Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2025-01-03T17:14:22.284Z Has data issue: false hasContentIssue false

12 - Symmetry, Modulation, and Nonlinear Waves

Published online by Cambridge University Press:  05 February 2016

By
Thomas J. Bridges
Edited by
Thomas J. Bridges ,
Mark D. Groves  and
David P. Nicholls
Thomas J. Bridges
Affiliation:
University of Surrey, Guildford
Thomas J. Bridges
Affiliation:
University of Surrey
Mark D. Groves
Affiliation:
Universität des Saarlandes, Saarbrücken, Germany
David P. Nicholls
Affiliation:
University of Illinois, Chicago
Get access

Summary

Abstract

These lecture notes provide an introduction to the theory of “modulation” and its role in the derivation of model equations, such as the KdV equation, Boussinesq equation, KP equation, and Whitham modulation equations, and their role in the theory of water waves. The classical theory of modulation, such as Whitham modulation theory, will be discussed, and a new approach will be introduced, based on modulation of background flow. Methodology that is key to the theory is symmetry and conservation laws, relative equilibria, Hamiltonian and Lagrangian structures, multiple scale perturbation theory, and elementary differential geometry. By basing the theory on modulation of relative equilibria, new settings are discovered for the emergence of KdV and other modulation equations. For example, it is shown that the KdV equation can be a valid model for deep water as well as shallow water. The lecture notes are introductory, and no prior knowledge is assumed.

Introduction

Modulation is one of the most widely used concepts in the theory of nonlinear waves. In linear theory modulation is normally the process of varying the envelope of a signal. In electronics this concept is expanded further to include digital modulation, analog modulation, pulse modulation, frequency modulation, and so on. In the theory of nonlinear waves it is used to describe “modulation equations,” which typically are nonlinear equations governing the envelope of a wave, although the term is much more widely used now, with any equation on a slow space and time scale called a modulation equation. The term modulation in these notes is closest in spirit to Whitham modulation theory [1]. The idea is that given a basic state, dependent on a phase and a parameter or parameters, the phase and parameters are treated as slowly-varying functions of space and time, and governing equations are derived for these slowly-varying functions.

An abstraction of the idea of a basic state dependent on a phase is a relative equilibrium. A central theme of these notes is modulation of relative equilibria (RE). A definition of RE is given in §12.2. The context throughout will be conservative (Lagrangian, Hamiltonian, multisymplectic).

Type
Chapter
Information
Lectures on the Theory of Water Waves , pp. 261 - 283
Publisher: Cambridge University Press
Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] G.B., Whitham. Linear and Nonlinear Waves, Wiley-Interscience: New York (1974).Google Scholar
[2] G.B., Whitham. Two-timing, variational principles and waves, J. Fluid Mech. 44 373-395 (1970).Google Scholar
[3] J.E., Marsden. Lectures on Mechanics, London Mathematical Society Lecture Notes 174, Cambridge University Press.
[4] A.E., Taylor & D.C., Lay. Introduction to Functional Analysis, Krieger Publishers (1986).Google Scholar
[5] T.J., Bridges & N.M., Donaldson. Degenerate periodic orbits and homoclinic torus bifurcation, Phys. Rev. Lett. 95 104301 (2005).Google Scholar
[6] T.J., Bridges. Emergence of unsteady dark solitary waves from coalescing spatially-periodic patterns, Proc. Roy. Soc. Lond. A 468 3784-3803 (2012).Google Scholar
[7] T.J., Bridges. Multi-symplectic structures and wave propagation, Math. Proc. Camb. Phil. Soc. 121 147-190 (1997).Google Scholar
[8] M.W., Dingemans. Water wave propagation over uneven bottoms. Part 2 - Non-linear wave propagation, World Scientific Publisher: Singapore (1997).Google Scholar
[9] T.J., Bridges. A universal form for the emergence of the Korteweg-de Vries equation, Proc. Roy. Soc. Lond. A 469 20120707 (2013).Google Scholar
[10] V.D., Djordjevic & L.G., Redekopp. On two-dimensional packets of capillary-gravity waves, J. Fluid Mech. 79 703-714 (1977).Google Scholar
[11] T.J., Bridges. Breakdown of the Whitham modulation theory and the emergence of dispersion, Stud. Appl. Math. doi: 10.1111/sapm. 12086 (2015).Google Scholar
[12] P.G., Saffman. The superharmonic instablility of finite-amplitude water waves, J. Fluid Mech. 159 169-174 (1985).Google Scholar
[13] P.A.E.M., Janssen. Stability of steep gravity waves and the average Lagrangian method, KNMI Preprint (1989).
[14] T., Kataoka. On the superharmonic instability of surface gravity waves on fluid of finite depth, J. Fluid Mech. 547 175-184 (2006).Google Scholar
[15] A., Doelman, B., Sandstede, A., Scheel & G., Schneider. The dynamics of modulated wave trains, AMS Memoirs 934, American Mathematical Society: Providence (2009).Google Scholar
[16] T.J., Bridges. Bifurcation from rolls to multi-pulse planforms via reduction to a parabolic Boussinesq model, Physica D 275 8-18 (2014).
[17] T.J., Bridges. Dimension breaking from spatially-periodic patterns to KdV planforms, J. Dyn. Diff. Eqns. DOI: 10.1007/s10884-014-9405-y (2014).Google Scholar
[18] J., Wilkening. An algorithmfor computing Jordan chains and inverting analytic matrix functions, Lin. Alg. Appl. 427 6-25 (2007).Google Scholar
[19] T.J., Bridges. Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow, J. Fluid Mech. 761 R1-R9 (2014).Google Scholar
[20] T.J., Bridges, P.E., Hydon, & J.K., Lawson. Multisymplectic structures and the variational bicomplex, Math. Proc. Camb. Phil. Soc. 148, 159-178 (2010).Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×