Bäcklund transformations for the sine–Gordon equations

Abstract

The generalized sine–Gordon equations z ,xt = F ( z ) in two independent variables x , t include the sine–Gordon z ,xt = sin z and the multiple sine–Gordon’s like z ,xt = sin z + ½ sin ½ z . Among other physical applications all these sine–Gordon’s are significant to the theory of intense ultra-short optical pulse propagation. The sine–Gordon itself has analytical multi-soliton solutions. It also has an infinity of polynomial conserved densities and has auto-Bäcklund transformations which generate a second solution of the sine–Gordon from a first solution – particularly from the solution z ≡ 0. We prove first that the generalized multi-dimensional sine–Gordon in two or more space variables x 1 , x 2 , . . . has no auto-Bäcklund transformations. Next we prove that the generalized sine–Gordon’s z ,xt = F ( z ) and z ' ,xt = G ( z ') have an invertible Bäcklund transformation between solutions z and z ' if and only if F and G are solutions of F ¨ = α 2 F , G ¨ = β 2 G where, in general, β = αh -1 , α is a complex number and h 2 (≠ 0) is real. In case h = 1 and F and G are the same function z ,xt = F ( z ) has an auto-Bäcklund transformation if and only if F ¨ = α 2 F . We exhibit the B. ts and a. B. ts in these cases as well as the other B. ts for the generalized sine–Gordon. We conclude that the multiple sine–Gordon’s do not have a. B. ts and infer that, despite the soliton character of the numerical solutions, the multiple sine–Gordon’s are not soluble by present simplest formulations of the two by two inverse scattering method.