5 research outputs found
Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
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
Symmetry operators of the two-component GrossβPitaevskii equation with a Manakov-type nonlocal nonlinearity
We consider an integro-differential 2-component multidimensional Gross-Pitaevskii equation with a Manakov-type cubic nonlocal nonlinearity. In the framework of the WKB-Maslov semiclassical formalism, we obtain a semiclassically reduced 2-component nonlocal Gross- Pitaevskii equation determining the leading term of the semiclassical asymptotic solution. For the reduced Gross-Pitaevskii equation we construct symmetry operators which transform arbitrary solution of the equation into another solution. Constructing the symmetry operator is based on the Cauchy problem solution technique and uses an intertwining operator which connects two solutions of the reduced Gross-Pitaevskii equation. General structure of the symmetry operator is illustrated with a 1D case for which a family of symmetry operators is found explicitly and a set of exact solutions is generated
Symmetry operators of the two-component GrossβPitaevskii equation with a Manakov-type nonlocal nonlinearity
We consider an integro-differential 2-component multidimensional Gross-Pitaevskii equation with a Manakov-type cubic nonlocal nonlinearity. In the framework of the WKB-Maslov semiclassical formalism, we obtain a semiclassically reduced 2-component nonlocal Gross- Pitaevskii equation determining the leading term of the semiclassical asymptotic solution. For the reduced Gross-Pitaevskii equation we construct symmetry operators which transform arbitrary solution of the equation into another solution. Constructing the symmetry operator is based on the Cauchy problem solution technique and uses an intertwining operator which connects two solutions of the reduced Gross-Pitaevskii equation. General structure of the symmetry operator is illustrated with a 1D case for which a family of symmetry operators is found explicitly and a set of exact solutions is generated