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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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