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On Bäcklund transformations for nonlinear partial differential equations. (English) Zbl 0828.35123
Summary: Bäcklund transformations for nonlinear partial differential equations are obtained without using any structure associated with the equations. We classify all the nonlinear partial differential equations of the form \[ u_{xxx} = {\mathcal F} (u,u_x,u_t) \] that have Bäcklund transformations whose definition involves only \(u\), \(u_x\), \(u_{xx}\), and a function determined by \(u\), \(u_x\), and \(u_{xx}\) via an integrable system of two first-order partial differential equations, and we obtain all such Bäcklund transformations.
In particular, a new nonlinear partial differential equation with a Bäcklund transformation (i.e., \(u_{xxx} = - {3 \over 2} \sin 2u\) \(u_x - {1 \over 2} u^3_x + u_t)\) is found, and the known Bäcklund transformation for the KdV equation, the MKdV equation, the potential KdV equation, and the potential MKdV equation are recovered without using any knowledge of these equations. Our method is applicable to many other classes of nonlinear partial differential equations.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q58 Other completely integrable PDE (MSC2000)
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs
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