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Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation

M. S. Bruzón, T. M. Garrido, E. Recio, R. de la Rosa

TL;DR

This work analyzes a generalized (2+1)-dimensional inhomogeneous Fisher-Kolmogorov equation $u_t = a\,u - M(x)N(y)\,u^2 + b\,(u_{xx}+u_{yy})$ with exponential inhomogeneities $M(x)=m_1 e^{p x}$ and $N(y)=n_1 e^{q y}$. Using the Lie group method, it classifies all point symmetries and then employs two-dimensional abelian subalgebras to reduce the PDE to ordinary differential equations, yielding new invariant solutions. One reduction leads to a travelling-wave type solution expressed via a Weierstrass elliptic function under a parameter constraint, while another reduction produces a rational-exponential solution showing shock-like transitions and exponential-sheet behavior. The results provide explicit exact solutions and insights into wave propagation, shocks, and blow-up phenomena in inhomogeneous reaction-diffusion systems, and set the stage for extensions to more general inhomogeneities.

Abstract

In this work we consider a Fisher-Kolmogorov equation depending on two exponential functions of the spatial variables. We study this equation from the point of view of symmetry reductions in partial differential equations. Through two-dimensional abelian subalgebras, the equation is reduced to ordinary differential equations. New solutions have been derived and interpreted.

Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation

TL;DR

This work analyzes a generalized (2+1)-dimensional inhomogeneous Fisher-Kolmogorov equation with exponential inhomogeneities and . Using the Lie group method, it classifies all point symmetries and then employs two-dimensional abelian subalgebras to reduce the PDE to ordinary differential equations, yielding new invariant solutions. One reduction leads to a travelling-wave type solution expressed via a Weierstrass elliptic function under a parameter constraint, while another reduction produces a rational-exponential solution showing shock-like transitions and exponential-sheet behavior. The results provide explicit exact solutions and insights into wave propagation, shocks, and blow-up phenomena in inhomogeneous reaction-diffusion systems, and set the stage for extensions to more general inhomogeneities.

Abstract

In this work we consider a Fisher-Kolmogorov equation depending on two exponential functions of the spatial variables. We study this equation from the point of view of symmetry reductions in partial differential equations. Through two-dimensional abelian subalgebras, the equation is reduced to ordinary differential equations. New solutions have been derived and interpreted.

Paper Structure

This paper contains 6 sections, 1 theorem, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Lie symmetries
  3. Reductions and some exact solutions
  4. Reduction under the two-dimensional abelian subalgebra $\left\{ X_1-\alpha X_2, X_{2'}+\beta X_2 \right\}$
  5. Reduction under the two-dimensional abelian subalgebra $\left\{ X_2, X_{2'} \right\}$
  6. Conclusions

Key Result

theorem 1

The complete classification of the point symmetries admitted by the (2+1)-dimensional inhomogeneous Fisher-Kolmogorov equation (edp) with $M(x)=m_1 e^{px}$, $N(y)=n_1 e^{qy}$ is given by:

Figures (4)

  • Figure 1: Shock solution (\ref{['solAr']}) at $(t,0,0)$ with $c_1=50$ (solid), $c_1=1$ (dash), $c_1=10^{-4}$ (dot).
  • Figure 2: Solution (\ref{['solAr']}) at $(t,0,0)$ which blows-up in finite times.
  • Figure 3: Shock transition solution (\ref{['solBr']}) $u(t,0,0)= \frac{7}{3 {e}^{-7t}+2}.$
  • Figure 4: Spatial distribution of solution (\ref{['solBr']})) $u(0,x,y)= \frac{7}{5} {e}^{-x-2y}.$

Theorems & Definitions (1)

  • theorem 1