Computer Physics Communications 177 (2007) 566–583 www.elsevier.com/locate/cpc Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions A. Aydın a , B. Karasözen b,∗ a Department of Mathematics, Atılım University, 06836 Ankara, Turkey b Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey Received 20 July 2006; received in revised form 18 May 2007; accepted 22 May 2007 Available online 5 June 2007 Abstract We consider for the integration of coupled nonlinear Schrödinger equations with periodic plane wave solutions a splitting method from the class of symplectic integrators and the multi-symplectic six-point scheme which is equivalent to the Preissman scheme. The numerical experiments show that both methods preserve very well the mass, energy and momentum in long-time evolution. The local errors in the energy are computed according to the discretizations in time and space for both methods. Due to its local nature, the multi-symplectic six-point scheme preserves the local invariants more accurately than the symplectic splitting method, but the global errors for conservation laws are almost the same.  2007 Elsevier B.V. All rights reserved. PACS: 02.70.Bf; 42.65.Sf Keywords: Coupled nonlinear Schrödinger equation; Periodic waves; Symplectic and multi-symplectic methods; Splitting 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 i ∂ψ 1 ∂t + α 1 ∂ 2 ψ 1 ∂x 2 + ( σ 1 |ψ 1 | 2 + v 12 |ψ 2 | 2 ) ψ 1 = 0, (1) i ∂ψ 2 ∂t + α 2 ∂ 2 ψ 2 ∂x 2 + ( σ 2 |ψ 2 | 2 + v 21 |ψ 1 | 2 ) ψ 2 = 0, where ψ 1 (x,t) and ψ 2 (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 [1,2] and geophysical fluid dynamics [3,4]. The parameters α j are the dispersion coefficients, σ j the Landau constants which describe the self-modulation of the wave packets, and v 12 and v 21 are the wave–wave interaction coefficients which describe the cross-modulations of the wave packets [5,6]. 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. α 1 = α 2 = 1, σ 1 = σ 2 = v 12 = v 21 . Other integrable cases are: α 1 = α 2 , σ 1 = σ 2 = v 12 = v 21 and α 1 =−α 2 , σ 1 = σ 2 =−v 12 =−v 21 which can be solved by the inverse scattering method [7]. * Corresponding author. E-mail address: bulent@metu.edu.tr (B. Karasözen). 0010-4655/$ – see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2007.05.010