Breather wave solutions on the Weierstrass elliptic periodic background for the (2 + 1)-dimensional generalized variable-coefficient KdV equation.

Abstract

In this paper, the Nth Darboux transformations for the (2+1)-dimensional generalized variable-coefficient Koretweg-de Vries (gvcKdV) equation are proposed. By using the Lamé function method, the generalized Lamé-type solutions for the linear spectral problem associated with the gvcKdV equation with the static and traveling Weierstrass elliptic ℘-function potentials are derived, respectively. Then, the nonlinear wave solutions for the gvcKdV equation on the static and traveling Weierstrass elliptic ℘-function periodic backgrounds under some constraint conditions are obtained, respectively, whose evolutions and dynamical properties are also discussed. The results show that the degenerate solutions on the periodic background can be obtained by taking the limits of the half-periods ω1,ω2 of ℘(x), and the evolution curves of nonlinear wave solutions on the periodic background are determined by the coefficients of the gvcKdV equations.