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Solitary wave interactions in dispersive equations using Manton's approach
P G Kevrekidis1, Avinash Khare, A Saxena
1Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
Summary
This study generalizes a method for calculating solitary wave interactions in dispersive equations. The approach is demonstrated on Korteweg-de Vries, nonlinear Schrödinger, and sine-Gordon equations.
Area of Science:
- Mathematical Physics
- Nonlinear Dynamics
- Soliton Theory
Background:
- Solitary waves are localized solutions to nonlinear partial differential equations.
- Understanding their interactions is crucial in various scientific fields.
- Existing methods may have limitations in generality.
Purpose of the Study:
- To generalize Manton's approach for computing solitary wave interactions.
- To provide a unified framework for analyzing diverse solitary wave types.
- To demonstrate the method's applicability across different nonlinear equations.
Main Methods:
- Generalization of Manton's method (1979).
- Application to translationally invariant, dispersive equations.
- Analysis of specific models: Korteweg-de Vries, nonlinear Schrödinger, and sine-Gordon equations.
Main Results:
- A generalized computational approach for solitary wave interactions.
- Successful application to solitons, standing waves, kinks, and breathers.
- Demonstration of the method's versatility across canonical nonlinear models.
Conclusions:
- The generalized method offers a powerful tool for studying solitary wave dynamics.
- This unified approach simplifies the analysis of localized solutions.
- The findings have implications for fields utilizing these nonlinear models.