The Spectral Problem for the Dispersionless Camassa–Holm Equation

2016 ◽  
pp. 67-90 ◽  
Author(s):  
C. Bennewitz ◽  
B. M. Brown ◽  
R. Weikard
Keyword(s):  
Spectral Problem ◽  
Holm Equation

scholarly journals The Inverse Spectral Problem for Periodic Conservative Multi-peakon Solutions of the Camassa–Holm Equation

2018 ◽  
Vol 2020 (16) ◽  
pp. 5126-5151
Author(s):  
Jonathan Eckhardt ◽  
Aleksey Kostenko
Keyword(s):  
Spectral Problem ◽  
Inverse Spectral Problem ◽  
Finite Dimensional ◽  
Holm Equation ◽  
Inverse Spectral

Abstract We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa–Holm equation. The corresponding isospectral sets can be identified with finite-dimensional tori.

scholarly journals Ermakov Equation and Camssa-Holm Waves

2016 ◽  
Vol 34 (1-2) ◽  
pp. 47-51
Author(s):  
Haret C. Rosu ◽  
Stefan C. Mancas
Keyword(s):  
Shallow Water ◽  
Spectral Problem ◽  
Water Waves ◽  
Shallow Water Waves ◽  
Important Ingredient ◽  
Interesting Issue ◽  
Holm Equation

Since the works of [1] and [2], it is known that the solution of the Ermakov equation is an important ingredient in the spectral problem of the Camassa-Holm equation. Here, we review this interesting issue and consider in addition more features of the Ermakov equation which have an impact on the behavior of the shallow water waves as described by the Camassa-Holm equation.

The string density problem and the Camassa–Holm equation

2007 ◽  
Vol 365 (1858) ◽  
pp. 2299-2312 ◽  
Author(s):  
Richard Beals ◽  
David H Sattinger ◽  
Jacek Szmigielski
Keyword(s):  
Inverse Problem ◽  
Special Reference ◽  
Shallow Water ◽  
Spectral Problem ◽  
Water Waves ◽  
Shallow Water Waves ◽  
Numerical Examples ◽  
Holm Equation ◽  
Density Problem

Recently, the string density problem, considered in the pioneering work of M. G. Krein, has arisen naturally in connection with the Camassa–Holm equation for shallow water waves. In this paper we review the forward and inverse string density problems, with some numerical examples, and relate it to the Camassa–Holm equation, with special reference to multi-peakon/anti-peakon solutions. Under stronger assumptions, the Camassa–Holm spectral problem and the string density problem can be transformed to the Schrödinger spectral problem and its inverse problem. Recent results exploiting this transformation are reviewed briefly.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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