A hybrid variational method of describing pulse splitting by dispersion management

Variational methods have become a widely used tool in the search for solitary waves in nonlinear systems. If a system exhibits fast oscillations in one of the degrees of freedom, its wavefunction can split into two or more separate parts and 'orthodox' variational descriptions are inadequate to include such nontrivial dynamics. A candidate for somewhat more realistic description is the hybrid variational method introduced by Edwards et al (2005 J. Phys. B: At. Mol. Opt. Phys. 38 363), which to our knowledge has not yet been used for a dynamical situation. Here we investigate an application of the hybrid method to a two-dimensional system with dispersion management, where pulse splitting is known to occur. By comparison with full numerical simulations, we conclude that in some cases these methods can give an improvement in the proper description of the evolution, but we also find an example where the hybrid methods fail. Edwards' method is thus evaluated in practical terms. It is hoped that this will encourage its wider use. The quest for multidimensional solitons in nonlinear optical systems and Bose–Einstein condensates has recently attracted some attention [1]. One of the most efficient methods used in the search for solitons in complex systems is the variational approximation (VA) [2]. This technique simplifies a numerical study of these nonlinear systems, since upon using it one has to deal with ordinary rather than partial differential equations. This simplification becomes crucial if one needs to numerically search the phase space for regions of stability. Moreover, in many cases the predictions obtained from a variational analysis agree very well with the full numerical simulations. The VA was first introduced for solitons in plasma physics by Bondeson, Lisak and Anderson [3]. Next it was applied for solitons in optical fibres in the paper by Anderson [4] and further developed in the more general context by Anderson, Lisak and Reichel [5]. These efforts became the basis for the rapid development of analytical methods in nonlinear optics based on the variational approximation. We illustrate the main steps of this approximation using the nonlinear Schrödinger equation i ∂u ∂z + 1 2 ∂ 2 u ∂t 2 + |u| 2 u = 0, (1)