Xinghua Fan

Stability of negative solitary waves for an integrable modified Camassa–Holm equation

In this paper, we prove that the modified Camassa–Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.

New exact travelling wave solutions for the K(2,2) equation with osmosis dispersion

Abstract In this paper, by using bifurcation method, we successfully find the K ( 2 , 2 ) equation with osmosis dispersion u t + ( u 2 ) x - ( u 2 ) xxx = 0 possess two new types of travelling wave solutions called kink-like wave solutions and antikink-like 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 of travelling wave system is made by Maple. The results show that our theoretical analysis agrees with the numerical simulation.

Soliton and Periodic Wave Solutions to the Osmosis Equation

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.

Traveling wave solutions to the (n+1)-dimensional sinh-cosh-Gordon equation

Traveling wave solutions for a generalized sinh-cosh-Gordon equation are studied. The equation is transformed into an auxiliary partial differential equation without any hyperbolic functions. By using the theory of planar dynamical system, the existence of different kinds of traveling wave solutions of the auxiliary equation is obtained, including smooth solitary wave, periodic wave, kink and antikink wave solutions. Some explicit expressions of the blow-up solution, kink-like solution, antikink-like solution and periodic wave solution to the generalized sinh-cosh-Gordon equation are given. Planar portraits of the solutions are shown.

Symmetric and non-symmetric waves in the osmosis K(2, 2) equation

We give an improved qualitative method to solve the osmosis K(2, 2) equation. This method combines several characteristics of other methods. Using this method, the existence of symmetric and non-symmetric wave solutions of the osmosis K(2, 2) equation is studied. Besides abundant symmetric forms such as smooth wave solutions, peaked waves, cusped waves, looped waves, stumpons and fractal-like waves, this equation also admits non-symmetric ones including breaking kink wave solutions, breaking anti-kink wave solutions and rampons. As regards this equation most of those solutions, either symmetric or non-symmetric solutions, have not appeared in the literature. We also study the limiting behavior of all periodic solutions as the parameters tend to some special values.