ABSTRACT. In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.

No abstract

No abstract

The generalized conditional symmetry method, which can be considered a generalization of the conditional symmetry method, is used to study the nonlinear diffusion᎐convection equations with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the convective term and the source term, which permit the generalized conditional symmetry reductions. A number of examples are considered and some exact solutions are constructed via the compatibility of the generalized conditional symmetry and the considered equation.

In this paper, a new integrable two-component system, m t = [m(u x v x − uv + uv x − u x v)] x , n t = [n(u x v x − uv + uv x − u x v)] x , where m = u − u xx and n = v − v xx , is proposed. Our system is a generalized version of the integrable system m t = [m(u 2 x − u 2)] x , which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained.