Published: 2018-12-28

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

Andrey D. Bulygin


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.

Full Text:



Balashov, A.D., Pergament A.Kh.: Mathematical modeling of femtosecond pulse propagation. Matem. Mod. 18(4), 3–18 (2006) (in Russian)

Bulygin, A.D., Zemlyanov, A.A.: Fully conservative numerical scheme for nonlinear Schrodinger equation with higher nonlinearities. Computational technologies 22(5), 3–13 (2017)

Tadmor E.: A review of numerical methods for nonlinear partial differential equations. Bulletin of the American Mathematical Society 49(4), 507–554 (2012), DOI: 10.1090/S0273-0979-2012-01379-4

Terekhov, A.V.: A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method. Parallel Computing 39(6–7), 245–258 (2013), DOI: 10.1016/j.parco.2013.03.003

Zemlyanov A.A., Bulygin A.D.: Analysis of Some Properties of the Nonlinear Schrodinger Equation Used for Filamentation Modeling. Russian Physics Journal 61(2), 357—363 (2018), DOI: 10.1007/s11182-018-1407-5

Publishing Center of South Ural State University (454080, Lenin prospekt, 76, Chelyabinsk, Russia)