Generalized mkdv equation, liouville equation,sine-gordon equation and sinh-gordon equation as well as a formal b?cklund transformation

摘要

A Lie algebra which consists of linear combinations of one basis of the Lie algebra A_1 is presented for which an isospectral Lax pair is exhibited. By using the zero curvature equation, the generalized mKdV equation, Liouville equation and sine-Gordon equation, sinh-Gordon equation are generated via polynomial expansions. Finally, we investigate a kind of formal B?cklund transformation for the generalized sine-Gordon equation. The explicit B?cklund transformation of the standard sine-Gordon equation is presented. The other equations given in the paper are obtained similarly.