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Trigonometric function periodic wave solutions and their limit forms for the KdV-like and the PC-like equations. (English) Zbl 1235.35259
Summary: We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation ${u}_{t}+a\left(1+bu\right)u{u}_{x}+{u}_{xxx}=0$, and PC-like equation ${v}_{tt}-{v}_{ttxx}-{\left({a}_{1}v+{a}_{2}{v}^{2}+{a}_{3}{v}^{3}\right)}_{xx}=0$, 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.
##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35B10 Periodic solutions of PDE 35B32 Bifurcation (PDE)
##### Keywords:
bifurcation method; KdV equation