Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

• Дата: 2013
• Автори: Lisok, A.L., Shapovalov, A.V., Trifonov, A.Y.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2013
• Назва видання: Symmetry, Integrability and Geometry: Methods and Applications
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/149358
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation / A.L. Lisok, A.V. Shapovalov, A.Y. Trifonov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 34 назв. — англ.

We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.