On nonlinear hyperbolic systems related by Bäcklund transforms

Abstract

In this work we describe pairs of nonlinear hyperbolic system of equations $u_{xy} = f(u, u_x, u_y)$, where $u^i_{xy} = f^i$, $i = 1,2, \dots n$, the linearizations of which are related by the first order Laplace transform. On the base of this Laplace transform we construct Bäcklund transforms relating the solutions of nonlinear systems. The classical Bäcklund transform is defined for a second-order nonlinear differential equation whose solution is a function of two independent variables. The Bäcklund transform for a pair of nonlinear equations is a system of relations involving functions and their first derivatives and it provides a transform of a solution of one equation into the solution of another and vice versa. The Bäcklund transforms preserve integrability. The Bäcklund problem is to list possible Bäcklund transforms and the equations admitting such transforms. The Laplace cascade integration method is one of the classical methods for integrating linear partial differential equations. The Laplace transform is a special case of the Bäcklund transform for linear equations. The method used in this paper was previously applied to nonlinear hyperbolic equations. In this paper, this method is employed to describe systems associated with Bäcklund transforms.