A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrödinger equation with wave operator

In this paper, we study the efficient solution of the nonlinear Schrödinger equation with wave operator, subject to periodic boundary conditions. In such a case, it is known that its solution conserves a related functional. By using a Fourier expansion in space, the problem is at first casted into Hamiltonian form, with the same Hamiltonian functional. A Fourier-Galerkin space semi-discretization then provides a large-size Hamiltonian ODE problem, whose solution in time is carried out by means of energy-conserving methods in the HBVM class (Hamiltonian Boundary Value Methods). The efficient implementation of the methods for the resulting problem is also considered and some numerical examples are reported.