1. Introduction

The nonlinear Schrödinger equation (NLS) arises as model equation with second-order dispersion and cubic nonlinearity for describing the dynamics of slowly varying wave packets in nonlinear optics and fluid dynamics. If there are two or more modes the coupled nonlinear Schrödinger (CNLS) system would be the relevant model. The two coupled nonlinear Schrödinger (CNLS) equations are given by

where ψ₁(x,t) and ψ₂(x,t) are complex amplitudes or ‘envelopes’ of two wave packets, i is the imaginary number, x and t are the space and time variables, respectively. The CNLS system has many applications including nonlinear optics[1] and [2] and geophysical fluid dynamics [3] and [4]. The parameters α_j are the dispersion coefficients, σ_j the Landau constants which describe the self-modulation of the wave packets, and v₁₂ and v₂₁ are the wave–wave interaction coefficients which describe the cross-modulations of the wave packets [5] and [6].

 

Analytical solutions can be obtained only for a few special integrable cases, like the Manakov model [1] where the self-modulation and wave–wave interaction coefficients are equal, i.e. α₁=α₂=1, σ₁=σ₂=v₁₂=v₂₁. Other integrable cases are: α₁=α₂,σ₁=σ₂=v₁₂=v₂₁ and α₁=−α₂, σ₁=σ₂=−v₁₂=−v₂₁ which can be solved by the inverse scattering method [7]. For nonintegrable cases, where the parameters are different, numerical methods have to be used in order to understand different nonlinear phenomena that arise by the interaction of stable and unstable wave packets in the CNLS system.

There has been a lot work done for the solitary wave solutions of (1). Symplectic and multi-symplectic methods are also used for the numerical solution of soliton collision[8] and [9]. Little attention has been paid to plane wave solutions under periodic boundary conditions. In this work we will consider the plane wave solutions of (1) under periodic boundary conditions with period L

with the dispersion relationswhere the amplitudes ψ₁₀ and ψ₂₀ are assumed to be real [5] and [6]. We takev₁₂=v₂₁=v because this choice of the parameters allows a multi-symplectic formulation of the CNLS system.

 

Besides the Manakov's case which corresponds to elliptical polarization we consider two other polarizations [5] and [6]:

•

linear polarization: v₁₂=v₂₁=2/3,

•

circular polarization: v₁₂=v₂₁=2

with α₁=α₂=1, σ₁=σ₂=1.

 

Several numerical studies have been carried out in recent years in order to understand the behavior of the periodic plane wave solutions of the CNLS system. In [5] a pseudospectral discretization was used in the space variables and the resulting system of ordinary differential equations (ODEs) were integrated using the standard fourth-order Runge–Kutta method. Because the method used in [5] is not conservative, no indication is given about the preservation of the conserved quantities of the CNLS system. In [6] the CNLS system with periodic boundary conditions was discretized in space using second-order finite differences and the resulting system of ordinary differential equations (ODEs) are integrated in time using the so-called Hopscotch method, a mixture of an explicit and an implicit method. The accuracy of the solutions were checked only using the norm conservation of each amplitude of the CNLS system. The choice of particular initial conditions may affect the preservation of some invariants. As mentioned in [10] in the context of multi-symplectic integrators usually symmetric initial conditions are used in the literature. Unsymmetric initial conditions may lead to an increase of errors in preserving some invariants like momentum over time [10].

The numerical solution of nonlinear wave equations using symplectic and multi-symplectic geometric integrators has been the subject of many studies in recent years (see for a recent review [11]). The NLS and CNLS systems represent an infinite-dimensional Hamiltonian system. After semi-discretization in the space variables one usually obtains a system of Hamiltonian ordinary differential equations, for which various symplectic integrators can be applied [8], [12] and [13]. In Section 2 we give a symplectic integrator based on the splitting of the semi-discretized Hamiltonian system in linear and nonlinear parts. Another way is to use the multi-symplectic structure of the NLS and CNLS equations and apply multi-symplectic integrators [9], [14] and [15]. A brief review of multi-symplectic integrators and their application to the CNLS system given in Section3. In Section 4 numerical results obtained with the symplectic splitting method and multi-symplectic six-point Preissman scheme are compared with results found in the literature for symmetric and unsymmetric initial conditions. Section 5 is devoted to concluding remarks.