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The zero dispersion limit for the Korteweg-deVries KdV equation
1Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012.
Summary
This study uses the inverse scattering method to find the weak limit of Korteweg-deVries equation solutions as dispersion approaches zero. The results are valid for all time and connect to quadratic programming and averaged equations.
Area of Science:
- Nonlinear partial differential equations
- Mathematical physics
- Fluid dynamics
Background:
- The Korteweg-deVries (KdV) equation models phenomena like shallow water waves.
- Understanding the behavior of KdV solutions as dispersion becomes negligible is crucial.
- Previous studies have explored approximations for KdV dynamics.
Purpose of the Study:
- To determine the weak limit of Korteweg-deVries equation solutions as dispersion tends to zero.
- To characterize this limit using a solvable mathematical problem.
- To analyze the long-time behavior of these solutions.
Main Methods:
- Application of the inverse scattering method.
- Characterization of the weak limit via a quadratic programming problem.
- Utilizing function theoretic methods for problem-solving.
Main Results:
- The weak limit of Korteweg-deVries solutions is determined for all time.
- The limit is expressible as a quadratic programming problem.
- For large times, solutions exhibit behavior described by Whitham's averaged equations and Flaschka et al.'s equations.
Conclusions:
- The inverse scattering method provides a framework for analyzing KdV equation limits.
- The derived limit offers insights into the transition from dispersive to non-dispersive regimes.
- The long-time dynamics reveal a complex interplay between different averaging approaches.