Approximation Theory Methods for Linear and Nonlinear Differential Equations with Deviating Arguments

Henry, Myron S.; Wiggins, Kenneth L

doi:10.1137/0512031

Cited by 3 publications

(1 citation statement)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the physical relevance of these two equations as models for the propagation of shallow water waves over a flat bottom one can consult, for example [5,6,7,8,9,10,11]. More about the physical applications, modifications and the type of solutions of the CH equation can also be found in [4,12,13,14,15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Equations of the Camassa-Holm hierarchy

Ivanov¹

2009

Theor Math Phys

View full text Add to dashboard Cite

The squared eigenfunctions of the spectral problem associated with the CamassaHolm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy, and their Hamiltonian structures, can be naturally expressed using the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. Using the GFT, we explicitly describe some members of the CH hierarchy, including integrable deformations for the CH equation. We also show that solutions of some (1 + 2) -dimensional members of the CH hierarchy can be constructed using results for the inverse scattering transform for the CH equation. We give an example of the peakon solution of one such equation.

show abstract