New second- and fourth-order accurate numerical schemes for the nonlinear cubic Schrödinger equation.

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. [ABSTRACT FROM AUTHOR]