Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

Abstract

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.

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• Supersolitons: Solitonic excitations in atomic soliton chains
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Cited by in corpus (7)

• Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians
• Intertwining operators for non self-adjoint Hamiltonians and bicoherent states
• Bi-squeezed states arising from pseudo-bosons
• Generalized Heisenberg algebra and (non linear) pseudo-bosons
• Non self-adjoint Hamiltonians with complex eigenvalues
• Gibbs states defined by biorthogonal sequences
• Heisenberg dynamics for non self-adjoint Hamiltonians: symmetries and derivations