On a class of discretizations of Hamiltonian nonlinear partial differential equations

Abstract

We present a new class of discretizations of partial differential equations (PDEs) that preserve a (momentum-like) integral of the motion. This results in an “effective” translational invariance for the dynamical problem and the absence of a Peierls–Nabarro barrier that is usually present in discretizations of Hamiltonian PDEs. A general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential–difference equations are analyzed in a number of different cases for nonlinear Klein–Gordon as well for nonlinear Schrödinger type systems. While for the former nonintegrability of the dynamical problem is evident, in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems. We also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones.