Patterns on liquid surfaces: cnoidal waves, compactons and scaling

Localized patterns and nonlinear oscillation formations on the bounded free surface of an ideal incompressible liquid are investigated. Cnoidal modes, solitons and compactons, as traveling non-axially symmetric shapes are discussed. A finite-difference differential generalized Korteweg-de Vries (KdV) equation is shown to describe the three-dimensional motion of the fluid surface, and in the limit of long and shallow channels one recovers the well-known KdV equation. A tentative expansion formula for the representation of the general solution of a nonlinear equation, for given initial conditions is introduced. The model is useful in multilayer fluid dynamics, cluster formation, and nuclear physics since, up to an overall scale, these systems display a free liquid surface behavior. Copyright (C) 1998 Elsevier Science B.V.