Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations. (English) Zbl 0697.58059

Beals, Richard; Rabelo, Mauro; Tenenblat, Keti

Stud. Appl. Math. 81, No. 2, 125-151 (1989).

The family of equations \[ (1)\quad \{u_ t-[\alpha g(u)+\beta]u_ x\}_ x=\epsilon g'(u), \] where \(g''+\mu g=\theta\), \(\epsilon =\pm 1\), \(\mu\),\(\alpha\) \(\beta\),\(\theta\in {\mathbb{R}}\), describes pseudospherical surfaces. It is shown that solutions of (1) correspond to solutions of \[ (2)\quad u_{yt}=\epsilon g'(u)\sqrt{\epsilon '-\alpha \epsilon u^ 2_ y,}\quad \epsilon '=\pm 1. \] Examples include sine-Gordon, sin h- Gordon and Liouville equations. A self-Bäcklund transformation for (2) is constructed based on a geometric method introduced by A. Cavalcante and by L. P. Jorge and the third author [J. Math. Phys. 29, 1044-1049 (1988; Zbl 0695.35038)] and by L. P. Jorge and the third author [Stud. Appl. Math. 77, 103-107 (1987; Zbl 0642.35017)].
Finally, solutions to equation (2), with \(\epsilon '=1\), \(g'(0)=0\) and \(g''(0)=\epsilon\) if \(\mu\neq 0\) are obtained using an inverse scattering method.

Reviewer: R.Racke

MSC:

58J72	Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35P25	Scattering theory for PDEs
35R30	Inverse problems for PDEs

Keywords:

pseudospherical surface; Bäcklund transformation; inverse scattering

Citations:

Zbl 0695.35038; Zbl 0642.35017

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[9]	u xt = u u x k u x k