Inverse scattering transform for the Camassa–Holm equation

Adrian Constantin, Vladimir S. Gerdjikov, Rossen I. Ivanov

Type: Article

Publication Date: 2006-10-20

Citations: 293

DOI: https://doi.org/10.1088/0266-5611/22/6/017

Abstract

An inverse scattering method is developed for the Camassa–Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard inverse scattering transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.

Locations

Inverse Problems - View
arXiv (Cornell University) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+ PDF Chat	On an inverse scattering algorithm for the Camassa-Holm equation	2008	Octavian G. Mustafa Donal O’Regan
+	On the inverse scattering approach to the Camassa-Holm equation	2004	Adrian Constantin Jonatan Lenells
+ PDF Chat	On the Inverse Scattering Approach to the Camassa-Holm Equation	2003	Adrian Constantin Jonatan Lenells
+	Solving the modified Camassa–Holm equation via the inverse scattering transform	2023	Hui Mao Qian Yu Yuanyuan Miao
+	The inverse spectral transform for the Camassa-Holm equation	2013
+	Generalized Fourier transform for the Camassa–Holm hierarchy	2007	Adrian Constantin Vladimir S. Gerdjikov Rossen I. Ivanov
+	The scattering approach for the Camassa-Holm equation	2003	Jonatan Lenells
+	Об уравнении Камасса-Холма с самосогласованным источником	2011	Балтаева Ирода Исмаиловна Уразбоев Гайрат Уразалиевич
+	On the scattering problem for the Camassa-Holm equation	2001	Adrian Constantin
+ PDF Chat	The Scattering Approach for the Camassa—Holm equation	2002	Jonatan Lenells
+	Scattering Problem for a Modified Camassa-Holm Equation	2013	Tingting Liu Lixin Tian
+	On the isospectral problem of the dispersionless Camassa–Holm equation	2013	Jonathan Eckhardt Gerald Teschl
+	Solving the modified Camassa-Holm equation via the inverse scattering transform	2023	Hui Mao Qian Yu Yuanyuan Miao
+ PDF Chat	On an inverse problem associated with an integrable equation of Camassa–Holm type: explicit formulas on the real axis	2011	Keivan Mohajer Jacek Szmigielski
+ PDF Chat	Integration of Camassa-Holm equation with a self-consistent source of integral type	2022	G. U. Urazboev I. I. Baltaeva
+	The short-wave model for the Camassa–Holm equation: a Riemann–Hilbert approach	2011	Anne Boutet de Monvel Dmitry Shepelsky Lech Zieliński
+	On the inverse scattering approach and action-angle variables for the Dullin–Gottwald–Holm equation	2008	Ognyan Christov Sevdzhan Hakkaev
+	Infinite propagation speed of the Camassa–Holm equation	2006	Jonatan Lenells
+	On the isospectral problem of the Camassa–Holm equation	2023	Yaru Dou Jieyu Han Gang Meng
+	Inverse scattering transform for integrable nonisospectral hierarchy associate with Camassa-Holm equation	2024	Hongyi Zhang Yufeng Zhang

Works That Cite This (271)

Action	Title	Year	Authors
+	Riemann--Hilbert problem for Camassa--Holm equation with step-like initial data	2014	Alexander Minakov
+	Breather waves, rogue waves and complexiton solutions for a Zakharov–Kuznetsov equation	2021	Hongye Wang Yan Wang
+	An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions	2018	Baoqiang Xia Zhijun Qiao Jibin Li
+	Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions	2018	Alexander Sakhnovich
+ PDF Chat	Reciprocal link for a coupled Camassa–Holm type equation	2016	Nianhua Li Jinshun Zhang Lihua Wu
+ PDF Chat	Local well-posedness and blow-up for a family of U(1)-invariant peakon equations	2020	Stephen C. Anco Huijun He Zhijun Qiao
+	Multi-component generalization of the Camassa–Holm equation	2016	Baoqiang Xia Zhijun Qiao
+	Well-posedness and analyticity for the Cauchy problem for the generalized Camassa–Holm equation	2013	Yongsheng Mi Chunlai Mu
+ PDF Chat	A dressing method for soliton solutions of the Camassa-Holm equation	2017	Rossen I. Ivanov Tony Lyons Nigel Orr
+	Nonlocal Symmetries of the Camassa-Holm Type Equations	2020	Lu Zhao Changzheng Qu

Works Cited by This (36)

Action	Title	Year	Authors
+	Symplectic structures, their Bäcklund transformations and hereditary symmetries	1981	Benno Fuchssteiner A. S. Fokas
+	On the Inverse Spectral Problem for the Camassa–Holm Equation	1998	Adrian Constantin
+	The Camassa–Holm equation for water waves moving over a shear flow	2003	R. S. Johnson
+ PDF Chat	Conformal properties and Bäcklund transform for the associated Camassa–Holm equation	2005	Rossen I. Ivanov
+ PDF Chat	Parametric Representation for the Multisoliton Solution of the Camassa–Holm Equation	2005	Yoshimasa Matsuno
+	Explicit solutions of the Camassa–Holm equation	2005	E.J. Parkes В.О. Вахненко
+	The multiple-soliton solution of the Camassa-Holm equation	2004	Yishen Li Jin E. Zhang
+	The associated Camassa-Holm equation and the KdV equation	1999	Andrew N. W. Hone
+	On the scattering problem for the Camassa-Holm equation	2001	Adrian Constantin
+	Evolution of the Scattering Coefficients of the Camassa–Holm Equation, for General Initial Data	2006	D. J. Kaup