Solitons, Introduction to

By M. A. Helal. Encyclopedia of Complexity and Systems Science, 2009

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Numerical Methods for Evolutionary Differential Equations (2008, background): “The equation has a long history and admits, in particular, solitary wave solutions [58].”
Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton (2010, background): “More generally, the N -soliton N of the KdV equation (see [2, 12]) is also a particular solution of the ZK equation which is smooth and bounded together with its time and space derivatives and behaves as a sum of solitons of velocities 4n2 1≤ n ≤ N when t → .”
The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations (2007, methodology): “We refer to [15] for a discussion of these properties in the context of the KdV model.”
On the Blow-up Phenomena for the Degasperis–Procesi Equation (2010, background): “Both the KdV equation [20] and the Camassa-Holm equation [1, 6, 8, 13, 14] are completely integrable models for the propagation of shallow water waves.”
Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions (2019, methodology): “In modern soliton theory, the isomonodromic and inverse scattering transform methods have been introduced for solving Cauchy problems for nonlinear ordinary and partial differential equations [1,6,41].”