Newell-Whitehead-Segel equation,A Simpler Proof
Luisiana X. Cundin
TL;DR
The paper addresses the nonlinear Newell-Whitehead-Segel equation, typically yielding an infinitely nested convolution solution that is hard to analyze, e.g. $\partial_t u - \nu \partial_{xx} u + \alpha u - \epsilon u^n = 0$. It adopts a simpler approach by representing $u$ as a convolution $u = (f * G e^{-lpha t})$ and applying the convolution exponent property $ (f * g)^n = (f * g^n) = (f^n * g)$ to obtain a Bernoulli-type ODE in the spectral domain, yielding $F$ in closed form. The authors show that the inverse Fourier transform of the spectral solution is a null function for all $n$ and parameter values, implying that the physically admissible solution is $u=0$. They discuss alternative representations such as Fujita-type equations and MOSV/Neumann expansions and demonstrate they yield zero or inconsistent results, all supported by a formal exponent-property theorem that underpins the simplifications. The result provides a streamlined framework for analyzing this class of nonlinear diffusion equations and cautions against relying on standard separation-based methods, indicating a universal null attractor for the NWS-type system.
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.