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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
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.
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.
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