Abstract
In this chapter, we present some examples of nonlinear evolution equations in one space dimension. We re-discuss the traditional Korteweg–de Vries (KdV) equation for the shallow water long channel case, and its cnoidal waves and soliton solutions. Then we briefly present the MKdV equation and some nonlinear dispersion extension of it. In the last sections, we discuss some possible dynamical generalizations of the shallow water models on compact intervals, for any depth of the fluid. The resulting equation is an infinite-order differential one, and it reduces to a finite difference differential equation. We show that this generalized KdV equation approaches the KdV, MKdV, and Camassa–Holm limiting equations, both at the equation and at the solution level, in the appropriate physical conditions. In the last part we discuss the Boussinesq equations on a circle.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsAuthor information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Ludu, A. (2011). Nonlinear Surface Waves in One Dimension. In: Nonlinear Waves and Solitons on Contours and Closed Surfaces. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22895-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-22895-7_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22894-0
Online ISBN: 978-3-642-22895-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)