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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.

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35K57 Reaction-diffusion equations
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