Lagrangian modelling of nonlinear waves in optical fibres

Abstract

The Lagrangian perturbation method for the NLS is revisited in the form of an equivalent direct problem. The analogy can be extended to arbitrarily perturbed systems. It is then possible to provide first order perturbation expansions for the fundamental soliton. The case of the damped NLS is considered and shown to fully comply with IST predictions. Subsequently the problem of NLS initial condition not corresponding to an exact soliton is examined. There are two issues that need to be considered: the location of the soliton solution and the modelling of the continuum. The location of the soliton solution is handled by considering the integrals of motion of the NLS. The improvement arises by the inclusion of the contributions due to the continuum. The results are compared with numerical calculations and are proved to be satisfactory provided that the initial pulse shape does not depart greatly from the Asech(z) functional form. The propagation problem is handled by considering the evolution of the soliton and the continuum separately and recombining them at the required time. Two cases are considered: the far field pattern and the position where the peak of the soliton lies. For the former the recombination of continuum with the soliton is achieved with the help of the inverse part of the IST. For the peak position a Bäcklund transform is considered. Results from both regimes are compared with numerical results and shown to agree satisfactorily