Traveling wave solutions for the generalized Burgers-Fisher equation
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911183837069312 |
|---|---|
| author | Iagar, Razvan Gabriel Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Sánchez, Ariel |
| contents | 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 $$ \lim\limits_{ξ\to-\infty}f^*(ξ)=\lim\limits_{ξ\to\infty}f^*(ξ)=0, \quad ξ=x+ct. $$ 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 $\lim\limits_{ξ\to-\infty}f(ξ)=0$ and $\lim\limits_{ξ\to\infty}f(ξ)=1$, while for $c>\max\{c^*,kn\}$ any traveling wave with speed $c$ satisfies $\lim\limits_{ξ\to-\infty}f(ξ)=1$ and $\lim\limits_{ξ\to\infty}f(ξ)=0$. 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 $k=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Traveling wave solutions for the generalized Burgers-Fisher equation Iagar, Razvan Gabriel Sánchez, Ariel Analysis of PDEs Dynamical Systems 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 $$ \lim\limits_{ξ\to-\infty}f^*(ξ)=\lim\limits_{ξ\to\infty}f^*(ξ)=0, \quad ξ=x+ct. $$ 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 $\lim\limits_{ξ\to-\infty}f(ξ)=0$ and $\lim\limits_{ξ\to\infty}f(ξ)=1$, while for $c>\max\{c^*,kn\}$ any traveling wave with speed $c$ satisfies $\lim\limits_{ξ\to-\infty}f(ξ)=1$ and $\lim\limits_{ξ\to\infty}f(ξ)=0$. 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 $k=0$. |
| title | Traveling wave solutions for the generalized Burgers-Fisher equation |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2509.24909 |