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

A type of bounded traveling wave solutions for the Fornberg-Whitham equation

In this paper, by using bifurcation method, we successfully find the Fornberg-Whitham equation has a type of traveling wave solutions called kink-like wave solutions and antikinklike wave solutions. They are defined on some semifinal bounded domains and possess properties of kink waves and anti-kink waves. Their implicit expressions are obtained. For some concrete data, the graphs of the implicit functions are displayed, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.Comment: 14 pages, 10 figure

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

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

Does solitary wave solution persist for the long wave equation with small perturbations?

In this paper, persistence of solitary wave solutions of the regularized long wave equation with small perturbations are investigated by the geometric singular perturbation theory. Two different kinds of the perturbations are considered in this paper: one is the weak backward diffusion and dissipation, the other is the Marangoni effects. Indeed, the solitary wave persists under small perturbations. Furthermore, the different perturbations do affect the proper wave speed ensuring the persistence of the solitary waves. Finally, numerical simulations are utilized to confirm the theoretical results

Stability of smooth solitary waves for the generalized Korteweg - de Vries equation with combined dispersion

The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation.Розглянуто задачу про орбітальну стійкість гладких відокремлених хвиль для узагальненого рівняння Кортевега – де Фріза з комбінованою дисперсією. Отримані результати показують, що гладкі відокремлені хвилі є стійкими при довільній швидкості поширення хвиль

Soliton and periodic wave solutions to the osmosis K(2, 2) equation

In this paper, two types of traveling wave solutions to the osmosis K(2, 2) equation are investigated. They are characterized by two parameters. The expresssions for the soliton and periodic wave solutions are obtained.Comment: 14 pages, 16 figure

Smooth and Peaked Solitons of the CH equation

The relations between smooth and peaked soliton solutions are reviewed for the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann-Hilbert problem. The momentum map from the action-angle scattering variables $T^*({\mathbb{T}^N})$ to the flow momentum ($\mathfrak{X}^*$) provides the Eulerian representation of the $N$-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The dispersionless limit of the CH equation and its resulting peakon solutions are examined by using an asymptotic expansion in the dispersion parameter. The peakon solutions of the dispersionless CH equation in one dimension are shown to generalize in higher dimensions to peakon wave-front solutions of the EPDiff equation whose associated momentum is supported on smoothly embedded subspaces. The Eulerian representations of the singular solutions of both CH and EPDiff are given by the (cotangent-lift) momentum maps arising from the left action of the diffeomorphisms on smoothly embedded subspaces.Comment: First version -- comments welcome! Submitted to JPhys