Related papers: Long-Time Dynamics of KdV Solitary Waves over a Va…
The nolinear hydrodynamic equations of the surface of a liquid drop are shown to be directly connected to Korteweg de Vries (KdV, MKdV) systems, giving traveling solutions that are cnoidal waves. They generate multiscale patterns ranging…
A novel geometric method is applied to the problem of describing traveling wave solutions of the generalized Korteweg--de Vries (gKdV) equation in the form $$ u_t + u_{xxx} + a(u)u_x = 0, $$ where $a(u)$ is a smooth function characterizing…
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…
We consider a class of pseudodifferential evolution equations of the form $$u_t + (n(u) + Lu)_x = 0,$$ in which $L$ is a linear smoothing operator and $n$ is at least quadratic near the origin; this class includes in particular the Whitham…
It has been observed that the dynamics of the Toda lattice can be well described by solutions of the Korteweg-de Vries (KdV) equation in the continuum limit. We show that, under the KdV scaling and a suitable translation, the solution of…
In this paper, we study the novel nonlinear wave structures of a (2+1)-dimensional variable-coefficient Korteweg-de Vries (KdV) system by its analytic solutions. Its $N$-soliton solution are obtained via Hirota's bilinear method, and in…
We provide an accurate description of the long time dynamics of the solutions of the generalized Korteweg-De Vries (gKdV) and Benjamin-Ono (gBO) equations on the one dimension torus, without external parameters, and that are issued from…
The dynamics of solitary gravity-capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time dependent solutions, we simplify the full potential…
The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such…
We prove special decay properties of solutions to the initial value problem associated to the $k$-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with…
Debris flows often exhibit coherent wave structures-shock-like roll waves on steeper slopes and weaker, more sinusoidal dispersive pulses on gentler slopes. Coarse-rich heads raise basal resistance, whereas fines-rich tails lower it; in…
The Korteweg-de Vries equation (KdV) and various generalized, most often semi- linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general…
The core focus of this research work is to obtain invariant solutions and conservation laws of the (3+1)-dimensional ZK equation, a higher-dimensional generalization of the Korteweg--de Vries (KdV) equation, which describes the phenomenon…
In this work we consider the initial-value problem associated with a coupled system of generalized Korteweg-de Vries equations. We present a relationship between the best constant for a Gagliardo-Nirenberg type inequality and a criterion…
Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only H\"older continuous…
We study the asymptotics of the complex modified Korteweg-de Vries equation $\partial_t u + \partial_x^3 u = -|u|^2 \partial_x u$, which can be used to model vortex filament dynamics. In the real-valued case, it is known that solutions with…
We apply a multiple-time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity-free perturbation theory, we show that the…
In this study, we give a survey of derivations of KdV-type equations with an uneven bottom for several cases when small (perturbation) parameters $\alpha, \beta, \delta$ are of different orders. Six different cases of such ordering are…
Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other…
In this paper we study uniqueness properties of solutions of the k-generalized Korteweg-de Vries equation. Our goal is to obtain sufficient conditions on the behavior of the difference $u_1-u_2$ of two solutions $u_1, u_2$ of the equation…