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We introduce a solvable model for phase transitions in solitons, demonstrating both first and second-kind symmetry-breaking bifurcations. This model provides exact solutions for solitons in combined linear-nonlinear potentials.

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Area of Science:

  • Nonlinear dynamics
  • Soliton physics
  • Phase transitions

Background:

  • Analytically solvable models are crucial for understanding phase transitions and bifurcations.
  • Existing models primarily address phase transitions in uniform media, not localized states like solitons.
  • Integrable equations producing solitons typically lack intrinsic phase transitions.

Purpose of the Study:

  • To introduce a novel solvable model for symmetry-breaking phase transitions of solitons.
  • To investigate both first-kind (subcritical) and second-kind (supercritical) bifurcations for solitons.
  • To analyze soliton behavior in a combined linear-nonlinear double-well potential.

Main Methods:

  • Development of an analytically solvable model.
  • Mathematical analysis of solitons in a potential represented by a symmetric pair of delta-functions.
  • Consideration of both self-focusing and self-defocusing nonlinearities.

Main Results:

  • Exact solutions derived for symmetric and asymmetric solitons in the self-focusing case.
  • Demonstration of a switch between first and second-kind symmetry-breaking transitions.
  • Observation of a second-kind transition breaking antisymmetry in the first excited state for the self-defocusing model.

Conclusions:

  • The developed model successfully describes symmetry-breaking phase transitions for solitons.
  • The model offers exact solutions, providing benchmarks for soliton dynamics in complex potentials.
  • This work extends the understanding of phase transitions to localized nonlinear phenomena.