Symmetry approach and the generalized Korteweg-de Vries equation with variable coefficients

Abstract

Using Lie symmetry arguments, we consider a class of nonlinear KdV-type equations which are usually denoted by K(m, n). We start with the 4-dimensional Lie algebra of the ordinary KdV equation to derive and classify K(m, n) equations with space-and time-dependent coefficients.

Time-dependent exact solutions for Rosenau–Hyman equations with variable coefficients

“…(2) are then defined. We followed this prescription for subalgebras fX 4 g; fX 1 ; X 2 ; X 3 ; X 4 g, and fX 3 g. The corresponding vcKðm; nÞ equations we have obtained are given by [38] …”

“…We followed along this line to study and classify some generalized Kðm; nÞ equations by choosing the classical KdV equation Lie-symmetry algebra. We have found three Rosenau-Hyman-like equations with variable coefficients (vcKðm; nÞ) [38]. Once Lie approach is a useful tool to search for solutions [26,34,[39][40][41][42][43][44], we also employed it here to obtain Lie-symmetry invariant time-dependent solutions for suited values of parameters m and n of vcKðm; nÞ equations.…”

“…In Section 2, we present the method according to a preliminary discussion we have carried out in a proceeding [38]. Invariant solutions are presented in Section 3.…”

Scaling Symmetries and Conservation Laws for Variable-coefficients Nonlinear Dispersive Equations

“…Besides, scaling invariance is closely related to the theory of fractals as well as to the general theory of dimensional analysis and renormalization [12]. By having these considerations in mind and motivated by remarkable features of the compacton K(m, n), introduced by Rosenau and Hyman [35], Souza and Silva [37,38] have recently employed the Lie symmetry machinery [9,33,39] to build up a generalized Rosenau-Hyman equation invariant under the scaling symmetry of standard KdV and obtained a variable-coefficients K(m, n) (vcK(m, n) hereafter) of the form…”