56 research outputs found
Cusped and Smooth Solitons for the Generalized Camassa-Holm Equation on the Nonzero Constant Pedestal
We investigate the traveling solitary wave solutions of the generalized Camassa-Holm equation ut - uxxt + 3u2ux=2uxuxx + uuxxx on the nonzero constant pedestal limξ→±∞uξ=A. Our procedure shows that the generalized Camassa-Holm equation with nonzero constant boundary has cusped and smooth soliton solutions. Mathematical analysis and numerical simulations are provided for these traveling soliton solutions of the generalized Camassa-Holm equation. Some exact explicit solutions are obtained. We show some graphs to explain our these solutions
Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations
In this paper we employ two recent analytical approaches to investigate the
possible classes of traveling wave solutions of some members of a
recently-derived integrable family of generalized Camassa-Holm (GCH) equations.
A recent, novel application of phase-plane analysis is employed to analyze the
singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible
non-smooth peakon and cuspon solutions. One of the considered GCH equations
supports both solitary (peakon) and periodic (cuspon) cusp waves in different
parameter regimes. The second equation does not support singular traveling
waves and the last one supports four-segmented, non-smooth -wave solutions.
Moreover, smooth traveling waves of the three GCH equations are considered.
Here, we use a recent technique to derive convergent multi-infinite series
solutions for the homoclinic orbits of their traveling-wave equations,
corresponding to pulse (kink or shock) solutions respectively of the original
PDEs. We perform many numerical tests in different parameter regime to pinpoint
real saddle equilibrium points of the corresponding GCH equations, as well as
ensure simultaneous convergence and continuity of the multi-infinite series
solutions for the homoclinic orbits anchored by these saddle points. Unlike the
majority of unaccelerated convergent series, high accuracy is attained with
relatively few terms. We also show the traveling wave nature of these pulse and
front solutions to the GCH NLPDEs
Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa- Holm equations
In this paper, we study peakon, cuspon, and pseudo-peakon solutions for two generalized Camassa-Holm equations. Based on the method of dynamical systems, the two generalized Camassa-Holm equations are shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, pseudo-peakons, and periodic cusp solutions. In particular, the pseudo-peakon solution is for the first time proposed in our paper. Moreover, when a traveling system has a singular straight line and a heteroclinic loop, under some parameter conditions, there must be peaked solitary wave solutions appearing
Negative order KdV equation with both solitons and kink wave solutions
In this paper, we report an interesting integrable equation that has both
solitons and kink solutions. The integrable equation we study is
, which actually comes from the negative KdV
hierarchy and could be transformed to the Camassa-Holm equation through a gauge
transform. The Lax pair of the equation is derived to guarantee its
integrability, and furthermore the equation is shown to have classical
solitons, periodic soliton and kink solutions
Solitons, peakons, and periodic cuspons of a generalized Degasperis-Procesi equation
We employ the bifurcation theory of planar dynamical systems to investigate
the exact travelling wave solutions of a generalized Degasperis-Procesi
equation. The implicit expression of smooth soliton solutions is given. The
explicit expressions of peaked soliton solutions and periodic cuspon solutions
are also obtained. Further, we show the relationship among the smooth soliton
solutions, the peaked soliton solutions, and the periodic cuspon solutions. The
physical relevance of the found solutions and the reasonwhy these solutions can
exist in this equation are also given.Comment: 14 pages, 41 figure
Stability of stationary solutions for nonintegrable peakon equations
The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions - "peakons" - with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter varies through the family.
In this article, we describe analytical results on one of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions - "leftons" - which are orbitally stable
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