Related papers: Compactons in PT-symmetric generalized Korteweg-de…
The distance between the solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations is estimated in appropriate Sobolev spaces. This family of equations includes, as…
An alternative way for the derivation of the new KdV-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It…
The Korteweg-de Vries (KdV) equation is known as a universal equation describing various long waves in dispersive systems. In this article, we prove that in a certain scaling regime, a large class of rough solutions to the Boussinesq…
The universal theory of weakly nonlinear wave packets given by the nonlinear Schr\"odinger equation is revisited. In the limit where the group and phase velocities are very close together, a multiple scale analysis carried out beyond all…
Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually…
We obtain a dispersive long-time decay in weighted energy norms for solutions of the 3D Klein-Gordon equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schredinger equation. For the proof…
We consider the modified Korteweg-de Vries equation (mKdV) and prove that given any sum P of solitons and breathers of (mKdV) (with distinct velocities), there exists a solution p of (mKdV) such that p(t) -- P(t) $\rightarrow$ 0 when t…
We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{k+1})=0$, where $k>4$ is an integer number and $\mu=\pm1$. We give an alternative proof of the Kenig, Ponce, and Vega result in…
We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for…
In this paper, we study the existence of periodic waves for the perturbed generalized defocusing mKdV equation using the theory of geometric singular perturbation. By Abelian integral and involution operation, we prove that the limit wave…
We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW…
The aim of this work is to study asymptotically and numerically the interaction of solitons with an external forcing with variable speed using the forced modified Korteweg-de Vries equation (mKdV). We show that the asymptotic predictions…
The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical study for many decades. However, analytical solution of such a special soliton is rarely considered. In this Letter, we study the…
We consider approximate, exact, and numerical solutions to the cylindrical Korteweg-de Vries equation. We show that there are different types of solitary waves and obtain the dependence of their parameters on distance. Then, we study the…
A numerical study of the nonlinear wave solutions of the Rosenau-Pikovsky K(cos) equation is presented. This equation supports at least two kind of solitary waves with compact support: compactons of varying amplitude and speed, both…
The existence of ``dispersion-managed solitons'', i.e., stable pulsating solitary-wave solutions to the nonlinear Schr\"{o}dinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our…
An optimal-velocity (OV) model describes car motion on a single lane road. In particular, near to the boundary signifying the onset of traffic jams, this model reduces to a perturbed Korteweg-de Vries (KdV) equation using asymptotic…
The solution of a coupled system consisting of generalized Korteweg-de Vries-type equations is obtained for all time where the initial data are analytic on a band in the complex plane. We show that the width of this band decreases…
A generalization of the KP equation involving higher-order dispersion is studied. This equation appears in several physical applications. As new results, the Lie point symmetries are obtained and used to derive conservation laws via…
We investigate the interaction of solitons with an external periodic field within the framework of the modified Korteweg-de Vries (mKdV) equation. In the case of small perturbation a simple dynamical system is used to describe the soliton…