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

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

Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method

The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions

Wave Structures and Nonlinear Balances in a Family of 1+1 Evolutionary PDEs

We study the following family of evolutionary 1+1 PDEs that describe the balance between convection and stretching for small viscosity in the dynamics of 1D nonlinear waves in fluids: m_t + \underbrace{um_x \} _{(-2mm)\hbox{convection}(-2mm)} + \underbrace{b u_xm \} _{(-2mm)\hbox{stretching}(-2mm)} = \underbrace{\nu m_{xx}\ }_{(-2mm)\hbox{viscosity}}, \quad\hbox{with}\quad u=g*m . Here $u=g*m$ denotes $u(x)=\int_{-\infty}^\infty g(x-y)m(y) dy .$ We study exchanges of stability in the dynamics of solitons, peakons, ramps/cliffs, leftons, stationary solutions and other solitary wave solutions associated with this equation under changes in the nonlinear balance parameter $b$.Comment: 69 pages, 26 figure

Soliton, kink and antikink solutions of a 2-component of the Degasperis-Procesi equation

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the traveling wave solutions of a 2-component of the Degasperis-Procesi equation. The expressions for smooth soliton, kink and antikink solutions are obtained.Comment: 16 pages, 18 figure

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 $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, 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

The Bifurcations of Traveling Wave Solutions of the Kundu Equation

We use the bifurcation method of dynamical systems to study the bifurcations of traveling wave solutions for the Kundu equation. Various explicit traveling wave solutions and their bifurcations are obtained. Via some special phase orbits, we obtain some new explicit traveling wave solutions. Our work extends some previous results