## 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.

### MSC:

 35Q51 Soliton equations
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### References:

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