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Compact and noncompact dispersive patterns. (English) Zbl 1115.35365
Summary: We discuss the pivotal role played by the nonlinear dispersion in shaping novel, compact and noncompact patterns. It is shown that if the normal velocity of a planar curve is $U=-{\left({k}^{n}\right)}_{s}$, $n>1$, where $k$ is the curvature, then the solitary disturbances may propagate like compactons. We extend the KP and the Boussinesq equations to include nonlinear dispersion to the effect that the new equations support compact and semi-compact solitary structures in higher dimensions. We also discuss the relations between equations sharing the same scaling. We show how compacton supporting equations may be cast into a strong formulation wherein one avoids dealing with weak solutions.
##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35Q51 Soliton-like equations