The Cauchy Problem

In this paper, we consider generalized Camassa–Holm equations and the generalized weakly dissipative Camassa–Holm equations and derive some new exact peaked solitary wave solutions. For m=3, where m is representative of the strength of the nonlinearity, we give two types new exact traveling wave solutions of the generalized weakly dissipative Camassa–Holm equations.

New peaked solitary wave solutions of the generalized Camassa-Holm equation

The analytic expressions of peaked solitary wave solutions and peaked periodic wave solutions of Camassa–Holm equation are obtained by using bifurcation method of planar dynamical systems. The convergence of the peaked periodic wave solutions is proved. Numerical simulation results show the consistence with the theoretical analysis. The results in this paper are wider than those already known.

Peaked wave solutions of Camassa–Holm equation

Under the travelling wave transformation, the Camassa–Holm equation with dispersion is reduced to an integrable ordinary differential equation (ODE), whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa–Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa–Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa–Holm equation with dispersion.

The classification of travelling wave solutions and superposition of multi-solutions to Camassa?Holm equation with dispersion

A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations

Camassa and Holm [1993] recently derived a new dispersive shallow water equation known as the Camassa–Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called "peakons". In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa–Holm equation \[ u_t+2ku_x-u_{xxt}+au^mu_x = 2u_xu_{xx} + uu_{xxx} \] with a>0, k∈ℝ, m∈ℕ and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For m=1, 2, 3, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.

Peakons and their bifurcation in a generalized camassa–holm equation

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

Tifei Qian

Papers

Peakons and their bifurcation in a generalized camassa–holm equation

Heteroclinic orbits for monotone twist maps through a variational principle