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Hidetsugu Sakaguchi1, Boris A Malomed2

  • 1Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan.

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Summary
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This study extends the concept of solitons to include weakly singular solutions in nonlinear Schrödinger equations (NLSEs). These novel singular solitons, arising from self-repulsion, exhibit surprising stability and offer new insights into nonlinear physics.

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Area of Science:

  • Nonlinear Optics
  • Mathematical Physics
  • Soliton Theory

Background:

  • Solitons are typically nonsingular solutions arising from nonlinear self-attraction and linear dispersion.
  • Existing models primarily focus on attractive nonlinearities for soliton formation.

Purpose of the Study:

  • To extend the definition of solitons to include weakly singular solutions.
  • To investigate the generation and stability of these singular solitons in various dimensions.
  • To provide a physical interpretation for these counterintuitive states.

Main Methods:

  • Analysis of one-, two-, and three-dimensional nonlinear Schrödinger equations (NLSEs) with septimal, quintic, and cubic nonlinear terms.
  • Numerical simulations to determine soliton shapes and stability.
  • Analytical calculations for asymptotic behavior.

Main Results:

  • Demonstrated the existence of stable, weakly singular solitons generated by self-repulsion.
  • Interpreted singular solitons as screened attractive potentials.
  • Found complete stability for singular modes in 1D and 3D, predicted by the anti-Vakhitov-Kolokolov criterion.
  • Identified unstable singular solitons with vorticity in 2D NLSEs.

Conclusions:

  • The concept of solitons can be broadened to encompass weakly singular states.
  • Self-repulsive nonlinearities can generate stable, singular soliton solutions.
  • Singular solitons provide a new avenue for understanding nonlinear wave phenomena.