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 $n\geq2$, $p&gt;q\geq1$ and $k&gt;0$, 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 $c<sup>*\in(0,\infty)$ such that there exists a unique soliton $f<sup>*$ with speed $c<sup>*$ and such that $$\lim\limits_{\xi\to-\infty}f^{<em>(\xi)=\lim\limits_{\xi\to\infty}f^</em>(\xi)=0,} \quad \xi=x+ct.$$ Moreover, if $n&lt;p+q+1$ then $c^*&lt;kn$ and if $n&gt;p+q+1$ then $c<sup>*&gt;kn$. For $c&lt;\min{c<sup>*,kn}$, 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 $c&gt;\max{c<sup>*,kn}$ 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 $c\in(0,c<sup>*)$, 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 $k=0$.