Approximate Nonlocal Symmetries for a Perturbed Schrödinger Equation with a Weak Infinite Power-Law Memory

A nonlocally perturbed linear Schrödinger equation with a small parameter was derived under the assumption of low-level fractionality by using one of the known general nonlocal wave equations with an infinite power-law memory. The problem of finding approximate symmetries for the equation is studied...

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Bibliographic Details
Published in:AppliedMath 2022-12, Vol.2 (4), p.585-608
Main Author: Lukashchuk, Stanislav Yu
Format: Article
Language:English
Subjects:
approximate nonlocal symmetry
infinite memory
nonlocal perturbation
Schrödinger equation
small parameter
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Summary:A nonlocally perturbed linear Schrödinger equation with a small parameter was derived under the assumption of low-level fractionality by using one of the known general nonlocal wave equations with an infinite power-law memory. The problem of finding approximate symmetries for the equation is studied here. It has been shown that the perturbed Schrödinger equation inherits all symmetries of the classical linear equation. It has also been proven that approximate symmetries corresponding to Galilean transformations and projective transformations of the unperturbed equation are nonlocal. In addition, a special class of nonlinear, nonlocally perturbed Schrödinger equations that admits an approximate nonlocal extension of the Galilei group is derived. An example of constructing an approximately invariant solution for the linear equation using approximate scaling symmetry is presented.
ISSN:2673-9909
2673-9909
DOI:10.3390/appliedmath2040034