Traveling wave solutions for the generalized Burgers-Fisher equation

Abstract

Traveling wave solutions, in the form $u(x,t)=f(x+ct)$, to the generalized Burgers-Fisher equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k>0$, are classified with respect to their speed $c\in(-\infty,\infty)$ and the behavior at $\pm\infty$. The existence and uniqueness of traveling waves with any speed $c\in\mathbb{R}$ is established and their behavior as $x\to\pm\infty$ is described. In particular, it is shown that there exists a unique $c^*\in(0,\infty)$ such that there exists a unique soliton $f^*$ with speed $c^*$ and such that $$ \lim\limits_{\xi\to-\infty}f^*(\xi)=\lim\limits_{\xi\to\infty}f^*(\xi)=0, \quad \xi=x+ct. $$ Moreover, if $n<p+q+1$ then $c^*<kn$ and if $n>p+q+1$ then $c^*>kn$. For $c<\min\{c^*,kn\}$, any traveling wave with speed $c$ satisfies $\lim\limits_{\xi\to-\infty}f(\xi)=0$ and $\lim\limits_{\xi\to\infty}f(\xi)=1$, while for $c>\max\{c^*,kn\}$ any traveling wave with speed $c$ satisfies $\lim\limits_{\xi\to-\infty}f(\xi)=1$ and $\lim\limits_{\xi\to\infty}f(\xi)=0$. In particular, for any speed $c\in(0,c^*)$, there are traveling wave solutions $u$ with speed $c$ such that $u(x,t)\to1$ as $t\to\infty$, in contrast to the non-convective case $k=0$.