The cnoidal wave/corner wave/breaking wave scenario: A one-sided infinite-dimension bifurcation

doi:10.1016/j.matcom.2005.01.002

Mathematics and Computers in Simulation

Volume 69, Issues 3–4, 24 June 2005, Pages 235-242

https://doi.org/10.1016/j.matcom.2005.01.002 Get rights and content

Abstract

Many wave species have families of travelling waves — cnoidal waves and solitons — which are bounded by a wave of maximum amplitude. Remarkably, for a great many different wave systems, the limiting wave has a discontinuous slope — a so-called “corner” wave. Blending in previously unpublished graphs and formulas, we review both progress and unresolved difficulties in understanding corner waves. Why are they so common? What is universal about the cnoidal/corner/breaking (CCB) scenario, and what features are unique to particular wave equations? The peakons and coshoidal waves of the Camassa–Holm equation and equatorially-trapped Kelvin waves in the ocean are used as specific examples.

Introduction

Many species of dispersive nonlinear waves exhibit the three regimes illustrated in Fig. 1. For small amplitude, there are steadily-translating solitary waves and spatially-periodic waves, which shall be dubbed “cnoidal waves” by analogue with water waves. For large amplitude, dispersion loses the battle with nonlinearity, and the wave breaks. If the wave equation is regularized by adding viscosity with a small coefficient, then the wave will remain single-valued, but “breaking” implies the development of a very narrow, near-discontinuous frontal zone. Without such regularization, the breaking wave will become triple-valued as shown.

The boundary between these two regimes, the limiting travelling wave, is a wave with discontinuous slope at the crest — a “corner” wave. This is a bifurcation very different from the usual sort of limit point or trans-critical bifurcation: a single branch of solutions simply stops. It does not blend smoothly into an upper branch, or cross another branch; there is only a single branch of solutions, and at a certain finite amplitude, it dies as a corner wave.

For ordinary water waves, this cnoidal/corner/breaking (CCB) scenario was known to Stokes [29]. One might suppose that a century-and-a-half of further study would have completely resolved all questions about corner waves, but this is not true.

It is known that many species of waves, not merely external gravity waves, exhibit the CCB scenario: a sample is given in Table 1. Fig. 2 shows some travelling waves and the limiting corner wave for a particular wave equation.

In the next two sections, we discuss a couple of particular examples that clearly have similarities to better understood cases of corner waves, and also pose unique, unresolved puzzles. In the final section, we discuss what is known about corner waves as a generic bifurcation and list a large number of open research questions.

Section snippets

Camassa–Holm equation

The letter of Camassa and Holm that derived the equation that now bears their name was cited more than 200 times in the first 10 years after publication. The time-dependent form is listed in the table; travelling waves of phase speed c satisfy $c u_{X X X} + (2 κ - c) u_{X} - 3 u u_{X} - 2 u_{X} u_{X X} - u u_{X X X} = 0 [Camassa–Holm equation]$ where $κ$ is a parameter. If $u (X; 1, c)$ is a solution, then so is $κ u (X; κ, c κ)$ for any $κ$ , so it is sufficient to specialize to $κ = 1$ . Similarly, by a trivial rescaling one can transform solutions whose mean

The equatorial Kelvin wave

The Kelvin wave is a very important oceanic mode which varies like a Gaussian in latitude but propagates freely east-west. As explained in [3], [6], the simplest physically-plausible model is the shallow water wave equations on the equatorial beta-plane, a set of three partial differential equations in latitude, longitude and time.

To identify particular wave modes in a general two-space dimensional solution, one can exploit the fact that when the shallow water equations are linearized about a

Summary

A steadily-translating wave is a balance between nonlinearity and dispersion — if these two competing influences are imbalanced, then the shape of the wave will change, and the time-dependence is no longer merely a steady translation. As the amplitude of a wave increases, advective nonlinearity obviously becomes stronger. A solitary or cnoidal wave is possible if and only if the dispersion can become stronger, too, to remain in balance. If the dispersion increases without bound as the wave

Acknowledgements

This work was supported by the National Science Foundation through grant OCE9521133. I also thank Professor Thiab Taha and the other organizers of the IMACS 2003 conference on “Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory”.

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The cnoidal wave/corner wave/breaking wave scenario: A one-sided infinite-dimension bifurcation

Abstract

Introduction

Section snippets

Camassa–Holm equation

The equatorial Kelvin wave

Summary

Acknowledgements

Dynam. Atmos. Oceans

Physica D

Appl. Math. Comput.

Dynam. Atmos. Oceans

J. Comput. Phys.

Chaos, Solitons Fractals

The geometry of peaked solitons and billiard solutions of a class of integrable PDEs

Lett. Math. Phys.

Advanced Mathematical Methods for Scientists and Engineers