On explicit solutions for coupled reaction-diffusion and Burgers-type equations with variable coefficients through a Riccati system

José M. Escorcia; Erwin Suazo

doi:10.3934/dcdss.2025030

Article Contents

2025, Volume 18, Issue 12: 3818-3845. Doi: 10.3934/dcdss.2025030

This issue Previous Article An anisotropic Dirichlet system including unbounded coefficients Next Article The exact limits of all order derivatives of the global smooth solution of a general Korteweg-de Vries-Burgers equation

On explicit solutions for coupled reaction-diffusion and Burgers-type equations with variable coefficients through a Riccati system

José M. Escorcia^1, , and
Erwin Suazo^2,

1.
Escuela de Ciencias Aplicadas e Ingeniería, Universidad EAFIT, Carrera 49 No. 7 Sur-50, Medellín 050022, Colombia
2.
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 W. University Drive, Edinburg, TX, 78539-2999, USA

^*Corresponding author: José M. Escorcia
^*Corresponding author: José M. Escorcia

Received: August 2024

Revised: February 2025

Early access: March 17, 2025

Published online: March 17, 2025

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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 the 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.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Solutions for the system (45)-(46) 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)

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Figure 2. Solutions for the system (49)-(50) 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)

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Figure 3. Solutions for the system (68)-(69). 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

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Figure 4. Figures (a) and (c) describe the traveling wave solutions for the system (84)-(85) 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)

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Figure 5. Solutions for the system (90)-(91) 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} $

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Figure 6. Solutions for the system (114)-(115) for the parameters $ b_1 = \frac{1}{8}, \ b_2 = 0 $. Here, (b) and (d) show the contours of $ \psi $ and $ \varphi $ functions, respectively

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Figure 7. Solutions for the system (119)-(121) for the parameters $ b_1 = \frac{1}{8}, \ b_2 = 0 $. The profiles of the functions $ \psi $ and $ \varphi $ are shown in (a) and (c). Figures (b) and (d) show the contours of $ \psi $ and $ \varphi $

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Figure 8. Solutions for the system (133)-(134) for the parameters $ b_1 = 2, \ b_2 = 1, \ c_1 = c_2 = 2 $, $ A = -\frac{111}{8}, $ $ B = -1, $ $ \beta(0) = 1, $ $ \gamma(0) = \varepsilon(0) = 0 $

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Figure 9. Solutions for the system (137)-(138) for the parameters $ b_1 = 2, \ b_2 = 1, \ c_1 = c_2 = 2 $, $ A = \frac{111}{8}, $ $ B = 1, $ $ \beta(0) = \gamma(0) = \varepsilon(0) = 1 $

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