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Exact special solutions with solitary patterns for the nonlinear dispersive $$K(m,n)$$ equations. (English) Zbl 1027.35115
Summary: We study the genuinely nonlinear dispersive $$K(m,n)$$ equation, $u_t-(u^m)_x+ (u^n)_{xxx}=0,$ which exhibits solutions with solitary patterns. Exact solutions that create solitary patterns having cusps or infinite slopes are developed. The nonlinear equation $$K(m,n)$$ is addressed for two different cases, namely when $$m=n=\text{odd}$$ integer and when $$m=n=\text{even}$$ integer. General formulas for the solutions of these cases of the $$K(m,n)$$ equations are established.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations 35K57 Reaction-diffusion equations
soliton
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