Similarity Analysis of Nonlinear Equations and Bases of Finite Wavelength Solitons

TL;DR

This paper presents a generalized similarity analysis method for nonlinear differential equations, enabling qualitative understanding of localized solutions like solitons and constructing a basis of finite wavelength functions with self-similar properties.

Contribution

It introduces a new similarity analysis technique applicable to various nonlinear structures and constructs a basis of finite wavelength functions with self-similarity.

Findings

Relations between amplitude, width, and velocity of solutions

Introduction of kink-antikink compact solutions

Construction of a basis of self-similar finite wavelength functions

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.