Integrability, existence of global solutions, and wave breaking criteria for a generalization of the Camassa–Holm equation

This paper is devoted to the continuation of solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L∞space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data [Formula: see text], and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.

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Wave breaking and propagation speed for the Camassa–Holm equation with

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2010.12.005 ◽

2011 ◽

Vol 12 (3) ◽

pp. 1875-1882 ◽

Cited By ~ 16

Author(s):

Yong Zhou ◽

Huiping Chen

Keyword(s):

Wave Breaking ◽

Propagation Speed ◽

Holm Equation

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Publisher Correction to: On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

Nonlinear Differential Equations and Applications NoDEA ◽

10.1007/s00030-021-00676-w ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Christian Rohde ◽

Hao Tang

Keyword(s):

Global Existence ◽

Wave Breaking ◽

Original Publication ◽

Holm Equation

During the typesetting process, some misprints have been introduced in the original publication of the article.

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Perspectives on the formation of peakons in the stochastic Camassa–Holm equation

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2021.0224 ◽

2021 ◽

Vol 477 (2250) ◽

pp. 20210224

Author(s):

Thomas M. Bendall ◽

Colin J. Cotter ◽

Darryl D. Holm

Keyword(s):

Finite Element ◽

Velocity Profile ◽

Wave Breaking ◽

Soliton Solutions ◽

Finite Element Discretization ◽

Stochastic Perturbations ◽

Element Discretization ◽

Holm Equation ◽

Stochastic Transport ◽

Inflection Points

A famous feature of the Camassa–Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

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