On nonanalytic solitary waves formed by a nonlinear dispersion. (English) Zbl 1052.35511

Summary: We study the prototypical, genuinely nonlinear, \(K(m,n)\) equation, \(u_t\pm a(u^m)_x+(u^n)_{xxx}=0\), \(a=\text{const}\), which exhibits a number of remarkable dispersive effects. In particular, the distinguished subclass wherein \(m=n+2\) is transformed into a new, purely dispersive equation free of convection. In addition to compactons, the \(K(m,n)\) can support both kinks and solitons with an infinite slope(s), periodic waves and dark solitons with cusp(s) all being manifestations of nonlinear dispersion in action. For \(n<0\) the enhanced dispersion at the tail may generate algebraically decaying patterns.


35Q51 Soliton equations
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