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Zbl 0697.58059
Beals, Richard; Rabelo, Mauro; Tenenblat, Keti
Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations.
(English)
[J] Stud. Appl. Math. 81, No.2, 125-151 (1989). ISSN 0022-2526; ISSN 1467-9590/e

The family of equations $$(1)\quad \{u\sb t-[\alpha g(u)+\beta]u\sb x\}\sb x=\epsilon g'(u),$$ where $g''+\mu g=\theta$, $\epsilon =\pm 1$, $\mu$,$\alpha$ $\beta$,$\theta\in {\bbfR}$, describes pseudospherical surfaces. It is shown that solutions of (1) correspond to solutions of $$(2)\quad u\sb{yt}=\epsilon g'(u)\sqrt{\epsilon '-\alpha \epsilon u\sp 2\sb y,}\quad \epsilon '=\pm 1.$$ 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 {\it A. Cavalcante} and by {\it L. P. Jorge} and the third author [J. Math. Phys. 29, 1044-1049 (1988; Zbl 0695.35038)] and by {\it L. P. Jorge} and the third author [Stud. Appl. Math. 77, 103-107 (1987; Zbl 0642.35017)]. \par Finally, solutions to equation (2), with $\epsilon '=1$, $g'(0)=0$ and $g''(0)=\epsilon$ if $\mu\ne 0$ are obtained using an inverse scattering method.
[R.Racke]
MSC 2000:
*58J72 Correspondences and other transformation methods
35P25 Scattering theory (PDE)
35R30 Inverse problems for PDE

Keywords: pseudospherical surface; Bäcklund transformation; inverse scattering

Citations: Zbl 0695.35038; Zbl 0642.35017

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