In this work, we established exact solutions for the combined KdV-MKdV equation. By constructing four new types of Jacobi elliptic functions solutions, the Jacobi elliptic functions expansion method will be extend. With the aid of symbolic computation system mathematica, obtain some new exact periodic solutions of nonlinear combined KdV-MKdV equation , and these solutions are degenerated to sol...

We present a simplified one-dimensional model for pulse wave propagation through fluid-filled tubes with elastic walls, which takes into account the elasticity of the wall as well as the tapering effect. The spatial dynamics in this model is governed by a variable coefficient KdV equation with conditions given at the inflow site. We discuss an existence theory for the associated evolution equat...

The KdV equation can be considered as a special case of the general equation ut + f(u)x − δg(uxx)x = 0, δ > 0, (0.1) where f is non-linear and g is linear, namely f(u) = u/2 and g(v) = v. As the parameter δ tends to 0, the dispersive behavior of the KdV equation has been throughly investigated (see, e.g., [11], [6], [2] and the references therein). We show through numerical evidence that a comp...

In this paper we study weak continuity of the dynamical systems for the KdV equation in H−3/4(R) and the modified KdV equation in H1/4(R). This topic should have significant applications in the study of other properties of these equations such as finite time blow-up and asymptotic stability and instability of solitary waves. The spaces considered here are borderline Sobolev spaces for the corre...

Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Airault–McKean–Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple coll...

An analytic study was conducted on coupled partial differential equations. We formally derived new solitary wave solutions of generalized coupled system of Zakharov-Kuznetsov (ZK) and KdV equations by using modified extended tanh method. The traveling wave solutions for each generalized coupled system of ZK and KdV equations are shown in form of periodic, dark, and bright solitary wave solution...

An extension of the Super KdV integrable system in terms of operator valued functions is obtained. Following the ideas of Gardner, a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented. The approach is applied to the above described system and infinitely many conserved quantities are constructed. In a particular case they reduce to ...

Algebro-geometric Poisson brackets for real, finite-zone solutions of the Korteweg–de Vries (KdV) equation were studied in [1]. The transfer of this theory to the Toda lattice and the sinh-Gordon equation is more or less obvious. The complex part of the finite-zone theory for the nonlinear Schrödinger equation (NS) and the sine-Gordon equation (SG) is analogous to KdV, but conditions that solut...

We prove in this paper the stability and asymptotic stability in H of a decoupled sum of N solitons for the subcritical generalized KdV equations ut + (uxx + u )x = 0 (1 < p < 5). The proof of the stability result is based on energy arguments and monotonicity of local L norm. Note that the result is new even for p = 2 (the KdV equation). The asymptotic stability result then follows directly fro...

We prove that one system of coupled KdV equations, recently claimed by Hirota, Hu and Tang [J. Math. Anal. Appl. 288:326–348 (2003)] to pass the Painlevé test for integrability, actually fails the test at the highest resonance of the generic branch. Introduction In Section 6 of their recent work [1], Hirota, Hu and Tang reported that the system of coupled KdV equations ∂ui ∂t + 6a ( N