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A universal separatrix map for weak interactions of solitary waves in generalized nonlinear Schrödinger equations

Yi Zhu Center for Applied Mathematics (ZCAM), Tsinghua University, Beijing 100084, China Richard Haberman Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA Jianke Yang Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, USA

TBD mathscidoc:2204.43002

2008.10
It is known that weak interactions of two solitary waves in generalized nonlinear Schrödinger (NLS) equations exhibit fractal dependence on initial conditions, and the dynamics of these interactions is governed by a universal two-degree-of-freedom ODE system [Y. Zhu J. Yang, Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations, Phys. Rev. E 75 (2007) 036605]. In this paper, this ODE system is analyzed comprehensively. Using asymptotic methods along separatrix orbits, a simple second-order map is derived. This map does not have any free parameters after variable rescalings, and thus is universal for all weak interactions of solitary waves in generalized NLS equations. Comparison between this map’s predictions and direct simulations of the ODE system shows that the map can capture the fractal-scattering phenomenon of the ODE system very well both qualitatively and quantitatively.
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@inproceedings{yi2008a,
  title={A universal separatrix map for weak interactions of solitary waves in generalized nonlinear Schrödinger equations},
  author={Yi Zhu, Richard Haberman, and Jianke Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220421110822147130078},
  year={2008},
}
Yi Zhu, Richard Haberman, and Jianke Yang. A universal separatrix map for weak interactions of solitary waves in generalized nonlinear Schrödinger equations. 2008. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220421110822147130078.
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