## Some new results on explicit traveling wave solutions of $$K(m, n)$$ equation.(English)Zbl 1195.35092

The author use the bifurcation method and numeric simulation approach of dynamical systems to study the traveling wave solutions of $$K(m,2)$$ $$(m=2,3,4)$$ equation
$u_t+a(u^m)_x+(u^n)_{xxx}=0.$
It is showed that when $$a>0$$, $$K(2,2)$$ equation has two type of explicit traveling wave solutions, the smooth periodic wave solution and the periodic-cusp wave solution, when $$a<0$$, $$K(2,2)$$ equation has three types of explicit traveling wave solutions, the periodic-cusp wave solution, the peakon wave solution and the blow-up solution. When $$a\neq 0$$, $$K(3,2)$$ equation has six types of traveling wave solutions, the smooth periodic wave solution, the periodic-cusp wave solution, the periodic-blow-up solution, the 1-blow-up solution, the smooth solitary wave solution and the peakon wave solution. When $$a>0$$, $$K(4,2)$$ equation has one type of explicit traveling wave solution, that is, the smooth periodic wave solution, finally, when $$a<0$$, $$K(4,2)$$ equation has three types of explicit traveling wave solution, that are, the peakon wave solution, the 1-blow-up and 2-blow-up solutions.

### MSC:

 35C07 Traveling wave solutions 35B32 Bifurcations in context of PDEs
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