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Classification of the travelling waves in the nonlinear dispersive KdV equation. (English) Zbl 1209.34038

The traveling wave solutions of the Korteweg-de Vries-like equation (\(K(2,2)\))
\[ u_t + a(u^2)_x + (u^2)_{xxx} =0 \]
modeling the formation of pattern in nonlinear dispersion, are investigated.
Besides the known compacton solutions (compactly supported solitons with nonsmooth fronts), the equation is shown to admit other different types of solutions too, such as cuspons, peakons, loopons, stumpons and fractal-like waves. A qualitative analysis of the traveling waves, as well as their classification, is carried on. Finally, some new explicit solutions are given.

MSC:

34B40 Boundary value problems on infinite intervals for ordinary differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
35C07 Traveling wave solutions
35C08 Soliton solutions
34A05 Explicit solutions, first integrals of ordinary differential equations
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References:

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