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.

Keywords

• two nonlinear coupled Schrödinger equations
• conservation laws
• finite-difference scheme
• split-step method
• approximation order
• Hamiltonian