### 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 | |

Publication status | Published - 1 Jan 2005 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Symmetry operators of a Hartree-type equation with quadratic potential.** / Lisok, Alexander Leonidovich; Trifonov, A. Yu; Shapovalov, Aleksandr Vasilievich.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Symmetry operators of a Hartree-type equation with quadratic potential

AU - Lisok, Alexander Leonidovich

AU - Trifonov, A. Yu

AU - Shapovalov, Aleksandr Vasilievich

PY - 2005/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=15044346408&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15044346408&partnerID=8YFLogxK

U2 - 10.1007/s11202-005-0013-2

DO - 10.1007/s11202-005-0013-2

M3 - Article

VL - 46

SP - 119

EP - 132

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 1

ER -