The I-method in its first version as developed by Colliander et al. in [2] is applied to prove that the Cauchy-problem for the generalised Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space H(R → R), provided s > − 1 42 .

In this work, we consider varying aspects of the stability of periodic traveling wave solutions to nonlinear dispersive equations. In particular, we are interested in deriving universal geometric criterion for the stability of particular third order nonlinear dispersive PDE’s. We begin by studying the spectral stability of such solutions to the generalized Korteweg-de Vries (gKdV) equation. Usi...

The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary d...

Abstract We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense and remain close to multi-solitons. show that these necessarily pure For equation (KdV) modified (mKdV) particular, we obtain a characterization multi-solitons multi-breathers terms dispersion.

with general C nonlinearity f . Under an explicit condition on f and c > 0, there exists a solution in the energy space H of (0.1) of the type u(t, x) = Qc(x − x0 − ct), called soliton. Stability theory for Qc is well-known. In [11], [14], we have proved that for f(u) = u, p = 2, 3, 4, the family of solitons is asymptotically stable in some local sense in H, i.e. if u(t) is close to Qc (for all...