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

The family of equations

(1){u t -[αg(u)+β]u x } x =ϵg ' (u),

where g '' +μg=θ, ϵ=±1, μ,α β,θ, describes pseudospherical surfaces. It is shown that solutions of (1) correspond to solutions of

(2)u yt =ϵg ' (u)ϵ ' -αϵu y 2 ,ϵ ' =±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 ϵ ' =1, g ' (0)=0 and g '' (0)=ϵ if μ0 are obtained using an inverse scattering method.

Reviewer: R.Racke
MSC:
58J72Correspondences and other transformation methods (PDE on manifolds)
35P25Scattering theory (PDE)
35R30Inverse problems for PDE