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.
In: Siberian Mathematical Journal, Vol. 46, No. 1, 01.01.2005, p. 119-132.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 -