k ζ +11 ,k − ζ det( M k )det( M ) E ( ζ +11 ,k ) + N − X k =1 C ,k ( ζ − ,k − ζ ) det( M N +11 + k )det( M ) E ( ζ − ,k ) , (A.14)Θ − ( ζ ) = Θ +2 ( ζ ) , (A.15)Θ − ( ζ ) = −G ( ζ )1 − N +11 X ℓ =1 G ,ℓ ( ζ ) det( M ℓ )det( M ) − N − X ℓ =1 G ,ℓ ( ζ ) det( M N +11 + ℓ )det( M ) . (A.16)Under this framework, these results scale up for the soliton solutions of the general n × n operator[31]. References [1] Ablowitz M.J., Kaup D.J., Newell A.C. and Segur H. 1974 The inverse scattering transform - Fourieranalysis for nonlinear problems, Studies in Appl. Math.
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