## Abstract

The generalized sine-Gordon equations $z_{,xt}=F(z)$ in two independent variables $x$, mi include the sine-Gordon $z_{,xt}=$sin z and the multiple sine-Gordon's like $z_{,xt}$ = sin $z+\frac{1}{2}$ sin $\frac{1}{2}z$. 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 multisoliton solutions. It also has an infinity of polynomial conserved densities and has auto-Backlund transformations which generate a second solution of the sine-Gordon from a first solution - particularly from the solution $z\equiv 0$. We prove first that the generalized multi-dimensional sine-Gordon in two or more space variables $x^{1},x^{2}$... has no auto-Backlund transformations. Next we prove that the generalized sine-Gordon's $z_{,xt}=F(z)$ and $z_{,xt}^{\prime}=G(z^{\prime})$ have an invertible Backlund transformation between solutions $z$ and $z^{\prime}$ if and only if $F$ and $G$ are solutions of $\ddot{F}=\alpha ^{2}F,\ddot{G}=\beta ^{2}G$ where, in general, $\beta =\alpha h^{-1},\alpha $ is a complex number and $h^{2}(\neq 0)$ is real. In case $h$ = 1 and $F$ and $G$ are the same function $z_{,xt}=F(z)$ has an auto-Backlund transformation if and only if $\ddot{F}=\alpha ^{2}F$. 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.