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On Explicit Solutions for Coupled Reaction-Diffusion and Burgers-Type Equations with Variable Coefficients Through a Riccati System

José M. Escorcia, Erwin Suazo

Abstract

This work is concerned with the study of explicit solutions for generalized coupled reaction-diffusion and Burgers-type systems with variable coefficients. Including nonlinear models with variable coefficients such as diffusive Lotka-Volterra model, the Gray-Scott model, the Burgers equations. The equations' integrability (via the explicit formulation of the solutions) is accomplished by using similarity transformations and requiring that the coefficients fulfill a Riccati system. We present traveling wave type solutions, as well as solutions with more complex dynamics and relevant features such as bending. A Mathematica file has been prepared as supplementary material, verifying the Riccati systems used in the construction of the solutions.

On Explicit Solutions for Coupled Reaction-Diffusion and Burgers-Type Equations with Variable Coefficients Through a Riccati System

Abstract

This work is concerned with the study of explicit solutions for generalized coupled reaction-diffusion and Burgers-type systems with variable coefficients. Including nonlinear models with variable coefficients such as diffusive Lotka-Volterra model, the Gray-Scott model, the Burgers equations. The equations' integrability (via the explicit formulation of the solutions) is accomplished by using similarity transformations and requiring that the coefficients fulfill a Riccati system. We present traveling wave type solutions, as well as solutions with more complex dynamics and relevant features such as bending. A Mathematica file has been prepared as supplementary material, verifying the Riccati systems used in the construction of the solutions.

Paper Structure

This paper contains 25 sections, 6 theorems, 156 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Classical Coupled Reaction-Diffusion and Burgers Systems
  3. The linear reaction-diffusion system
  4. The diffusive Lotka-Volterra system
  5. The Gray-Scott model
  6. The coupled Burgers system
  7. The Generalized Coupled Reaction-Diffusion and Burgers-type Systems
  8. Solutions for the generalized linear reaction-diffusion system
  9. Constant diffusivity and an interaction coefficient with exponential decay
  10. Periodic diffusivity and an interaction coefficient with rational decay
  11. Constant diffusivity and a periodic interaction coefficient
  12. Solutions for the generalized Lotka-Volterra system
  13. Constant diffusivity and tanh-type interaction coefficients
  14. Constant diffusivity and interaction coefficients with periodic central axes
  15. Constant diffusivity and interaction coefficients with exponential growth
  16. ...and 10 more sections

Key Result

Theorem \oldthetheorem

The variable coefficients reaction-diffusion system can be transformed into the constant coefficients system

Figures (9)

  • Figure 1: Solutions for the system (\ref{['Ex1a']})-(\ref{['Ex1b']}) for the parameters $a_1 = 1,$$b_1 = 100,$ and $b_2 = 1$. Here, (a) and (c) describe the profiles of the functions $\psi$ and $\varphi$ respectively. The corresponding contours of $\psi$ and $\varphi$ are shown in (b) and (d).
  • Figure 2: Solutions for the system (\ref{['Ex2a']})-(\ref{['Ex2b']}) for the parameters $a_1 = 1,$$b_1 = 25,$ and $b_2 = 1$. Here, (a) and (c) describe the profiles of the functions $\psi$ and $\varphi$ respectively. The corresponding contours of $\psi$ and $\varphi$ are shown in (b) and (d).
  • Figure 3: Solutions for the system (\ref{['Ex2aa']})-(\ref{['Ex2bb']}). The profiles of the functions $\psi$ and $\varphi$ are shown in (a) and (c). In the contours of $\psi$ and $\varphi$, Figures (b) and (d), the bending dynamics are clearly observed.
  • Figure 4: Figures (a) and (c) describe the traveling wave solutions for the system (\ref{['Ex3a']})-(\ref{['Ex3b']}) with $a_1 = c_2 = c_1 = 2,$$a_2 = 1,$$b_1 = b_2 = \sqrt{2}$, $\nu_0 = \frac{1}{2},$$\nu_1 = -\frac{\sqrt{2}}{4}$, $A = 1,$$B = \frac{\sqrt{2}}{2}$, $\kappa(0) = 1$, and $\varepsilon(0) = 2$. The contours of these solutions are shown in (b) and (d).
  • Figure 5: Solutions for the system (\ref{['Ex4a']})-(\ref{['Ex4b']}) for the parameters $a_1 = c_2 = c_1 = 2,$$a_2 = 1,$$b_1 = b_2 = \sqrt{2}$, $\nu_0 = \frac{1}{2},$$\nu_1 = -\frac{\sqrt{2}}{4}$, $A = 1,$$B = \frac{\sqrt{2}}{2}$.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Theorem \oldthetheorem: Generalized Linear Reaction-Diffusion System
  • proof
  • Theorem \oldthetheorem: Exponential-type Solutions for the Linear Reaction-Diffusion System
  • proof
  • Theorem \oldthetheorem: Generalized Diffusive Lotka-Volterra System
  • proof
  • Proposition 1: Generalized Three-Component Lotka-Volterra System
  • proof
  • Theorem \oldthetheorem: Generalized Gray-Scott Model
  • proof
  • ...and 2 more