Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lie symmetry classification and exact solutions of a diffusive Lotka-Volterra system with convection

Roman Cherniha, Vasyl' Davydovych

TL;DR

The paper addresses a nonlinear reaction–diffusion–convection system modeling viscous fingering from A+B→C by reducing a five-component model to a three-component diffusive Lotka–Volterra system with convection via a stream function. Using complete Lie symmetry analysis, it identifies admissible stream-function forms that maximize symmetry, derives equivalence transformations, and enumerates 11 symmetry extensions, with the widest invariance occurring for a linear velocity field, e.g. $U=(2y,-2x)$. It then constructs rich families of exact solutions through symmetry reductions, including Weierstrass-function-based and radially symmetric forms, providing explicit expressions for the concentrations and illustrating spatiotemporal evolution of reactants and product. These results yield analytic benchmarks for numerical simulations and deepen understanding of symmetry structures in multicomponent reaction–diffusion–convection models relevant to viscous fingering.

Abstract

A mathematical model for description of the viscous fingering induced by a chemical reaction is under study. This complicated five-component model is reduced to a three-component diffusive Lotka-Volterra system with convection by introducing a stream function. The system obtained is examined by the classical Lie method. A complete Lie symmetry classification is derived via a rigorous algorithm. In particular, it is proved that the widest Lie algebras of invariance occur when the stream function generate a linear velocity field. The most interesting cases (from the symmetry and applicability point of view) are further studied in order to derive exact solutions. A wide range of exact solutions are constructed for radially-symmetric stream functions. These solutions include time-dependent and radially symmetric solutions as well as more complicated solutions expressed in terms of the Weierstrass function. It was shown that some of exact solutions can be used for demonstration of spatiotemporal evolution of concentrations corresponding to two reactants and their product.

Lie symmetry classification and exact solutions of a diffusive Lotka-Volterra system with convection

TL;DR

The paper addresses a nonlinear reaction–diffusion–convection system modeling viscous fingering from A+B→C by reducing a five-component model to a three-component diffusive Lotka–Volterra system with convection via a stream function. Using complete Lie symmetry analysis, it identifies admissible stream-function forms that maximize symmetry, derives equivalence transformations, and enumerates 11 symmetry extensions, with the widest invariance occurring for a linear velocity field, e.g. . It then constructs rich families of exact solutions through symmetry reductions, including Weierstrass-function-based and radially symmetric forms, providing explicit expressions for the concentrations and illustrating spatiotemporal evolution of reactants and product. These results yield analytic benchmarks for numerical simulations and deepen understanding of symmetry structures in multicomponent reaction–diffusion–convection models relevant to viscous fingering.

Abstract

A mathematical model for description of the viscous fingering induced by a chemical reaction is under study. This complicated five-component model is reduced to a three-component diffusive Lotka-Volterra system with convection by introducing a stream function. The system obtained is examined by the classical Lie method. A complete Lie symmetry classification is derived via a rigorous algorithm. In particular, it is proved that the widest Lie algebras of invariance occur when the stream function generate a linear velocity field. The most interesting cases (from the symmetry and applicability point of view) are further studied in order to derive exact solutions. A wide range of exact solutions are constructed for radially-symmetric stream functions. These solutions include time-dependent and radially symmetric solutions as well as more complicated solutions expressed in terms of the Weierstrass function. It was shown that some of exact solutions can be used for demonstration of spatiotemporal evolution of concentrations corresponding to two reactants and their product.

Paper Structure

This paper contains 7 sections, 3 theorems, 104 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Lie symmetry classification
  3. Exact solutions
  4. Exact solutions of the diffusive Lotka--Volterra type system with a specific stream function
  5. Exact solutions of the diffusive Lotka--Volterra type system with an arbitrary radially-symmetric stream function
  6. Conclusions
  7. Acknowledgments

Key Result

Theorem 1

System (2-1) can be transformed into a system of the same structure using equivalence transformations and/or where $\alpha_0>0, \ \alpha_1, \ \alpha_2, \ t_0, \ x_0, \ y_0$ and $\alpha_3>0$ are the real group parameters, $H(t,x,y)$ is an arbitrary solution of the linear equation

Figures (3)

  • Figure 1: Surfaces representing the functions $u(t_0,x,y)$ (blue), $v(t_0,x,y)$ (yellow) and $w(t_0,x,y)$ (green) from the solution (\ref{['3-56']}) of system (\ref{['2-23']}) with the parameters $d_1=1, d_2=2, \ d_3=3, \ \alpha_1=\frac{1}{2}, \ \alpha_2=\frac{1}{4}, C_1=2, \ C_3=-15, \ C_4=55$ and $t_0=0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{2}$.
  • Figure 2: Surfaces representing the functions $u(t,x,y_0)$ (blue), $v(t,x,y_0)$ (yellow) and $w(t,x,y_0)$ (green) from the solution (\ref{['3-56']}) of system (\ref{['2-23']}) with the parameters $d_1=1, d_2=2, \ d_3=3, \ \alpha_1=\frac{1}{2}, \ \alpha_2=\frac{1}{4}, C_1=2, \ C_3=-15, \ C_4=55$ and $y_0=-1,0,1$.
  • Figure 3: Surfaces representing the functions $u(t,x,y_0)$ (blue), $v(t,x,y_0)$ (yellow) and $w(t,x,y_0)$ (green) from the solution (\ref{['3-56']}) of system (\ref{['2-23']}) with the parameters $d_1=1, d_2=2, \ d_3=3, \ \alpha_1=\frac{1}{2}, \ \alpha_2=\frac{1}{4}, C_1=2, \ C_3=5, \ C_4=10$ and $y_0=-1,0,1$.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Remark 3