Exact Peakon, Compacton, Solitary Wave, and Periodic Wave Solutions for a Generalized KdV Equation

We employ the approaches of both dynamical system and numerical simulation to investigate a generalized KdV equation, which is presented by Yin (2012). Some peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions are obtained, and the planar graphs of the compactons and the periodic cusp waves are simulated.

Bifurcation of Traveling Wave Solutions for a Two-Component Generalizedθ-Equation

We study the bifurcation of traveling wave solutions for a two-component generalizedθ-equation. We show all the explicit bifurcation parametric conditions and all possible phase portraits of the system. Especially, the explicit conditions, under which there exist kink (or antikink) solutions, are given. Additionally, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves with explicit expressions, are obtained.

Extension on Bifurcations of Traveling Wave Solutions for a Two-Component Fornberg-Whitham Equation

Fan et al. studied the bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. They gave a part of possible phase portraits and obtained some uncertain parametric conditions for solitons and kink (antikink) solutions. However, the exact explicit parametric conditions have not been given for the existence of solitons and kink (antikink) solutions. In this paper, we study the bifurcations for the two-component Fornberg-Whitham equation in detalis, present all possible phase portraits, and give the exact explicit parametric conditions for various solutions. In addition, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves are obtained. Our results extend the previous study.

EXACT PERIODIC WAVE SOLUTIONS OF A SINGULAR INTEGRABLE EQUATION

In this paper, theory of dynamical systems is employed to investigate periodic waves of a singular integrable equation. These periodic waves contain smooth periodic waves, periodic cusp waves and periodic cusp loop waves. Under fixed parameter conditions, their exact parametric expressions are given.

PERIODIC WAVES AND THEIR LIMITS FOR THE CAMASSA–HOLM EQUATION

In this paper, the bifurcation method of dynamical systems is employed to study the Camassa–Holm equation [Formula: see text] We investigate the periodic wave solutions of form u = φ(ξ) which satisfy φ(ξ + T) = φ(ξ), here ξ = x - ct and c, T are constants. Their six implicit expressions and two explicit expressions are obtained. We point out that when the initial values are changed, the periodic waves may become three waves, periodic cusp waves, smooth solitary waves and peakons. Our results give an explanation to the appearance of periodic cusp waves and peakons. Moreover, three sets of graphs of the implicit functions are drawn, and three sets of numerical simulations are displayed. The identity of these graphs and simulations imply the correctness of our theoretical results.