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In this paper, we focus on zero-filter limit problem for the Camassa-Holm equation in the more general Besov spaces. We prove that the solution of the Camassa-Holm equation converges strongly in L ∞ (0 , T ; B 2 , r s (R)) to the inviscid Burgers equation as the filter parameter α tends to zero with the given initial data u 0 ∈ B 2 , r s (R) . Moreover, we also show that the zero-filter limit for the Camassa-Holm equation does not converges uniformly with respect to the initial data in B 2 , r s (R) . [ABSTRACT FROM AUTHOR]

In this short note, we prove that given initial data u 0 ∈ H s (R) with s > 3 2 and for some T > 0 , the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in L ∞ (0 , T ; H s (R)) to the inviscid Burgers equation as the filter parameter α tends to zero. This is a complement of our recent result on the zero-filter limit. [ABSTRACT FROM AUTHOR]

The generalized modified Camassa-Holm equation possesses well-known peaked solitary waves that are called peakons. Their stability has been established by Z. Guo et al. (J Differ Equ 266:7749–7779, 2019) by using the Constantin-Strauss approach. In this paper, using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm, we prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space. [ABSTRACT FROM AUTHOR]

The paper aims at developing the Riemann–Hilbert (RH) approach for the modified Camassa–Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at different sides of the line. We present detailed properties of spectral functions associated with the initial data for the Cauchy problem for the mCH equation and obtain a representation for the solution of this problem in terms of the solution of an associated RH problem. [ABSTRACT FROM AUTHOR]

In the present paper we prove the validity of the Camassa-Holm equation as a long wave limit to the double dispersion equation which describes the propagation of bidirectional weakly nonlinear and dispersive waves in an infinite elastic medium. First we show formally that the right-going wave solutions of the double dispersion equation can be approximated by the solutions of the Camassa-Holm equation in the long wave limit. Then we rigorously prove that the solutions of the double dispersion and the Camassa-Holm equations remain close over a long time interval, determined by two small parameters measuring the effects of nonlinearity and dispersion. [ABSTRACT FROM AUTHOR]

In this paper, we study the orbital stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation. We first verify that the QCHE has global peakon and periodic peakon solutions. Then by the invariants of the equation and controlling the extrema of the solution, we prove that the shapes of the peakons and periodic peakons are stable under small perturbations in the energy space. [ABSTRACT FROM AUTHOR]

This paper deals with the Cauchy problem for the interacting system of the Camassa-Holm and Degasperis-Procesi equations mt=-3m(2ux+vx)-mx(2u+v),nt=-2n(2ux+vx)-nx(2u+v),where m=u-uxx and n=v-vxx. By the transport equations theory and the classical Friedrichs regularization method, the local well-posedness of solutions for this system in nonhomogeneous Besov spaces Bp,rs×Bp,rs with 1≤p,r≤+∞ and s>max2+1p,52 is obtained, and the local well-posedness in critical Besov space B2,15/2×B2,15/2 is also established. Moreover, by the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for this equation, which are bounded in the Sobolev space Hs(R)×Hs(R) with s>5/2, and the distance between the two sequences is lower-bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous. [ABSTRACT FROM AUTHOR]