Bell polynomial approach to an extended Korteweg-de Vries equation

摘要

Under investigation in this paper is an extended Korteweg-de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV-SK-type equation and KdV-Lax-type equation. On the one hand, N-soliton solutions, bilinear B?cklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV-Lax-type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV-SK-type equation is also presented, and the exact relation between the one periodicwave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit.