Symmetry operators of a Hartree-type equation with quadratic potential

Abstract

We study the symmetry properties of a nonstationary one-dimensional Hartree-type equation with quadratic periodic potential and nonlocal nonlinearity. We find an explicit form of a nonlinear evolution operator for this equation and obtain a solution to a Cauchy problem in the class of semiclassically concentrated functions. We find parametric families of nonlinear symmetry operators of a Hartree-type equation (keeping invariant the set of solutions to this equation). Using the symmetry operators, we construct families of exact solutions to the equation. This approach constructively extends the ideas and methods of group analysis to the case of nonlinear integro-differential equations.

Original language	English
Pages (from-to)	119-132
Number of pages	14
Journal	Siberian Mathematical Journal
Volume	46
Issue number	1
DOIs	https://doi.org/10.1007/s11202-005-0013-2
Publication status	Published - 1 Jan 2005

Keywords

Evolution operator
Hartree-type equation
Nonlinear equations
Semi-classical concentrated states
Symmetry operators

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1007/s11202-005-0013-2

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Cite this

Lisok, A. L., Trifonov, A. Y., & Shapovalov, A. V. (2005). Symmetry operators of a Hartree-type equation with quadratic potential. Siberian Mathematical Journal, 46(1), 119-132. https://doi.org/10.1007/s11202-005-0013-2

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abstract = "We study the symmetry properties of a nonstationary one-dimensional Hartree-type equation with quadratic periodic potential and nonlocal nonlinearity. We find an explicit form of a nonlinear evolution operator for this equation and obtain a solution to a Cauchy problem in the class of semiclassically concentrated functions. We find parametric families of nonlinear symmetry operators of a Hartree-type equation (keeping invariant the set of solutions to this equation). Using the symmetry operators, we construct families of exact solutions to the equation. This approach constructively extends the ideas and methods of group analysis to the case of nonlinear integro-differential equations.",

keywords = "Evolution operator, Hartree-type equation, Nonlinear equations, Semi-classical concentrated states, Symmetry operators",

author = "Lisok, {Alexander Leonidovich} and Trifonov, {A. Yu} and Shapovalov, {Aleksandr Vasilievich}",

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AU - Trifonov, A. Yu

AU - Shapovalov, Aleksandr Vasilievich

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Y1 - 2005/1/1

N2 - We study the symmetry properties of a nonstationary one-dimensional Hartree-type equation with quadratic periodic potential and nonlocal nonlinearity. We find an explicit form of a nonlinear evolution operator for this equation and obtain a solution to a Cauchy problem in the class of semiclassically concentrated functions. We find parametric families of nonlinear symmetry operators of a Hartree-type equation (keeping invariant the set of solutions to this equation). Using the symmetry operators, we construct families of exact solutions to the equation. This approach constructively extends the ideas and methods of group analysis to the case of nonlinear integro-differential equations.

AB - We study the symmetry properties of a nonstationary one-dimensional Hartree-type equation with quadratic periodic potential and nonlocal nonlinearity. We find an explicit form of a nonlinear evolution operator for this equation and obtain a solution to a Cauchy problem in the class of semiclassically concentrated functions. We find parametric families of nonlinear symmetry operators of a Hartree-type equation (keeping invariant the set of solutions to this equation). Using the symmetry operators, we construct families of exact solutions to the equation. This approach constructively extends the ideas and methods of group analysis to the case of nonlinear integro-differential equations.

KW - Evolution operator

KW - Hartree-type equation

KW - Nonlinear equations

KW - Semi-classical concentrated states

KW - Symmetry operators

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