Integrability and rational soliton solutions for gauge invariant derivative nonlinear Schrödinger equations
IIntegrability and rational soliton solutions for gaugeinvariant derivative nonlinear Schr¨odinger equations
Paz Albares Departamento de F´ısica Fundamental, Universidad de Salamanca, [email protected]
Abstract
The present work addresses the study and characterization of the integrabilityof three famous nonlinear Schr¨odinger equations with derivative-type nonlin-earities in 1 + 1 dimensions. Lax pairs for these three equations are success-fully obtained by means of a Miura transformation and the singular manifoldmethod. After implementing the associated binary Darboux transformations,we are able to construct rational soliton-like solutions for those systems.
MSC class : 35C08, 35Q55, 37J35
Keywords : integrability, derivative nonlinear Schr¨odinger equation, Lax pair, ratio-nal solitons.
1. Introduction
The nonlinear Schr¨odinger (NLS) equation is one of the most famous integrable equa-tions in soliton theory and mathematical physics [1]. Among the several integrablegeneralizations of NLS, we are interested in the study of modified NLS systems withderivative-type nonlinearities in 1 + 1 dimensions, which are known as derivative non-linear Schr¨odinger (DNLS) equations. There exist three celebrated equations of thiskind, i. e. the Kaup-Newell (KN) system [2], im t − m xx − i (cid:0) | m | m (cid:1) x = 0 (1)the Chen-Lee-Liu (CLL) equation [3], im t − m xx − i | m | m x = 0 (2)and the Gerdjikov-Ivanov (GI) equation [4] im t − m xx + im m x − | m | m = 0 (3) Based on the contribution presented at the “Third BYMAT Conference: Bringing Young Math-ematicians Together”, December 1–3, 2020, Valencia, Spain. To appear in the Proceedings of theThird BYMAT Conference. a r X i v : . [ n li n . S I] F e b here m is a complex valued function and m denotes the complex conjugate of m .It is already known that these three equations are equivalent via a U (1)-gauge trans-formation [5]. If m ( x, t ) is a solution of the KN system (1), it is easy to find that thenew field M ( x, t ) M ( x, t ) = m ( x, t ) e iγ θ ( x,t ) , with θ x = | m | , θ t = i ( mm x − mm x ) + 32 | m | (4)satisfies the CLL equation for γ = 1, and the GI equation for γ = 2.Gauge transformations constitute an useful tool to link integrable evolution equationsin soliton theory, since they provide B¨acklund transformations between those equa-tions as well as the relation of their associated linear problems [6]. In this contributionwe exploit this gauge invariance property to construct a Lax pair and rational solitonsolutions for these three equations. For a detailed analysis and explicit calculations,we refer the reader to [7].
2. Integrability and Lax pair
The Painlev´e test [8] has been proved to be a powerful criterion for the identification ofintegrable partial differential equations (PDEs). A PDE is said to posses the Painlev´eproperty, frequently considered as a proof of integrability, when its solutions aresingled-valued about the movable singularity manifolds. This requires the generalizedLaurent expansion for the field m ( x, t ) = (cid:80) ∞ j =0 a j ( x, t ) φ ( x, t ) j − µ , where φ ( x, t ) is anarbitrary function called the singular manifold and the index µ ∈ N is an integer.The Painlev´e test is unable to check the integrability of any DNLS equation sincethe leading index is not integer, µ = . This fact allow us to introduce two new realfields α ( x, t ) , β ( x, t ) m ( x, t ) = √ α x e i β ( x,t ) , with α x = 12 | m | , β = (2 γ − α + (cid:90) α t α x dx (5)with γ = 0 for the KN system, γ = 1 for the CLL equation and γ = 2 for the GIequation. This ansatz yields an identical differential equation for α in each case,expressed in the conservative form (cid:2) α x − α t (cid:3) t = (cid:20) α xxx + α x − α t + α xx α x (cid:21) x (6)From expression (4), it can be easily seen that the probability density θ x = | m | = | M | is invariant under a U (1)-gauge transformation, indeed it constitutes the firstconservation law for these systems. Due to this symmetry, it is straightforward tosee that once we obtain a soliton solution for a particular DNLS equation, it isimmediate to derive soliton solutions for any DNLS equation linked by a U (1)-gaugetransformation. 