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A proof of validity for multiphase Whitham modulation theory
Thomas J Bridges1, Anna Kostianko1,2, Guido Schneider3
1Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK.
Proceedings. Mathematical, Physical, and Engineering Sciences
|December 28, 2020
Summary
Approximations from multiphase Whitham modulation equations accurately model coupled nonlinear Schrödinger equations. This study rigorously proves these approximations remain close to original solutions over time.
Area of Science:
- Nonlinear Dynamics
- Mathematical Physics
- Wave Propagation
Background:
- Coupled nonlinear Schrödinger equations model complex wave phenomena.
- These equations are not generally integrable, posing analytical challenges.
- Multiphase Whitham modulation equations offer approximations for these systems.
Purpose of the Study:
- To rigorously compare solutions of coupled nonlinear Schrödinger equations with their multiphase Whitham modulation approximations.
- To establish the validity of Whitham theory approximations for a specific class of nonlinear wave equations.
- To analyze the behavior of wave solutions across different equation types (elliptic, hyperbolic, mixed).
Main Methods:
- Utilizing Gevrey spaces for function set-up due to type changes in modulation equations.
- Proving a Cauchy-Kowalevskaya-like existence and uniqueness theorem for initial data.
- Developing higher-order approximations based on Whitham theory.
- Rigorous mathematical comparison of solution sets.
Main Results:
- Demonstrated that approximations derived from multiphase Whitham modulation equations stay close to the original solutions on a natural time scale.
- Established existence and uniqueness of solutions in Gevrey spaces.
- Confirmed the applicability of Whitham theory for analyzing coupled nonlinear Schrödinger equations.
Conclusions:
- The multiphase Whitham modulation equations provide a valid and accurate approximation framework for coupled nonlinear Schrödinger equations.
- The established mathematical framework supports the analysis of complex nonlinear wave phenomena.
- This work bridges rigorous analysis with approximate models in nonlinear wave theory.
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