Recurrence in semidiscrete approximations of the nonlinear Schro¨dinger equation
SIAM Journal on Scientific and Statistical Computing
Numerical simulation of nonlinear Schro¨dinger systems: a new conservative scheme
Applied Mathematics and Computation
Persistence of travelling wave solutions of a fourth order diffusion system
Journal of Computational and Applied Mathematics
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New second-and fourth-order accurate semianalytical methods are introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. These methods are based on the combination of the Method of Lines and the Lanczos's Tau Method. The methods are self-starting and proved to be stable, accurate, and energy conservative for long time-integration periods. Approximate solutions are sought, on segmented sets of parallel lines, as finite expansions in terms of a given orthogonal polynomial basis. We have carried out numerical application concerning several cases for the propagation, collision and the bound states of N solitons, 2 ≤ N ≤ 5. Accurate results have been obtained using both shifted Chebyshev and Legendre polynomials. These results compare competitively with some other published results obtained using different methods.