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Some new results on explicit traveling wave solutions of $K(m, n)$ equation
In this paper, we investigate the traveling wave solutions of
$K(m, n)$ equation $ u_t+a(u^m)_{x}+(u^n)_{x x x}=0$ by using the
bifurcation method and numerical simulation approach of dynamical
systems. We obtain some new results as follows: (i) For
$K(2, 2)$ equation, we extend the expressions of the smooth
periodic wave solutions and obtain a new solution, the
periodic-cusp wave solution. Further, we demonstrate that the
periodic-cusp wave solution may become the peakon wave solution.
(ii) For $K(3, 2)$ equation, we extend the expression of the
elliptic smooth periodic wave solution and obtain a new solution,
the elliptic periodic-blow-up solution. From the limit forms of
the two solutions, we get other three types of new solutions, the
smooth solitary wave solutions, the hyperbolic 1-blow-up solutions
and the trigonometric periodic-blow-up solutions. (iii) For
$K(4, 2)$ equation, we construct two new solutions, the 1-blow-up
and 2-blow-up solutions.