Quasiparticle solutions to the 1D nonlocal Fisher--KPP equation with a fractal time derivative in the weak diffusion approximation

Abstract

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher--Kolmogorov--Petrovskii--Piskunov (Fisher--KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order $\alpha$, where $0<\alpha\leq 1$. This approach is based on the semiclassical approximation and the principles of the Maslov method. The fractal time derivative is introduced in the framework of $F^{\alpha}$-calculus. The Fisher--KPP equation is decomposed into a system of nonlinear equations that describe the dynamics of interacting quasiparticles within classes of trajectory-concentrated functions. A key element in constructing approximate quasiparticle solutions is the interplay between the dynamical system of quasiparticle moments and an auxiliary linear system of equations, which is coupled with the nonlinear system. General constructions are illustrated through examples that examine the effect of the fractal parameter $\alpha$ on quasiparticle behavior.