Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

Abstract

Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be different. The influence of each of both invariants on the computer simulation result accuracy is analyzed while solving the problem describing the third optical harmonic generation process. Two implicit conservative FDSs are developed for a numerical solution of this problem. One of them preserves a difference analog of the energy invariant (or the Hamiltonian) accurately, while the Hamiltonian (or the energy’s invariant) is preserved with the second order of accuracy. Both FDSs possess the second order of approximation at a smooth enough solution of the differential problem. Computer simulations demonstrate advantages of the implicit FDS preserving the Hamiltonian. To illustrate the advantages of the developed FDSs, a comparison of the computer simulation results with those obtained applying the Strang method, based on either an implicit scheme or the Runge–Kutta method, is made. The corresponding theorems, which claim the second order of approximation for preserving invariants for the FDSs under consideration, are stated.

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

“…It is demonstrated that, to avoid the development of the nonphysical solution's instability, it is necessary to consider the spectral invariant for providing computer simulations. In [61,62], the comparison of the efficiency of several conservative and splitting FDSs is provided. In [61], conservative FDSs are constructed for the problem of femtosecond pulse propagation in an optical fiber, considering the time derivative of the medium nonlinear response.…”

“…The proposed FDSs preserve several invariants involving both the intensity profile of the beam and the phase distribution of the beam. In [62], for the set of 1D-coupled NLSEs, two conservative FDSs are constructed. Each of them preserves one of two differential problem's invariants (the energy's invariant or the Hamiltonian), while another is preserved with the second order of approximation.…”