Bäcklund Transformations for the Camassa–Holm Equation

Alexander G. Rasin; Jeremy Schiff

doi:10.1007/s00332-016-9325-6

Bäcklund Transformations for the Camassa–Holm Equation

Alexander G. Rasin, Jeremy Schiff

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

28 Scopus citations

Abstract

The Bäcklund transformation (BT) for the Camassa–Holm (CH) equation is presented and discussed. Unlike the vast majority of BTs studied in the past, for CH the transformation acts on both the dependent and (one of) the independent variables. Superposition principles are given for the action of double BTs on the variables of the CH and the potential CH equations. Applications of the BT and its superposition principles are presented, specifically the construction of travelling wave solutions, a new method to construct multisoliton, multicuspon and soliton–cuspon solutions, and a derivation of generating functions for the local symmetries and conservation laws of the CH hierarchy.

Original language	English
Pages (from-to)	45-69
Number of pages	25
Journal	Journal of Nonlinear Science
Volume	27
Issue number	1
DOIs	https://doi.org/10.1007/s00332-016-9325-6
State	Published - 1 Feb 2017

Keywords

Bäcklund transformations
Camassa-Holm equation
Conservation laws
Superposition principle
Symmetries
Wave solutions

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10.1007/s00332-016-9325-6

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abstract = "The B{\"a}cklund transformation (BT) for the Camassa–Holm (CH) equation is presented and discussed. Unlike the vast majority of BTs studied in the past, for CH the transformation acts on both the dependent and (one of) the independent variables. Superposition principles are given for the action of double BTs on the variables of the CH and the potential CH equations. Applications of the BT and its superposition principles are presented, specifically the construction of travelling wave solutions, a new method to construct multisoliton, multicuspon and soliton–cuspon solutions, and a derivation of generating functions for the local symmetries and conservation laws of the CH hierarchy.",

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AB - The Bäcklund transformation (BT) for the Camassa–Holm (CH) equation is presented and discussed. Unlike the vast majority of BTs studied in the past, for CH the transformation acts on both the dependent and (one of) the independent variables. Superposition principles are given for the action of double BTs on the variables of the CH and the potential CH equations. Applications of the BT and its superposition principles are presented, specifically the construction of travelling wave solutions, a new method to construct multisoliton, multicuspon and soliton–cuspon solutions, and a derivation of generating functions for the local symmetries and conservation laws of the CH hierarchy.

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Bäcklund Transformations for the Camassa–Holm Equation

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