A convergent numerical scheme for the Camassa--Holm equation based on multipeakons

By Helge Holden, Xavier Raynaud. 2005

109 citations recorded by the source corpus.

The Camassa{Holm equation utiuxxt+3uuxi2uxuxxiuuxxx = 0 enjoys special solutions of the form u(x;t) = n=1 pi(t)e ijxiqi(t)j , denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data ujt=0 = u0 in H 1 (R) such that u i uxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in L 1 (R;H 1 (R)) to the unique global solution of the Camassa{Holm equation. The approach also provides a convergent, energy preserving nondissipative nu- merical method which is illustrated on several examples.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c162f34657e34be64a5e2a8c6c2c6ccf5cfcbeee

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In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Global Conservative Solutions of the Camassa–Holm Equation (2007, background): “for t ∈ [0 ,T) . But (6.6) shows that p(0 )>H 0, which by (6.7)‐(6.8) yields T =∞ . Consequently, no wave breaking takes place in this case (see also [ 15 ]).”
Wave-Breaking and Peakons for a Modified Camassa–Holm Equation (2013, unknown): “2) can be found in [30].”
Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View (2007, methodology): “The periodic case is treated in Holden and Raynaud (2006c).”
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].”
A note on the Cauchy problem of the Novikov equation (2013, background): “The ODE-system for the peakons has to be understood in the sense of Caratheodory solutions (see the discussion in [24]).”