Non-self-adjoint Dirac-type Systems and Related Nonlinear Equations: Wave Functions, Solutions, and Explicit Formulas

Abstract

A version of the Bäcklund-Darboux transformation, where Darboux matrix takes the form of the transfer matrix function from the system theory, for the non-self-adjoint Dirac-type system is considered. Related nonlinear Schrödinger equations (coupled and multi-component), self-induced transparency equation, and non-Abelian sine-Gordon equation are treated. Explicit formulas for the wave functions and solutions are obtained. Mathematics Subject Classification (2000). Primary 47A55, 47A48; Secondary 35Q51.

Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation

“…In proposition 2.1 we obtain a solution formula both for the focusing and defocusing NLS. For related work we refer to [5,12,14,24]. As another advantage we point out that formulas analogous to those used here give a natural access to the MPSs of the KdV.…”

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

“…Thus, GBDT is a convenient tool to construct wave functions and explicit solutions of the nonlinear wave equations as well as to solve various direct and inverse problems. GBDT and its applications were treated or included as important examples in the papers [22,23,48,55,56,57,58,59,61,63,64,66,67,68,70,71,72,73,75] (see also [28,29,30,31,32,33,37]). Here we consider self-adjoint and skew-self-adjoint Dirac-type systems including the singular case corresponding to soliton-positon interaction and solve direct and inverse problems.…”

Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions

“…Note that equality (4.13) for the solution of the nonisospectral CNLS formally coincides with equality (3.13) [29] for the solution of the isospectral CNLS, and equality (4.50) for the solution of the nonisospectral KdV formally coincides with the equality (5.4) [16] for the solution of the isospectral KdV. However, the dependence on x and t and asymptotic behavior of the terms on the right-hand sides of the equality (4.13) in our paper and (3.13) in [29] is quite different. The same is true for the right-hand sides of (4.50) and (5.4) [16].…”

“…Afterwards we apply to the constructed equations GBDT, which is a version of the Bäcklund-Darboux transformation developed by the authors in [24][25][26][27][28][29][30][31][32]. The Bäcklund-Darboux transformation is a well-known and fruitful tool to construct explicit solutions of the integrable equations and some classical linear equations as well.…”