Smooth Periodic Wave Solutions Research Articles

Travelling wave solutions for higher-order wave equations of KDV type (III)

By using the theory of planar dynamical systems to the travelling wave equation of a higher order nonlinear wave equations of KdV type, the existence of smooth solitary wave, kink wave and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some conditions, exact explicit parametric representations of these waves are obtain.

Bifurcations of travelling wave solutions for the Boussinesq equations

By using the theory of bifurcations of dynamical systems to the generalized Boussinesq equations, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Bifurcations of travelling wave solutions in a model of the hydrogen-bonded systems

By using the theory of bifucations of dynamical systems to a model of the Hydrogen-bonded systems, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. In some simple parametric conditions, exact explicit and implicit solution formulas are listed.

Travelling wave solutions for a higher order wave equations of KdV type (I)

Abstract Using the method of planar dynamical systems to a higher order wave equations of KdV type, the existence of smooth solitary wave and uncountably infinite many smooth and non-smooth periodic wave solutions is proved. In different regions of the parametric plane, the sufficient conditions to guarantee the existence of the above solutions are given.

Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation

The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation ( B ( m , n ) equation), u tt − u xx − a ( u n ) xx + b ( u m ) xxxx =0, is studied by using bifurcation theory of dynamical system . As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave , kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given.

Bifurcations of travelling wave solutions for the generalization form of the modified KdV equation

By using the theory of bifurcations of dynamical systems to the generalization form of the modified KdV equation, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Travelling wave solutions in a model of the helix polypeptide chains

By using the theory of bifucations of dynamical systems to a model of the helix polypeptide chains, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. In some simple conditions, exact explicit and implicit solution formulas are listed.

Bifurcations of travelling wave solutions for the generalized Kadomtsev–Petviashili equation

By using the theory of bifurcations of dynamical systems to the generalized Kadomtsev–Petviashili equation, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Peakons and periodic cusp waves in a generalized Camassa–Holm equation

We study the peakons and the periodic cusp wave solutions of the following equation: u t+2ku x−u xxt+auu x=2u xu xx+uu xxx with a,k∈ R , which we will call the generalized Camassa–Holm equation, or simply the GCH equation, for when a=3 it was the so-called CH equation given by R. Camassa and D.D. Holm [Phys. Rev. Lett. 71 (11) (1993) 1661–1664]. They showed that the CH equation has a class of new solitary wave solutions called “peakons”. J.P. Boyd [Appl. Math. Comput. 81 (2–3) (1997) 173–187] studied another class of new periodic wave solutions called “coshoidal waves”. Using the bifurcation method of the phase plane, we first construct peakons and show that a=3 is the peakon bifurcation parameter value for the GCH equation. Then we construct some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions, and show their convergence when either the parameter a or the wave speed c varies. We also illustrate how to identify the existence of peakons and periodic cusp waves from the phase portraits. It seems that the GCH equation is a good example to understand the relationships among peakons, periodic cusp waves, oscillatory solitary waves and smooth periodic wave solutions.