Publication:
Método conservativo de diferencias finitas para sistemas de ecuaciones de Schrödinger no lineales
Método conservativo de diferencias finitas para sistemas de ecuaciones de Schrödinger no lineales
dc.contributor.advisor | Castillo, Paul | |
dc.contributor.author | Aguilera Martinez, Axi Fabricio | |
dc.contributor.college | College of Arts and Sciences - Sciences | en_US |
dc.contributor.committee | Cruz, Angel | |
dc.contributor.committee | Ramos, Rafael | |
dc.contributor.department | Department of Mathematics | en_US |
dc.contributor.representative | Irizarry, Zollianne | |
dc.date.accessioned | 2022-07-05T13:39:18Z | |
dc.date.available | 2022-07-05T13:39:18Z | |
dc.date.issued | 2022-06-28 | |
dc.description.abstract | 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. | en_US |
dc.description.abstract | Presentamos un método de diferencias nitas de alto orden para aproximar la solución de un sistema general de ecuaciones no lineales de Schrödinger fuertemente acoplado. Conservación exacta de dos importantes cantidades físicas, potencia de cada componente y Hamiltoniano del sistema, es demostrada para el problema completamente discreto a partir de la simetría del Laplaciano discreto y de una discretizaci ón especial del potencial no lineal. La técnica de avance en tiempo se basa en una modi cación del esquema conservativo de Crank-Nicolson, descomponiendo el sistema original en una secuencia de problemas no lineales de menor tamaño, asociados a cada componente del campo complejo. Para un sistema birrefrigente de potencial cuadrático, se prueba formalmente que un método, basado en aproximaciones para la segunda derivada mediante una clase de fórmulas simétricas de diferencias nitas, converge con orden τ + h2p, con p = 1, ..., 4. Utilizando la teor ía de composición de métodos numéricos para ecuaciones diferenciales, se derivan esquemas de orden τ 2q + h2p, con q = 1, ..., 4. La conservación de invariantes discretos; la precisión del método propuesto, y de distintos métodos compuestos, se valida mediante una serie de experimentos numéricos para sistemas con diferentes no linealidades. | en_US |
dc.description.graduationSemester | Spring | en_US |
dc.description.graduationYear | 2023 | en_US |
dc.identifier.uri | https://hdl.handle.net/20.500.11801/2919 | |
dc.language.iso | es | en_US |
dc.rights.holder | (c) 2022 Axi Fabricio Aguilera Martinez | en_US |
dc.subject | Coupled nonlinear Schrödinger systems | en_US |
dc.subject | Coupled Gross-Pitaevskii equations | en_US |
dc.subject | Conservative methods | en_US |
dc.subject | High order finite differences | en_US |
dc.subject | Power (mass) and Hamiltonian (energy) conservation | en_US |
dc.subject | Splitting and composition methods | en_US |
dc.subject.lcsh | Schrödinger equation | en_US |
dc.subject.lcsh | Gross-Pitaevskii equations | en_US |
dc.subject.lcsh | Hamilton-Jacobi equations | en_US |
dc.subject.lcsh | Differential equations, Nonlinear--Numerical solutions | en_US |
dc.subject.lcsh | Partial differential equations (second and higher orders) | en_US |
dc.title | Método conservativo de diferencias finitas para sistemas de ecuaciones de Schrödinger no lineales | en_US |
dc.type | Thesis | en_US |
dspace.entity.type | Publication | |
thesis.degree.discipline | Applied Mathematics | en_US |
thesis.degree.level | M.S. | en_US |
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