N-soliton-like solutions and Bäcklund transformations for a non-isospectral and variable-coefficient modified Korteweg-de Vries equation

A non-isospectral and variable-coefficient modified Kortewegde Vries (mKdV) equation is investigated in this paper. Starting from the AblowitzKaupNewellSegur procedure, the Lax pair is established and the Bäcklund transformation in original variables is also derived. By a dependent variable transformation, the non-isospectral and variable-coefficient mKdV equation is transformed into bilinear equations, by virtue of which the N-soliton-like solution is obtained. In addition, the bilinear Bäcklund transformation gives a one-soliton-like solution from a vacuum one. Furthermore, the N-soliton-like solution in the Wronskian form is constructed and verified via the Wronskian technique.