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

A. V. Shapovalov, S. A. Siniukov

TL;DR

This work develops a semiclassical, quasiparticle-based asymptotic framework for the nonlocal Fisher--KPP equation with a fractal time derivative in the weak diffusion regime. By employing trajectory-concentrated functions within the $F^α$-calculus and the Maslov method, it derives a fractal Einstein--Ehrenfest system (FlEES) for the second-order moments of interacting quasiparticles and constructs associated fractal linear equations (AFlLE) that linearize the nonlinear problem. A Green-function approach yields explicit leading-order solutions and a concrete two-quasiparticle example illustrating how the fractal parameter $α$ controls diffusion and growth on a Cantor-type time set. The framework—including moment dynamics, prolongation of the fractal derivative, and Green-function representation—provides a generalizable pathway to analyze fractal reaction-diffusion models and can be extended to higher dimensions and fractal spatial operators. These results illuminate how memory and fractal time structures influence propagation through a principled quasiparticle lens, with potential applications in complex biological and physical systems.

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 $α$, where $0<α\leq 1$. 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 $F^α$-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 $α$ on quasiparticle behavior.

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

TL;DR

This work develops a semiclassical, quasiparticle-based asymptotic framework for the nonlocal Fisher--KPP equation with a fractal time derivative in the weak diffusion regime. By employing trajectory-concentrated functions within the -calculus and the Maslov method, it derives a fractal Einstein--Ehrenfest system (FlEES) for the second-order moments of interacting quasiparticles and constructs associated fractal linear equations (AFlLE) that linearize the nonlinear problem. A Green-function approach yields explicit leading-order solutions and a concrete two-quasiparticle example illustrating how the fractal parameter controls diffusion and growth on a Cantor-type time set. The framework—including moment dynamics, prolongation of the fractal derivative, and Green-function representation—provides a generalizable pathway to analyze fractal reaction-diffusion models and can be extended to higher dimensions and fractal spatial operators. These results illuminate how memory and fractal time structures influence propagation through a principled quasiparticle lens, with potential applications in complex biological and physical systems.

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 , where . 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 -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 on quasiparticle behavior.

Paper Structure

This paper contains 15 sections, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Fisher--KPP equation with fractal time derivative
  3. Trajectory concentrated functions and decomposition of the equation
  4. Estimates of operators and moments
  5. The moments
  6. Expansion of equation coefficients
  7. The dynamical system of moments
  8. The fractal Einstein--Ehrenfest system of the second-order
  9. On the "prolongation" of the fractal time derivative
  10. Associated fractal linear equations
  11. The linear operator $\varepsilon\hat{L}_s$ expansion accurate to $O(\varepsilon^{5/2})$
  12. The Cauchy problem
  13. The Green function
  14. Example
  15. Conclusion

Figures (4)

  • Figure 1: The zeroth- and first-order moments, $\mu_1(\alpha,t), \mu_2(\alpha,t)$, and $x_1(\alpha,t), x_2(\alpha,t)$, respectively, for two quasiparticles with the parameter $\alpha=0.5$, $0.63$, $0.75$, $0.88$, $0.95$, $1$, are given for $t \in [0,1]$ and $F=C^{\alpha} \subset [0,1]$.
  • Figure 2: The zeroth-order moments $\mu(\alpha,t)=\mu_1(\alpha,t)+\mu_2(\alpha,t)$ for parameter $\alpha=0.5, 0.63, 0.75, 0.88, 0.95, 1$ and $F=C^{\alpha}\subset[0,1], \, t \in [0,1]$.
  • Figure 3: The asymptotic solutions $u(\alpha,x,t)$ from equation \ref{['sol1a']} corresponding to \ref{['solEx']} of equation \ref{['flfkpp1']}, are shown for a set of parameters given in \ref{['parametr']}, and for $\alpha=0.5$, $0.63$, $0.75$, $0.88$, $0.95$, $1$, at the following time points: $a) t=0$, $b) t=0.3$, $c) t=0.6$, $d) t=1$, where $F=C^{\alpha} \subset [0,1]$ and $t \in [0,1]$.
  • Figure :