### Algorithm of the Parallel Sweep Method for Numerical Solution of the Gross–Pitaevskii Equation with Highest Nonlinearities

#### Abstract

In this paper, we for the first time introduce a numerical scheme the solution of a nonlinear equation of the Gross–Pitaevskii type (GP) or the nonlinear Schrodinger equation (NLSE) with highest nonlinearities, which provides implementation of a complete set of motion integrals. This scheme was parallelly implemented on a non-uniform grid. Propagation of a ring laser beam with non-zero angular momentum in the filamentation mode is studied using the implemented numerical scheme. It is shown, that filaments under exposure to centrifugal forces escape to the periphery. Based on a number of numerical experiments, we have found the universal property of motion integrals in the non-conservative case for a given class of equations. Research of dynamics of angular momentum for a dissipative case are also presented. We found, that angular moment, particularly normed by initial energy during filamentation process, is quasi-constant.

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