### 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 |
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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)