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Resolution of a shock in hyperbolic systems modified by weak dispersion
1Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom. g.el@lboro.ac.uk
Chaos (Woodbury, N.Y.)
|October 29, 2005
Summary
This study introduces a method for analyzing dispersive shock transitions in non-integrable systems. The approach models these transitions using expansion fan solutions of the modulation system, providing new transition conditions.
Area of Science:
- Nonlinear Dynamics
- Fluid Mechanics
- Plasma Physics
Background:
- Dispersive shock transitions occur in nonlinear systems with dispersion.
- Analyzing these transitions in non-integrable systems is challenging.
- Existing methods often require complete integrability or full numerical solutions.
Purpose of the Study:
- To develop a method for analyzing dispersion-dominated shock-type transitions in non-integrable systems.
- To derive transition conditions for dispersive shocks without full integration of modulation equations.
- To validate the method using examples like KdV, mKdV, and ion-acoustic waves.
Main Methods:
- Modeling dispersive shock transitions using expansion fan solutions of the modulation (Whitham) system.
- Assuming hyperbolicity of the Whitham system for relevant solutions.
- Utilizing Whitham averaging properties and Cauchy data on characteristics.
- Applying the method to single-wave and bidirectional systems.
Main Results:
- A set of transition conditions for dispersive shocks was derived.
- The method successfully bypasses the full integration of modulation equations.
- Results align with previous analytical and numerical findings for KdV, mKdV, and ion-acoustic waves.
Conclusions:
- The presented method provides an effective way to analyze dispersive shock transitions in a broad class of nonlinear systems.
- The derived transition conditions are applicable to both integrable and non-integrable systems.
- This work offers a significant advancement in understanding nonlinear wave phenomena.
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