Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 019, 17 pages      math-ph/0511081

Transverse Evolution Operator for the Gross-Pitaevskii Equation in Semiclassical Approximation

Alexey Borisov a, Alexander Shapovalov a, b, c and Andrey Trifonov b, c
a) Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
b) Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia
c) Math. Phys. Laboratory, Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia

Received July 27, 2005, in final form November 13, 2005; Published online November 22, 2005

The Gross-Pitaevskii equation with a local cubic nonlinearity that describes a many-dimensional system in an external field is considered in the framework of the complex WKB-Maslov method. Analytic asymptotic solutions are constructed in semiclassical approximation in a small parameter h, h -> 0, in the class of functions concentrated in the neighborhood of an unclosed surface associated with the phase curve that describes the evolution of surface vertex. The functions of this class are of the one-soliton form along the direction of the surface normal. The general constructions are illustrated by examples.

Key words: WKB-Maslov complex germ method; semiclassical asymptotics; Gross-Pitaevskii equation; solitons; symmetry operators.

pdf (641 kb)   ps (661 kb)   tex (643 kb)


  1. Gross E.P., Structure of a quantized vortex in boson systems, Nuovo Cimento, 1961, V.20, N 3, 454-477.
  2. Pitaevskii L.P., Vortex lines in an imperfect Bose gas, Zh. Eksper. Teor. Fiz., 1961, V.40, 646-651.
  3. Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevsky L.P., Theory of solitons: the inverse scattering method, Moscow, Nauka, 1980 (English transl.: New York, Plenum, 1984).
  4. Zakharov V.E., Shabat A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Zh. Eksper. Teor. Fiz., 1971, V.61, 118-134 (English transl.: Soviet Physics JETP, 1972, V.34, 62-69).
  5. Hasegawa A., Tappert F., Transmission of stationary nonlinear optical pulse in dispersive dielectric fibres, Appl. Phys. Lett., 1973, V.23, 171-172.
  6. Mollenauer L.F., Stolen R.H., Gordon J.P., Experimental observation of picosecond pulse narrowing and solitons in optical fibres, Phys. Rev. Lett., 1980, V.45, 1095-1098.
  7. Yuen H.C., Lake B.M., Nonlinear dynamics of deep-water gravity waves, Advances in Applied Mechanics, 1982, V.22, 67-229.
  8. Dalfovo F., Giorgini S., Pitaevskii L., Stringary S., Theory of Bose-Einstein condensation in traped gases, Rev. Mod. Phys., 1999, V.71, N 3, 463-512.
  9. Zakharov V.E., Synakh V.S., On the character of a singularity under self-focusing, Zh. Eksper. Teor. Fiz., 1975, V.68, 940-947.
  10. Karasev M.V., Maslov V.P., Nonlinear Poisson brackets: geometry and quantization, Moscow, Nauka, 1991 (English transl.: Ser. Traslations of Mathematical Monographs, Vol.119, Providence, RI, Amer. Math. Soc., 1993).
  11. Belov V.V., Trifonov A.Yu., Shapovalov A.V., The trajectory-coherent approximation and the system of moments for the Hartree type equation, Int. J. Math. and Math. Sci., 2002, V.32, N 6, 325-370.
  12. Lisok A.L., Trifonov A.Yu., Shapovalov A.V., The evolution operator of the Hartree-type equation with a quadratic potential, J. Phys. A.: Math. Gen., 2004, V.37, 4535-4556.
  13. Maslov V.P., Complex Markov chains and the Feynman path integral, Moscow, Nauka, 1976.
