On the scattering problem for the Camassa-Holm equation

By Adrian Constantin. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2001

744 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3d05800ec35af9da7c3959ca61a1e3f32f68b511

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Citations in the corpus

Riemann-Hilbert approach for the Camassa-Holm equation on the line (2006, background): “Il est connu [4] que, si l’on pose m = u − u xx , l’hypothèse initiale m(x, 0 ) + ω > 0 pour tout x implique l’existence de m(x,t) pour tout temps t > 0, ainsi que la positivité m(x,t) + ω > 0, ce qui justiﬁe alors la forme équivalente (2) de l’équation CH.”
Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation (2006, background): “It has a bi-Hamiltonian structure [33, 26] and is completely integrable [3, 9 ]. Its solitary waves are smooth if c0 > 0 and peaked in the limiting case c0 = 0 [4].”
Traveling wave solutions of the Degasperis-Procesi equation (2005, background): “3) is known to be integrable for a large class of initial data via the inv scattering procedure [1,5,8,13], for (1.”
An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation (2005, background): “Here one can choose a constant C2 uniformly valid as (u, � ),(u′, � ′) range over bounded subsets of D. Now let t 7→ u(t) be a solution of the Cauchy problem (1.1), (1.13), satisfying all the required conditions.”
Conservation laws of the Camassa–Holm equation (2005, background): “It has solutions that are global in time [5] as well as solutions modelling wave breaking [4, 7, 8].”
Global weak solutions for the Novikov equation (2011, background): “It has a bi-Hamiltonian structure [16], a Lax pair based on a linear spectral problem of second order and is completely integrable [5].”
Global weak solutions and blow-up structure for the Degasperis–Procesi equation (2006, background): “It has a bi-Hamiltonian structure [25,32] and is completely integrable [3,8].”
Compactly Supported Solutions of the Camassa-Holm Equation (2005, background): “1) is an integrable infinite-dimensional Hamiltonian system (see [5, 7, 12, 19]).”
A note on the Cauchy problem of the Novikov equation (2013, methodology): “To begin with, the Camassa–Holm equation approximates unidirectional fluid flow in Euler’s equations [5] at the next order beyond the KdV equation, it has a bi-Hamiltonian structure [6] and is completely integrable [7], and with a Lax pair based on a linear spectral problem of second order.”
Traveling wave solutions of the Camassa-Holm equation (2005, background): “1) is an infinitedimensional completely integrable Hamiltonian system: by means of an isospectral problem one can convert the equation into a finite or infinite sequence of linear ordinary differential equations which can be trivially integrated (for the periodic case see [8,20], and for aspects of ”
Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces (2020, background): “The CH equation is completely integrable [4, 7] with a bi-Hamiltonian structure [6, 21] and infinitely many conservation laws [4, 21].”