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On the continuum limit for a semidiscrete Hirota equation.

Andrew Pickering1, Hai-Qiong Zhao2, Zuo-Nong Zhu3

  • 1Área de Matemática Aplicada, ESCET , Universidad Rey Juan Carlos , C/ Tulipán s/n, 28933 Móstoles, Madrid, Spain.

Proceedings. Mathematical, Physical, and Engineering Sciences
|December 14, 2016
PubMed
Summary

This study introduces a new semidiscrete Hirota equation, analyzing how the discrete space step impacts soliton solutions. The findings confirm its continuum limit accurately reproduces the original Hirota equation results.

Keywords:
Darboux transformationintegrable discrete Hirota equationsoliton solutions

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Area of Science:

  • Nonlinear partial differential equations
  • Soliton theory
  • Computational physics

Background:

  • The Hirota equation is a key model in nonlinear science, often studied in its continuous form.
  • Understanding discrete approximations is crucial for numerical simulations and theoretical analysis.
  • Previous studies have explored various discrete versions of integrable equations.

Purpose of the Study:

  • To propose and analyze a novel semidiscrete Hirota equation.
  • To investigate the influence of the discrete space step (δ) on soliton solutions.
  • To establish the continuum limit of the semidiscrete model and its associated mathematical tools.

Main Methods:

  • Construction of the semidiscrete Hirota equation.
  • Application of the Darboux transformation for explicit solutions.
  • Analysis of the continuum limit, including the Lax pair and solution methods.

Main Results:

  • A new semidiscrete Hirota equation is successfully formulated.
  • The effect of the discrete space step (δ) on soliton simulations is elucidated.
  • The continuum limit rigorously yields the standard Hirota equation, including its Lax pair and Darboux transformation.

Conclusions:

  • The proposed semidiscrete Hirota equation serves as a valid discrete model.
  • The study provides a method for simulating Hirota equation solitons with discrete approximations.
  • The established continuum limit validates the discrete model's connection to the continuous Hirota equation.