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We show that wave breaking occurs with positive probability for the Stochastic Camassa-Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space-time paths.
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In this paper we review the recent progress in the (indefinite) string density problem and its applications to the Camassa--Holm equation.
In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.
We present a non-chiral version of the Intermediate Long Wave (ILW) equation that can model nonlinear waves propagating on two opposite edges of a quantum Hall system, taking into account inter-edge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws for this new equation. We also present a periodic non-chiral ILW equation, together with its soliton solutions governed by the elliptic CMS model.
We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.
In this paper, we study orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa-Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function $h(t,,x)$ and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Comm. Math. Phys., 322:967-997, 2013) successfully to the higher-order case, and is helpful to understand how higher-order nonlinearities affect the dispersion dynamics.
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