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An integrable shallow water equation with linear and nonlinear dispersion
H R Dullin1, G A Gottwald, D D Holm
1Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom. h.r.dullin@lboro.ac.uk
Physical Review Letters
|November 3, 2001
Summary
We derived a new nonlinear wave equation combining Korteweg-deVries (KdV) and Camassa-Holm (CH) properties. This integrable equation offers improved accuracy and includes KdV solitons and CH peakons as special cases.
Area of Science:
- Fluid dynamics
- Nonlinear wave phenomena
- Mathematical physics
Background:
- The Korteweg-deVries (KdV) equation describes shallow water waves with linear dispersion.
- The Camassa-Holm (CH) equation models waves with nonlinear and nonlocal dispersion.
- A need exists for more accurate nonlinear wave models.
Purpose of the Study:
- To derive a novel 1+1 unidirectional nonlinear wave equation.
- To integrate linear dispersion (KdV) with nonlinear/nonlocal dispersion (CH).
- To develop an equation that is asymptotically more accurate than KdV.
Main Methods:
- Asymptotic analysis
- Near-identity normal form transformation
- Water wave theory
- Inverse scattering transform method for integrability analysis
Main Results:
- A new 1+1 unidirectional nonlinear wave equation was derived.
- The equation combines KdV linear dispersion and CH nonlinear/nonlocal dispersion.
- The derived equation is one order more accurate than KdV.
- Complete integrability was proven using the inverse scattering transform method.
- Traveling wave solutions encompass KdV solitons and CH peakons.
Conclusions:
- The new equation offers a more accurate description of nonlinear waves.
- It unifies features of both KdV and CH equations.
- The model retains complete integrability, facilitating further analytical study.