Soliton solutions to generalized discrete Korteweg-de Vries equations

New discrete equations of the simplest three-point form are considered that generalize the discrete Korteweg-de Vries equation. The properties of solitons, kinks, and oscillatory waves are numerically examined for three types of interactions between neighboring chain elements. An analogy with solutions to limiting continual equations is drawn.     

Weighted identities for solutions of generalized Korteweg-de Vries equations

Consider the Korteweg-de Vries equation u _t + u _xxx + uu _x = 0 and its generalization u _t + u _xxx + f(u)x = 0. For the solutions of these equations, weighted identities (differential and integral) are obtained. These identities make it possible to establish the blow-up (in finite time) of the solutions of certain boundary-value problems.     

Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non dispersion.     

Bounded solutions of the Korteweg-de Vries equation

New types of bounded nondecreasing solutions of the equation are found and it is proved that they are limits of N-soliton solutions.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 59–70, 1988.     

Jordan algebras and generalized Korteweg-de Vries equations

Institute of Mathematics and Computing Center, Urals Branch, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 87, No. 3, pp. 391–403, June, 1991     

Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations

This paper considers the coupled complex modified Korteweg-de Vries (mKdV) equations and presents a binary Darboux transformation for the equations. As a direct application, we give a classification of general soliton solutions derived from vanishing and non-vanishing backgrounds, on the basis of the dynamical behavior of the solutions. Special types of solutions in the presented solutions include breathers, bright-bright solitons, bright-dark solitons, bright-W-shaped solitons, and rogue wave solutions. Furthermore, dynamics and interactions of vector bright solitons are exhibited.     

Exact solutions of the modified Korteweg-de Vries equation

We use the inverse scattering method to obtain a formula for certain exact solutions of the modified Korteweg-de Vries (mKdV) equation. Using matrix exponentials, we write the kernel of the relevant Marchenko integral equation as W( x + y;t ) = Ce ^{- ( x + y )A} e^{8A3 t} B\Omega \left( {x + y;t} \right) = Ce^{ - \left( {x + y} \right)A} e^{8A^3 t} BB, where the real matrix triplet (A,B,C) consists of a constant p×p matrix A with eigenvalues having positive real parts, a constant p×1 matrix B, and a constant 1× p matrix C for a positive integer p. Using separation of variables, we explicitly solve the Marchenko integral equation, yielding exact solutions of the mKdV equation. These solutions are constructed in terms of the unique solution P of the Sylvester equation AP + PA = BC or in terms of the unique solutions Q and N of the Lyapunov equations A^°Q + QA = C^°C and AN + NA^° = BB^°, where B^°denotes the conjugate transposed matrix. We consider two interesting examples.     

On the singular solutions of the Korteweg-de Vries equation

We consider two classes of singular solutions of the KdV equation: singular solutions of the Cauchy problem and singular traveling waves. In both cases, we establish sufficient conditions for their existence.     

Korteweg-de Vries and nonlinear equations related to the Toda lattice

The purpose of this work is to discuss the sense in which certain nonlinear equations related to the Toda lattice are discrete versions of the Korteweg-de Vries equation.     

Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solutions

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg-de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the x/t plane splits into g+1 soliton regions which are interlaced by g + 1 oscillatory regions, where g + 1 is the number of spectral gaps.     

Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations

This work is concerned with stability properties of periodic traveling waves solutions of the focusing Schrödinger equation

iut+uxx+²|u|u=0