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Nonintegrable semidiscrete Hirota equation: gauge-equivalent structures and dynamical properties.

Li-Yuan Ma1, Zuo-Nong Zhu1

  • 1Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, P. R. China.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 15, 2014
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Summary
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This study explores nonintegrable semidiscrete Hirota equations, revealing their gauge equivalence to Heisenberg ferromagnet equations. Exact spatial periodic solutions were found, demonstrating richer dynamics than nonlinear Schrödinger equations.

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Area of Science:

  • Integrable systems
  • Mathematical physics
  • Nonlinear dynamics

Background:

  • The study of nonintegrable systems is crucial for understanding complex phenomena.
  • Hirota equations are fundamental in nonlinear science.
  • Gauge equivalence offers a powerful tool for analyzing dynamical systems.

Purpose of the Study:

  • To investigate gauge-equivalent structures and dynamical behaviors of nonintegrable semidiscrete Hirota equations.
  • To establish gauge equivalence between Hirota equations and Heisenberg ferromagnet equations.
  • To explore the preservation of spatial periodicity under discrete gauge transformations.

Main Methods:

  • Utilizing the concept of prescribed discrete curvature.
  • Applying discrete gauge transformations.
  • Constructing periodic orbits of stationary discrete Hirota equations.
  • Performing numerical simulations.

Main Results:

  • Demonstrated gauge equivalence between nonintegrable semidiscrete Hirota(-) and generalized semidiscrete modified Heisenberg ferromagnet equations.
  • Showcased gauge equivalence between nonintegrable semidiscrete Hirota(+) and generalized semidiscrete Heisenberg ferromagnet equations.
  • Proved the reversibility of discrete gauge transformations.
  • Obtained exact spatial periodic solutions for the Hirota equations.
  • Found that the dynamics of stationary discrete Hirota equations are richer than those of stationary discrete nonlinear Schrödinger equations.

Conclusions:

  • Discrete gauge transformations provide a method to analyze and solve nonintegrable semidiscrete Hirota equations.
  • The spatial periodicity property is preserved under these transformations.
  • The findings offer new insights into the complex dynamics of these nonlinear systems.