Aguilera Martinez, Axi Fabricio

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Método conservativo de diferencias finitas para sistemas de ecuaciones de Schrödinger no lineales
(2022-06-28) Aguilera Martinez, Axi Fabricio; Castillo, Paul; College of Arts and Sciences - Sciences; Cruz, Angel; Ramos, Rafael; Department of Mathematics; Irizarry, Zollianne
We present a high-order finite difference method to approximate the solution of a strongly coupled general system of nonlinear Schrödinger equations. From the symmetry of the discrete Laplacian and a special discretization of the nonlinear potential, exact conservation of two important physical quantities, power of each component and system's Hamiltonian, is proved for the fully discrete problem. The time-advance technique is based on a modification of the conservative Crank-Nicolson scheme by decomposing the original system into a sequence of smaller nonlinear problems, associated to each component of the complex field. For a quadratic potential birefringent system, a method, based on approximations for the second derivative by a class of symmetric finite difference formulas, is formally proven to converge with order $\tau + h^{2p}$, with $p = 1, ..., 4$. Using the composition theory of numerical methods for differential equations, schemes of order $\tau^{2q} + h^{2p}$, with $q = 1, ..., 4$, are derived. The conservation of discrete invariants; accuracy of the proposed method, and various composite methods, is validated by a series of numerical experiments for systems with different nonlinearities.