Exact solutions and symmetry operators for the nonlocal Gross-Pitaevskii equation with quadratic potential

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The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross- Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution) are obtained for the equation under consideration. General constructions are illustrated by examples.

Original languageEnglish
Article number007
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Publication statusPublished - 1 Jan 2005



  • Gross-pitaevskii equation
  • Nonlinear evolution operator
  • Semiclassical asymptotics
  • Symmetry operators
  • The cauchy problem
  • Trajectory concentrated functions
  • Wkb-maslov complex germ method

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Geometry and Topology

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