Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV‐Like and the PC‐Like Equations

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV‐like equation ut + a(1 + bu)uux + uxxx = 0, and PC‐like equation vtt - vttxx - (a1v+a2v2+a3v3)xx=0, respectively. Via some special phase orbits, we obtain some new explicit periodic wave solutions which are called trigonometric function periodic wave solutions because they are expressed in terms of trigonometric functions. We also show that the trigonometric function periodic wave solutions can be obtained from the limits of elliptic function periodic wave solutions. It is very interesting that the two equations have similar periodic wave solutions. Our work extend previous some results.