Related papers: On connection between the splitting parameters of …
The soliton resolution for the focusing modified Korteweg-de vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation…
We study the Korteweg-de Vries-type equation dt u=-dx(dx^2 u+f(u)-B(t,x)u), where B is a small and bounded, slowly varying function and f is a nonlinearity. Many variable coefficient KdV-type equations can be rescaled into this equation. We…
By applying Darboux-Crum transformations to a Lax pair formulation of the Korteweg-de Vries (KdV) equation, we construct new sets of multi-soliton solutions to it as well as to the modified Korteweg-de Vries (mKdV) equation. The obtained…
We address two pressing questions in the theory of the Korteweg--de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity…
We prove the instability of a ``critical'' solitary wave of the generalized Korteweg -- de Vries equation, the one with the speed at the border between the stability and instability regions. The instability mechanism involved is ``purely…
Dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of the coupled one-dimensional nonlinear Schr\"odinger (NLS) equations. We consider the small amplitude limit in which the coupled…
We study the variable bottom generalized Korteweg-de Vries (bKdV) equation dt u=-dx(dx^2 u+f(u)-b(t,x)u), where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water.…
We consider the vector generalization of the modified Korteweg-de Vries equation. We develop the inverse scattering transform for solving this equation. We construct the solitons and the breather solutions and investigate the processes of…
We propose a numerical method for finding solitary wave solutions of generalized Korteweg-de Vries equations by solving the nonlinear eigenvalue problem on an unbounded domain. The artificial boundary conditions are obtained to make the…
In an earlier paper Cooper, Shepard, and Sodano introduced a generalized KdV equation that can exhibit the kinds of compacton solitary waves that were first seen in equations studied by Rosenau and Hyman. This paper considers the…
We study the class of generalized Korteweg-DeVries equations derivable from the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - { {(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right) dx, $ where the…
We consider the generalized Korteweg-de Vries equation in the supercritical case, and we are interested in solutions which converge to a soliton in large time in H^1. In the subcritical case, such solutions are forced to be exactly solitons…
We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special "condensate" limit. We prove that in this limit the integro-differential kinetic equation for the spectral density…
Acoustic solitons obtained through a reductive perturbation scheme are normally governed by a Korteweg-de Vries (KdV) equation. In multispecies plasmas at critical compositions the coefficient of the quadratic nonlinearity vanishes.…
We found, through analytical and numerical methods, new towers of infinite number of asymptotically conserved charges for deformations of the Korteweg-de Vries equation (KdV). It is shown analytically that the standard KdV also exhibits…
The Navier-Stokes-Voigt equations are a regularization of the Navier-Stokes equations that share some of its asymptotic and statistical properties and have been used in direct numerical simulations of the latter. In this article we…
The KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Backlund transformation for the restricted flows (by V.B.…
Differential invariants for the maximal Lie invariance group of the Korteweg-de Vries equation are computed using the moving frame method and compared with existing results. Closed forms of differential invariants of any order are presented…
Assuming that the nucleus can be treated as a perfect fluid we study the conditions for the formation and propagation of Korteweg-de Vries (KdV) solitons in nuclear matter. The KdV equation is obtained from the Euler and continuity…
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for $H^s$ initial data, $s>-1/2$, and for any $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$…