Scaling, Self-similarity and Superposition

TL;DR

This paper introduces a new method for combining self-similar solutions of the nonlinear heat equation, highlighting how nonlinearity affects superposition and solution exactness.

Contribution

It presents a novel procedure for nonlinear superposition of self-similar solutions and analyzes the impact of boundary conditions and nonlinearity on solution exactness.

Findings

Superposition of solutions is exact in linear cases.

Nonlinearity couples with scale, preventing exact superposition.

Boundary conditions conflict with self-similarity in nonlinear superpositions.

Abstract

A novel procedure for the nonlinear superposition of two self-similar solutions of the heat conduction equation with power-law nonlinearity is introduced. It is shown how the boundary conditions of the superposed state conflicts with self-similarity, rendering the nonlinearly superposed state to be a non-exact solution. It is argued that the nonlinearity couples with the presence of the scale so that the superposition in the linear case can give an exact solution.