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-type solitary waves and the stability analysis for the KdV-mKdV equation.
Zhi-Guo Liu1, Muhua Liu2,3, Jinliang Zhang1
1School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471000, China.
Scientific Reports
|July 15, 2024
Summary
This study reveals new solitary wave solutions for the KdV-mKdV equation, including novel -type waves. These findings enrich the equation's dynamics and offer methods for analyzing complex solitary wave behaviors.
Area of Science:
- Nonlinear Dynamics
- Mathematical Physics
- Wave Phenomena
Background:
- The Korteweg-de Vries (KdV) and modified KdV (mKdV) equations are fundamental in describing nonlinear wave phenomena.
- Investigating solitary wave solutions is crucial for understanding the complex dynamics of these systems.
Purpose of the Study:
- To explore and obtain closed-form analytical solutions for solitary waves in the KdV-mKdV equation.
- To identify and characterize novel -type solitary wave solutions beyond the standard sech-type.
- To demonstrate methods for generating stable multiple -type solitary waves.
Main Methods:
- Hirota's bilinear method for deriving analytical solutions.
- Qualitative analysis of solitary waveforms.
- Trial functions method for obtaining -type solutions.
- Split-Step Fourier Transform method for stability verification.
Main Results:
- Closed-form analytical single and multiple solitary wave solutions were successfully obtained.
- The existence of -type solitary waves, in addition to sech-type, was discovered.
- Stable double and triple -type solitary waves were excited through wave collisions.
- The proposed method allows for the excitation of stable multiple solitary waves.
Conclusions:
- The discovered solitary wave solutions significantly enrich the dynamic behavior of the KdV-mKdV equation.
- The study provides valuable methods for solving and analyzing -type solitary waves.
- These findings hold significant theoretical value for nonlinear wave research.
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