Newell-Whitehead-Segel equation,A Simpler Proof

TL;DR

This paper simplifies the analysis of the Newell-Whitehead-Segel equation, confirming that its best solution is null through more straightforward methods and alternative representations.

Contribution

It provides a simpler proof that the solution to the equation is null, using recent insights into convolution integrals and inverse Fourier transforms.

Findings

The solution is confirmed to be null.

Simplified proof avoids complex nested integrals.

Alternative solution representations also yield null solutions.

Abstract

Previous analysis of the Newell-Whitehead-Segel equation proved the best solution is null; although, the method of solution generated complex nested integrals, therefore, difficult to analyze \cite{NWSgen,NWS2020}. Recent insights into the properties of the convolution integral enable considerable simplification of the solution in the codomain, producing much simpler representations. The inverse Fourier transform of the spectral solution proves to be a non-bijective, null solution, therefore, confirming previous suspicions. Alternative representations of the solution, either expansions or Fujita type solutions, all prove the solution to be a null function.