  14. Maslov V.P., Equations of the self-consistent field, Itogi Nauki Tekhn. Ser. Sovrem. Probl. Mat., 1978, V.11, Moscow, VINITI, 153-234 (English transl.: J. Soviet Math., 1979, V.11, 123-195).
  15. Karasev M.V., Maslov V.P., Algebras with general commutation relations and their applications. II. Unitary-nonlinear operator equations, Itogi Nauki Tekhn. Ser. Sovrem. Probl. Mat., 1979, V.13, Moscow, VINITI, 145-267 (English transl.: J. Soviet Math., 1981, V.15, 273-368).
  16. Maslov V.P., Quantization of thermodynamics and ultrasecondary quantization, Moscow, Computer Sciences Institute Publ., 2001.
  17. Maslov V.P., Shvedov O.Yu., Quantization in the vicinity of the classical solutions in N-particle problem and superfluidity, Teor. Mat. Fiz., 1994, V.98, N 2, 266-288.
  18. Maslov V.P., Shvedov O.Yu., The complex germ method for the Fock space. I. Asymptotics like wave packets, Teor. Mat. Fiz., 1995, V.104, N 2, 310-330.
  19. Maslov V.P., Shvedov O.Yu., The complex germ method for the Fock space. II. Asymptotics, corresponding to finite-dimensional isotropic manifolds, Teor. Mat. Fiz., 1995, V.104, N 3, 479-508.
  20. Maslov V.P., Shvedov O.Yu., The complex germ method in multiparticle problems and in quantum field theory, Moscow, URSS, 2000.
  21. Maslov V.P., The canonical operator on the Lagrangian manifold with a complex germ and a regularizer for pseudodifferential operators and difference schemes, Dokl. Akad. Nauk SSSR, 1970, V.195, N 3, 551-554.
  22. Maslov V.P., The complex WKB method for nonlinear equations, Moscow, Nauka, 1977 (English transl.: The complex WKB method for nonlinear equations. I. Linear theory, Basel - Boston - Berlin, Birkhauser Verlag, 1994).
  23. Maslov V.P., Operational methods, Moscow, Nauka, 1973 (English transl.: Moscow, Mir, 1976).
  24. Belov V.V., Dobrokhotov S.Yu., Semiclassical Maslov asymptotics with complex phases. I. General appoach, Teor. Mat. Fiz., 1992, V.130, N 2, 215-254 (English transl.: Theor. Math. Phys., 1992, V.92, N 2, 843-868).
  25. Bagrov V.G., Belov V. V., Ternov I.M. Quasiclassical trajectory-coherent states of a nonrelativistic particle in an arbitrary electromagnetic field, Teor. Mat. Fiz., 1982, V.50, 390-396.
  26. Bagrov V.G., Belov V.V., Ternov I.M., Quasiclassical trajectory-coherent states of a particle in an arbitrary electromagnetic field, J. Math. Phys., 1983, V.24, N 12, 2855-2859.
  27. Bagrov V.G., Belov V.V., Trifonov A.Yu., Semiclassical trajectory-coherent approximation in quantum mechanics: I. High order corrections to multidimensional time-dependent equations of Schrödinger type, Ann. Phys. (NY), 1996, V.246, N 2, 231-280.
  28. Shapovalov A.V., Trifonov A.Yu., Semiclassical solutions of the nonlinear Schrödinger equation, J. Nonlinear Math. Phys., 1999, V.6, N 2, 1-12.
  29. Malkin M.A., Man'ko V.I., Dynamic symmetries and coherent states of quantum systems, Moscow, Nauka, 1979.
  30. Popov M.M., Green functions for Schrödinger equation with quadratic potential, Problemy Mat. Fiz., 1973, N 6, 119-125.
  31. Dodonov V.V., Malkin I.A., Man'ko V.I., Integrals of motion, Green functions and coherent states of dynamic systems, Intern. J. Theor. Phys., 1975, V.14, N 1, 37-54.
  32. Bang O., Krolikowski W., Wyller J., Rasmussen J.J., Collapse arrest and soliton stabilization in nonlocal nonlinear media, nlin.PS/0201036.
  33. Shvedov O.Yu., Semiclasical symmetries, Ann. Phys., 2002, V.296, 51-89.

Previous article   Next article   Contents of Volume 1 (2005)