Inverse scattering transform for the Camassa–Holm equation

By Adrian Constantin, V. Gerdjikov, Rossen I. Ivanov. Inverse Problems, 2006

432 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d59a708e3d0565e27505bddfd7e3f8d54b0e5f24

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

The Ermakov equation: A commentary (2008, background): “Further applications are found in Cosmology [39–41, 75, 94, 95, 97, 99, 100], partial differential equations of Mathematical Physics such as the Korteweg – de Vries and Camassa – Holm equations [19, 20, 22, 42–46], Elasticity [77–79, 94, 99], Quantum Mechanics [48] and nonlinear systems in general [”
Riemann-Hilbert approach for the Camassa-Holm equation on the line (2006, methodology): “An alternative inverse scattering method based on an additive Riemann–Hilbert problem formulation for the associated eigenfunctions is presented in [5].”
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.”
Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval (2018, methodology): “From then on, the IST method can be often used to study the initial value problems for integrable evolution equations on the line [1, 5, 6, 16].”
Blow-up of solutions to the DGH equation (2007, background): “It is worth to point out that a fundamental aspect of the Camassa–Holm equation, the fact that it is a completely integrable system, was shown in [6,15] for the periodic case and [1,8,19] for the nonperiodic case.”
Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition (2007, methodology): “More deeper mathematical attributes on the CH equation was shown by the constraint method and r-matrix structure [14, 15], inverse spectral theory [3], Riemann–Hilbert problem [12], and the global conservative solution [1].”
A three-component generalization of Camassa–Holm equation with N-peakon solutions (2011, background): “For example, it is shown that the CH equation is integrable by the inverse scattering transformation, it possesses the Lax pair formalism, multi-soliton solutions, algebro-geometric solutions and other remarkable properties of integrable equations [8,21,16,22,17,19,10,13,35,32,34,38,25, 26].”
The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation (2010, methodology): “It was shown in Constantin, Gerdjikov and Ivanov [11], and Constantin and McKean [12] that the inverse spectral or scattering approach was a powerful tool to handle Camassa–Holm equation and analyzed its various fluid dynamics.”
The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations (2007, background): “In [7] a Lax pair formulation of (8) was found, a fact which lies at the core of showing via direct and inverse scattering [8, 10] that (8) is a completely integrable Hamiltonian system: for a large class of initial data, solving (8) amounts to integrating an infinite number of linear first-order or”