Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation

In this paper, a generalized variable-coefficient Gardner equation arising in nonlinear lattice, plasma physics and ocean dynamics is investigated. With symbolic computation, the Lax pair and Bäcklund transformation are explicitly obtained when the coefficient functions obey the Painlevé-integrable conditions. Meanwhile, under the constraint conditions, two transformations from such an equation either to the constant-coefficient Gardner or modified Korteweg-de Vries (mKdV) equation are proposed. Via the two transformations, the investigations on the variable-coefficient Gardner equation can be based on the constant-coefficient ones. The N-soliton-like solution is presented and discussed through the figures for some sample solutions. It is shown in the discussions that the variable-coefficient Gardner equation possesses the right- and left-travelling soliton-like waves, which involve abundant temporally-inhomogeneous features.