Traveling wave solutions of the Camassa-Holm equation

By Jonatan Lenells. 2005

335 citations recorded by the source corpus.

All weak traveling wave solutions of the Camassa-Holm equation are classified. We show that, in addition to smooth solutions, there are a multitude of traveling waves with singularities: peakons, cuspons, stumpons, and composite waves.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/bad0f3ee399f5b458f05e0120a0eff1b28d99bb9

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In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms (2008, background): “1 [18] Let p(t) be a polynomial with real coefficients.”
Single and multi-solitary wave solutions to a class of nonlinear evolution equations (2008, methodology): “By the way, Lenells [27] has proposed an effective method to give the classifications of TWSs of the Camassa–Holm equation, Degasperis–Procesi equation and other nonlinear dispersive equations.”
Non-Uniform Dependence for the Periodic CH Equation (2010, methodology): “[22] Lenells, J. (2005).”
Wave Breaking and Global Existence for a Generalized Two-Component Camassa–Holm System (2010, background): “The first is the presence of solutions in the form of peaked solitary waves or “peakons” [7, 2, 39]: u(t, x) = ce−|x−ct|, c 6= 0, which are smooth except at the crests, where they are continuous, but have a jump discontinuity in the first derivative.”
Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions (2009, result): “r at points where ’= cproduces the complete collection of traveling wave solutions of (6.1). The analysis proceeds along the same lines and yields similar results as for CH (or as for HS if = 0) cf. [Le]. Here we choose to consider the peaked solutions. For completeness we point out that the equatio”
Controllability and asymptotic stabilization of the Camassa–Holm equation (2008, background): “The Camassa-Holm equation has been intensively investigated lately from the point of view of the Cauchy problem, see for instance [2, 4, 5, 6, 7, 15, 16, 17, 22, 23, 24, 27, 33] and references therein.”
A three-component generalization of Camassa–Holm equation with N-peakon solutions (2011, background): “4) have to be interpreted in the weak sense [33,28,6].”
Conservation laws of the Camassa–Holm equation (2005, background): “The equation admits, in addition to smooth waves, a multitude of travelling wave solutions with singularities—peakons, cuspons, stumpons and composite waves [2, 17, 18].”
On a novel integrable generalization of the nonlinear Schrödinger equation (2008, background): “3) was introduced by Fuchssteiner and one of the authors in 1981 [11], it began to attract a lot of interest (see, for example [1, 4, 5, 14, 16, 18]) only after its rederivation by Camassa and Holm [3].”
Exactly Solvable Model for Nonlinear Pulse Propagation in Optical Fibers (2008, neutral): “[23].”