Similarity Analysis of Nonlinear Equations and Bases of Finite Wavelength Solitons

Abstract

We introduce a generalized similarity analysis which grants a qualitative description of the localised solutions of any nonlinear differential equation. This procedure provides relations between amplitude, width, and velocity of the solutions, and it is shown to be useful in analysing nonlinear structures like solitons, dublets, triplets, compact supported solitons and other patterns. We also introduce kink-antikink compact solutions for a nonlinear-nonlinear dispersion equation, and we construct a basis of finite wavelength functions having self-similar properties.

Ice Spiral Patterns on the Ocean Surface

“…Multiple turns spirals are obtained by using multiple-fronts kinks (topological solitons) solutions of the sine-Gordon equation, built by gluing together shifted kinks on plateau of arbitrary width of other base kinks, and so on. Such solutions were obtained in the case of nonlinear dispersion equations, [31] (sometimes referred to as kovatons, [30]) whose stability is the same as for regular kinks, since multiple-fronts kinks have the same minimum value of energy as kinks, being calculated from their Hamiltonian (see e.g. [24], pp.…”