Traveling wave solutions of the Burgers-Huxley equations

TL;DR

This work addresses traveling wave solutions of the Burgers-Huxley equation by reducing the PDE to a planar polynomial ODE system via a traveling-wave reduction $\xi=z-ct$, and then analyzing global dynamics through the Poincaré disk with blow-up techniques. It proves that seven distinct global phase portraits arise, depending on $n\in\{1,2\}$, $k\in\mathbb{N}$, and $c>0$, and that traveling waves correspond to heteroclinic connections between $E_1=(1,0)$ and $E_0=(0,0)$. The nonexistence of limit cycles for $c\ge1$ is established, while traveling-wave solutions are shown to exist for $c\ge2$ by locating trajectories with $0<x<1$ that connect the two equilibria. Overall, the paper provides a geometric, global-dynamical perspective on wave propagation in Burgers-Huxley-type reaction-diffusion systems under the specified nonlinearity.

Abstract

We study the traveling wave solutions of the Burgers-Huxley equation from a geometric point of view via the qualitative theory of ordinary differential equations. By using the Poincaré compactification we study the global phase portraits of a family of polynomial ordinary differential equations in the plane related to the Burgers-Huxley equation. We obtain the traveling wave solutions and their asymptotic behaviors from the orbits that connect equilibrium points taking into account the restrictions of the studied equation.

Figures (4)

• Figure 1: Global phase portraits of systems \ref{['maineq']}. See Theorem \ref{['thm:01']}. The orbits in red give rise to the traveling wave solutions of \ref{['eq1']}. See Theorem \ref{['thm:traveling']}.
• Figure 2: $(a)$ Topological local phase portraits at $I_1$ and ${\tilde{I}}_1$ of systems \ref{['U1c1']}. $(b)$ Topological local phase portraits at $I_3$ of systems \ref{['U2c1']} if $k=1$.
• Figure 3: $(a)$ Desingularization of systems \ref{['U2c1']} with $k>1$ odd using polar blow-ups. $(b)$ Topological local phase portraits at the origin of systems \ref{['U2c1']} with $k>1$ odd. $(c)$ Dynamics close to the infinity of systems \ref{['maineq']} with $n=1$ and $k\geq1$ odd in the Poincaré disk. $(d)$ Desingularization of systems \ref{['U2c1']} using polar blow-ups whit $k>1$ even. $(e)$ Topological local phase portraits at the origin of systems \ref{['U2c1']} with $k>1$ even. $(f)$ Dynamics close to the infinity of systems \ref{['maineq']} with $n=1$ and $k\geq1$ even in the Poincaré disk.
• Figure 4: $(a)$ Desingularization of systems \ref{['U2c2']} using polar blow-ups. $(b)$ Topological local phase portraits at the origin of systems \ref{['U2c2']}. $(c)$ Dynamics close to the infinity of systems \ref{['maineq']} with $n=2$ and $k=1$ in the Poincaré disk. $(d)$ Desingularization of systems \ref{['U2c3']} with $k>1$ odd using polar blow-ups. $(e)$ Topological local phase portraits at the origin of systems \ref{['U2c3']} with $k>1$ odd. $(f)$ Dynamics close to the infinity of systems \ref{['maineq']} with $n=2$ and $k>1$ odd in the Poincaré disk. $(g)$ Desingularization of systems \ref{['U2c3']} with $k>1$ even using polar blow-ups. $(h)$ Topological local phase portraits at the origin of systems \ref{['U2c3']} with $k>1$ even. $(i)$ Dynamics close to the infinity of systems \ref{['maineq']} with $n=2$ and $k>1$ even in the Poincaré disk.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3