Chemical solitons from cubic autocatalysis: a KdV-based reduction and exact solutions

Abstract

We consider a one-dimensional cubic autocatalytic reaction–diffusion–advection system based on the scheme \(A+2B\rightarrow 3B\), \(B\rightarrow C\). Focusing on perturbations of a travelling reaction front, we study the regime of weak nonlinearity and long wavelengths in which a controlled asymptotic reduction is possible. Using a multiple-scale expansion near a marginally stable front, we obtain an effective Korteweg–de Vries (KdV)-type amplitude equation governing small, localized modulations of the autocatalyst concentration. Within this asymptotic framework, classical KdV soliton solutions provide a coarse-grained description of localized chemical pulses. Standard soliton invariants and phase shifts are interpreted in chemically meaningful terms, including excess autocatalyst content, effective pulse energy, and front displacement during pairwise interactions. Numerical simulations of the full reaction–diffusion system show quantitative agreement with the KdV approximation within its range of validity, confirming that the reduced description accurately captures the shape, propagation, and elastic interaction of localized pulses in the weakly nonlinear regime.