In most of the studies concerning nonlinear wave equations Korteweg-de Vries type, authors focus on waves elevation. Such have general form ~$u_{\text{u}}(x,t)=A f(x-vt)$, where ~$A>0$. this communication we show that if ~$u_{\text{up}}(x,t)=A f(x-vt)$ is solution a given equation, then $u_{\text{down}}(x,t)=-A is, an inverted same but with changed sign parameter ~$\alpha$. This property common...

We consider the initial value problem associated to a large class of fifth order nonlinear dispersive equations. This class includes several models arising in the study of different physical phenomena. Our aim is to establish special (space) decay properties of solutions to these systems. These properties complement previous unique continuation results and in some case, show that they are optim...

In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for a Boussinesq equation. The equations governing the corrections to the KdV approximation are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation al...

The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to 1 + 2 dimensions. Perturbation theory is the ...

The Korteweg-de Vries (KdV) equation is first derived from a general system of partial differential equations. An analysis of the linearized KdV equation satisfied by the higher order amplitudes shows that the secular-producing terms in this equation are the derivatives of the conserved densities of KdV. Using the multi-time formalism, we prove that the propagation on very long distances is gov...

The authors of the paper "Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension" \cite{Fok} claim that they derived equation which generalizes KdV to two space dimensions both in first second order small parameters. Moreover, obtain soliton solution (2+1)-dimension equation. has been obtained by applying perturbation method \cite{burde...

In this Letter, based on the idea of homogeneous balance (HB) method and with help of MATHEMATICA, we obtain a new auto-Bäcklund transformation for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. Then based on the Bäcklund transformation, some solutions for these two equations are derived.  2002 Elsevier Science B.V. All rights reserved.

By considering the set of coupled KdV differential equations as a zero curvature representation of some fourth order linear differential equation and factorizing the linear differential equation, the hierarchy of solutions of the coupled KdV differential equations have been obtained from the eigen spectrum of constant potentials.