Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be differe...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Vyacheslav Trofimov, Maria Loginova
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	two nonlinear coupled Schrödinger equations conservation laws finite-difference scheme split-step method approximation order Hamiltonian Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/2a930017d61443daa7baa0cddcf28190
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:2a930017d61443daa7baa0cddcf28190
record_format	dspace
spelling	oai:doaj.org-article:2a930017d61443daa7baa0cddcf281902021-11-11T18:16:20ZConservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants10.3390/math92127162227-7390https://doaj.org/article/2a930017d61443daa7baa0cddcf281902021-10-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2716https://doaj.org/toc/2227-7390Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be different. The influence of each of both invariants on the computer simulation result accuracy is analyzed while solving the problem describing the third optical harmonic generation process. Two implicit conservative FDSs are developed for a numerical solution of this problem. One of them preserves a difference analog of the energy invariant (or the Hamiltonian) accurately, while the Hamiltonian (or the energy’s invariant) is preserved with the second order of accuracy. Both FDSs possess the second order of approximation at a smooth enough solution of the differential problem. Computer simulations demonstrate advantages of the implicit FDS preserving the Hamiltonian. To illustrate the advantages of the developed FDSs, a comparison of the computer simulation results with those obtained applying the Strang method, based on either an implicit scheme or the Runge–Kutta method, is made. The corresponding theorems, which claim the second order of approximation for preserving invariants for the FDSs under consideration, are stated.Vyacheslav TrofimovMaria LoginovaMDPI AGarticletwo nonlinear coupled Schrödinger equationsconservation lawsfinite-difference schemesplit-step methodapproximation orderHamiltonianMathematicsQA1-939ENMathematics, Vol 9, Iss 2716, p 2716 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	two nonlinear coupled Schrödinger equations conservation laws finite-difference scheme split-step method approximation order Hamiltonian Mathematics QA1-939
spellingShingle	two nonlinear coupled Schrödinger equations conservation laws finite-difference scheme split-step method approximation order Hamiltonian Mathematics QA1-939 Vyacheslav Trofimov Maria Loginova Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants
description	Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be different. The influence of each of both invariants on the computer simulation result accuracy is analyzed while solving the problem describing the third optical harmonic generation process. Two implicit conservative FDSs are developed for a numerical solution of this problem. One of them preserves a difference analog of the energy invariant (or the Hamiltonian) accurately, while the Hamiltonian (or the energy’s invariant) is preserved with the second order of accuracy. Both FDSs possess the second order of approximation at a smooth enough solution of the differential problem. Computer simulations demonstrate advantages of the implicit FDS preserving the Hamiltonian. To illustrate the advantages of the developed FDSs, a comparison of the computer simulation results with those obtained applying the Strang method, based on either an implicit scheme or the Runge–Kutta method, is made. The corresponding theorems, which claim the second order of approximation for preserving invariants for the FDSs under consideration, are stated.
format	article
author	Vyacheslav Trofimov Maria Loginova
author_facet	Vyacheslav Trofimov Maria Loginova
author_sort	Vyacheslav Trofimov
title	Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants
title_short	Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants
title_full	Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants
title_fullStr	Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants
title_full_unstemmed	Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants
title_sort	conservative finite-difference schemes for two nonlinear schrödinger equations describing frequency tripling in a medium with cubic nonlinearity: competition of invariants
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/2a930017d61443daa7baa0cddcf28190
work_keys_str_mv	AT vyacheslavtrofimov conservativefinitedifferenceschemesfortwononlinearschrodingerequationsdescribingfrequencytriplinginamediumwithcubicnonlinearitycompetitionofinvariants AT marialoginova conservativefinitedifferenceschemesfortwononlinearschrodingerequationsdescribingfrequencytriplinginamediumwithcubicnonlinearitycompetitionofinvariants
_version_	1718431872448462848

Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

Ejemplares similares