Related papers: Multisolitonic solutions from a B\"acklund transfo…
The N=1 supersymmetric modified Korteweg-de Vries (SmKdV) system is transformed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV (BSmKdV) passes the Painlev\'{e} test and allows a set of…
We derive the Kadomtsev-Petviashvili (KP) equation defined over a general associative algebra and construct its N-soliton solution. For the example of the Moyal algebra, we find multi-soliton solutions for arbitrary space-space…
We present a multiparameter generalization of the St\"ackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized St\"ackel transform preserves the Liouville…
Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides…
We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, implementing a Backlund transformation of SL equation. In this…
In this paper, based on the regular KdV system, we study negative order KdV (NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many conservation laws, and explicit multi-soliton and multi-kink wave solutions thorough…
We establish the nonlinear stability of $N$-soliton solutions of the modified Korteweg-de Vries (mKdV) equation. The $N$-soliton solutions are global solutions of mKdV behaving at (positive and negative) time infinity as sums of…
The existence of strong solutions to general class of strongly coupled parabolic systems will be discussed. These systems can be degenerate or singular as boundedness of theirs solutions are unavailable and not assummed. The results greatly…
We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave…
In the present paper the non-Noether symmetries of the Toda model, nonlinear Schodinger equation and Korteweg-de Vries equations (KdV and mKdV) are discussed. It appears that these symmetries yield the complete sets of conservation laws in…
We solve the Gardner deformation problem for the N=2 supersymmetric a=4 Korteweg-de Vries equation (P. Mathieu, 1988). We show that a known zero-curvature representation for this superequation yields the system of new nonlocal variables…
We review some of the fundamental notions associated to the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary…
We present the hierarchy and soliton solutions associated to a multi-component generalisation of the modified Korteweg-de Vries equation. A recursive formula for obtaining the Lax operators associated to the higher flows of the hierarchy is…
We analytically study the large-time asymptotics of the solution of the defocusing modified Korteweg-de Vries (mKdV) equation under a symmetric non-vanishing background, which supports the emergence of solitons. It is demonstrated that the…
The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink…
Using the bicomplex approach we discuss a noncommutative system in two--dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine--Gordon equation when the noncommutation parameter is removed,…
A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV system. It is shown that this Darboux transformation can be constructed by means of two methods: Painlev\'{e} analysis and reduction of a binary Darboux…
We revisit the symmetry structure of integrable PDEs, looking at the specific example of the KdV equation. We identify 4 nonlocal symmetries of KdV depending on a parameter, which we call generating symmetries. We explain that since these…
The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with…
We study the dynamics of multi-component Bose gas described by the Vector Nonlinear Schr\"{o}dinger Equation (VNLS), aka the Vector Gross--Pitaevskii Equation (VGPE) . Through a Madelung transformation, the VNLS can be reduced to coupled…