Quasiperiodic waves and asymptotic behavior for the nonisospectral and variable-coefficient KdV equation

Abstract

In this paper, quasiperiodic waves and asymptotic behavior for the nonisospectral and variable-coefficient KdV (nvcKdV) equation are considered. The Hirota bilinear method is extended to explicitly construct multiperiodic (quasiperiodic) wave solutions for the nvcKdV equation. And a limiting procedure is presented to analyze asymptotic behavior of the one- and two-periodic waves in details. The exact relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a small amplitude limit.