Discrete symmetries and solitons (Q1918860)

The paper discusses the relation between the discrete symmetries associated with some nonlinear evolution equations and the corresponding spectral problems. In particular, it shows the connection between the Schrödinger equation \(L\psi= -\psi'' + u(x) \psi = \lambda \psi\) and the nonlinear lattice equations \[ (f_n+f_{n+1})' =f^2_n-f^2_{n+1} + \alpha_n, \] where \(n=0\), \(\pm 1, \dots\) and \(\alpha_n\) are constants. It is also shown how Bäcklund transformations and \(N\)-soliton solutions can be obtained directly from the symmetries.