2ince α x = θ x , we may conclude that equation (6) is the representative equationfor the probability density of any DNLS equation. Equation (6) passes the Painlev´etest, but it possesses two branches of expansion. The best method to overcome thisinconvenience requires the splitting of the field α as α = i ( u − u ) , α x − α t = u xx + u xx (7)The combination of equations in (7) yields two Miura transformations for { u, u } andthe coupling condition u xx = 12 (cid:0) α x − α t − iα xx (cid:1) , u xx = 12 (cid:0) α x − α t + iα xx (cid:1) ,iu t + u xx − iu t + u xx + ( u x − u x ) = 0 (8)which finally lead to the nonlocal Boussinesq equation [9] for u ( x, t ) of the form (cid:20) u tt + u xxxx + 2 u xx − u xt + u xxx u xx (cid:21) x = 0 (9)where it may be easily checked that u ( x, t ) satisfies the same equation. Equation(9) has the Painlev´e property with an unique branch of expansion. Hence, thisequation turns out to be integrable in the Painlev´e sense and it is possible to derivean equivalent linear spectral problem associated to the nonlinear equation (9). Thisaim may be achieved by means of the so-called singular manifold method (SMM).The SMM [8] focuses on solutions which emerge from the truncated Painlev´e series,and act as auto-B¨acklund transformations, of the form u [1] = u [0] + log( φ ). Thus, thesingular manifold φ is no longer an arbitrary function, since it satisfies the singularmanifold equations. The associated linear problem arises from the linearization ofthese equations, and it can be demonstrated that the Lax pair for u reads [7] ψ xx = (cid:32) u [0] xxx − iu [0] xt u [0] xx − iλ (cid:33) ψ x − u [0] xx ψ, ψ t = iψ xx − λψ x + i (cid:0) u [0] xx + λ (cid:1) ψχ xx = (cid:32) u [0] xxx + iu [0] xt u [0] xx + iλ (cid:33) χ x − u [0] xx χ, χ t = − iχ xx − λχ x − i (cid:0) u [0] xx + λ (cid:1) χ (10)where { χ, ψ } are two complex conjugated eigenfunctions satisfying ψ x χ x ψχ + u [0] xx = 0and λ is the spectral parameter. From (10), we may compute the Lax pair for theDNLS equations, obtaining χ xx = (cid:34) iλ − i ( γ − (cid:12)(cid:12) m [0] (cid:12)(cid:12) + m [0] x m [0] (cid:35) χ x + 12 (cid:20) im [0] m [0] x − γ − (cid:12)(cid:12) m [0] (cid:12)(cid:12) (cid:21) χχ t = iχ xx − (cid:34) ( γ − (cid:12)(cid:12) m [0] (cid:12)(cid:12) + 2 im [0] x m [0] (cid:35) χ x − iλ χ (11)and its complex conjugate, for the corresponding value of γ in each case. It isworthwhile to remark that the coupling condition for the Lax pair in u gives riseto ψ x χ x ψχ − i m [0] m [0] x + γ − (cid:12)(cid:12) m [0] (cid:12)(cid:12) = 0, which allows us to determine an additional butcompletely equivalent Lax pair for those systems.3 . Rational soliton solutions Once the Lax pair have been obtained for a given PDE by means of the SMM, binaryDarboux transformations can be constructed in order to obtain iterated solutions forthat PDE. We implement the Darboux transformation formalism over the spectralproblem (10) so as to provide a general iterative procedure to compute up to the n th iteration for u . By virtue of expressions (4), (5) and (7), solutions for the DNLSequations can be forthrightly established. Thus, soliton solutions for DNLS equa-tions may be derived by considering a suitable choice for the seed solution and theeigenfunctions in the Lax pair.In the following lines we summarize the main results regarding this procedure, ori-ented to the obtention of rational soliton solutions. Further details and a generalrigorous analysis may be found in [7].We start from a polynomial seed solution u [0] for (9) and binary exponential eigen-functions for (10), u [0] = − j (cid:20) j z x (cid:16) x j ( z + 1) t (cid:17) + i (cid:18) x + j (cid:18) z + 12 (cid:19) t (cid:19)(cid:21) ,χ σ = e i j z σ (cid:104) x + j (cid:16) − σ z ( z +7 z +1)+3( z +1) (cid:17) t (cid:105) , ψ σ = χ σ (12)where j and z are arbitrary parameters, σ = ± λ σ = j (2 σz − ( z + 1)). Thefirst and second iterations u [ j ] , j = 1 , (cid:12)(cid:12) m [ j ] (cid:12)(cid:12) = 2 i ( u [ j ] x − u [ j ] x ). The results are displayed inFigure 1.The first iteration ( j = 1) provides a rational soliton-like travelling wave along the x − j ( σz − ( z + 1)) t direction and constant amplitude, of expression (cid:12)(cid:12) m [1] σ (cid:12)(cid:12) = j − j z ( σ − z ) (cid:104) ( x − j ( σz − ( z + 1)) t ) + j z ( σ − z ) (cid:105) , σ = ± j = 2), we get the two-soliton solution (cid:12)(cid:12) m [2] (cid:12)(cid:12) = j + 8 (cid:104) ( x + j ( z + 2) t ) + j ( z − t + j ( z − (cid:105) j ( z − (cid:20)(cid:16) ( x + j ( z + 1) t ) − j z t − j ( z − (cid:17) + ( x + j ( z +2) t ) j ( z − (cid:21) (14)leading to a two asymptotically travelling rational solitons of the form (13) (for σ = 1and σ = −
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