Computational Methods for Three Coupled Nonlinear Schrödinger Equations

Abstract

In this work, we will derive numerical schemes for solving 3-coupled nonlinear Schrödinger equations using finite difference method and time splitting method combined with finite difference method. The resulting schemes are highly accurate, unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we use these methods to study the interaction dynamics of two solitons. It is found that both elastic and inelastic collision can take place under suitable parametric conditions. We have noticed that the inelastic collision of single solitons occurs in two different manners: enhancement or suppression of the amplitude.

Sixth Order Finite Difference Method for Three Coupled Nonlinear Schrödinger Equations

“…The interaction of two solitons is discussed in detail for different parameters to produce elastic and inelastic interactions. This behavior is agreeing with [1] [2] [3] with the highest accuracy. The derived method can be used to solve similar nonlinear problems.…”

“…In recent years the concept of soliton has been receiving considerable attention in optical communications. Since soliton is capable of propagating over long distances without change of shape and velocity, it has been found that the soliton propagating through optical fiber arrays is governed by a set of equations related to the coupled nonlinear Schrödinger equation [1] [2] [3].…”

Crank-Nicolson Scheme for Solving the Modified Nonlinear Schrodinger Equation

“…In addition, it is noticed that there is a symmetry in momentum about d = 1, where it becomes approximately equal to zero. Therefore, the relationship between momentum and d and g is highlighted as in Figure 5, (Alamri, 2003;Ismail et al, 2016;Antikainen et al, 2012)…”

Efficient energy‐preserving scheme of the three‐coupled nonlinear Schrödinger equation

“…However, to the best of my knowledge, effort put in T‐CNLS equation is very limited. () Aydin developed a multisymplectic scheme for N ‐coupled NLS equation with symmetric coefficients in the nonlinear terms . Ismail et al presented a splitting finite difference scheme for it which is unconditionally stable and accurate .…”

“…() Aydin developed a multisymplectic scheme for N ‐coupled NLS equation with symmetric coefficients in the nonlinear terms . Ismail et al presented a splitting finite difference scheme for it which is unconditionally stable and accurate . Liang et al studied a central difference and quadratic spline approximation‐based exponential time difference Crank‐Nicolson method .…”