Bäcklund transformations induced by symmetries. Application: Discrete mKdV

Abstract

We give a general method for the derivation of Backlund transformations induced by symmetries. We demonstrate this method on two easy examples, the KdV equation and the discrete mKdV equation ∂qn ∂t = (1 + q n)(qn+1 − qn−1) with simple point symmetries. In connection with symplectic numerical algorithms we are particularly interested in discrete equations. 1 Outline of the Method We consider a nonlinear partial differential equation in the form ut = F (u, ux, uxx, ...) , u = u(x, t) (1.1) or in discrete form ∂un ∂t = F (un, un+i, ...) , n, i ∈ Z. (1.2) We assume that this partial differential equation (1.1) is associated with a linear system { Φx = MΦ Φt = NΦ (1.3) which means that the compatibility condition Φxt = Φtx is equivalent of u being a solution of the partial differential equation (1.1) , where Φ =