Abstract
This chapter represents a mathematical annex. We briefly remember the properties of the Riccati equation and of some elliptic functions used in soliton theory. We also describe the one-soliton solutions of the KdV and MKdV equations. In the end we present a simple procedure, the so called nonlinear dispersion relation approach, through which one can find information about the relations between amplitude, half-width and speed of a soliton solution of any nonlinear equation (scalar, vector, or system, no matter of the nature of the nonlinearity) without actually solve the equation, providing such an equation admits soliton solutions. Several examples on well known cases are also given in order to illustrate how this procedure works.
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Notes
- 1.
I am indebted to Dr. Panayotis Kevrekidis for the existence of this section.
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© 2011 Springer-Verlag Berlin Heidelberg
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Ludu, A. (2011). Mathematical Annex. 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_18
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DOI: https://doi.org/10.1007/978-3-642-22895-7_18
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22894-0
Online ISBN: 978-3-642-22895-7
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