## Peakon, solitary and smooth periodic wave solutions for a modification of the $$K(2,2)$$ equation.(English)Zbl 1406.35321

Summary: We consider a modification of the $$K(2,2)$$ equation $$u_t=2uu_{xxx}+2ku_xu_{xx}+2uu_x$$ using the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, some peakons, solitary, and smooth periodic waves are found and their exact parametric representations are presented. Also, the coexistence of peakon and solitary wave solutions is investigated.

### MSC:

 35Q51 Soliton equations 35B10 Periodic solutions to PDEs 35B32 Bifurcations in context of PDEs 35C08 Soliton solutions
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