Traveling wave solutions, in the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u(x,t)=f(x+ct)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, to the generalized Burgers-Fisher equation <Equation ID="Equ62"> <EquationSource Format="TEX">$ \partial _tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in \mathbb {R}\times (0,\infty ), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>=</mo> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>+</mo> <mi>k</mi> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </msub> <mo>+</mo> <msup> <mi>u</mi> <mi>p</mi> </msup> <mo>-</mo> <msup> <mi>u</mi> <mi>q</mi> </msup> <mo>,</mo> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p>q\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mi>q</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, are classified with respect to their speed <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c\in (-\infty ,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mi>∞</mi> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the behavior at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\pm \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. The existence and uniqueness of traveling waves with any speed <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$c\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is established and their behavior as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x\rightarrow \pm \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mo>±</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> is described. In particular, it is shown that there exists a unique <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$c^\in (0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>c</mi> <mo>∗</mo> </msup> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that there exists a unique soliton <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$f^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> with speed <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$c^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>c</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> and such that <Equation ID="Equ63"> <EquationSource Format="TEX">$ \lim \limits _{\xi \rightarrow -\infty }f^(\xi )=\lim \limits _{\xi \rightarrow \infty }f^(\xi )=0, \quad \xi =x+ct. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>ξ</mi> <mo stretchy="false">→</mo> <mo>-</mo> <mi>∞</mi> </mrow> </munder> <msup> <mi>f</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>ξ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <msup> <mi>f</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>ξ</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mi>t</mi> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Moreover, if <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$n<p+q+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> then <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$c^<kn$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>c</mi> <mo>∗</mo> </msup> <mo><</mo> <mi>k</mi> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> and if <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$n>p+q+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> then <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$c^>kn$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>c</mi> <mo>∗</mo> </msup> <mo>></mo> <mi>k</mi> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$c<\min \{c^,kn\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo><</mo> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <msup> <mi>c</mi> <mo>∗</mo> </msup> <mo>,</mo> <mi>k</mi> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, any traveling wave with speed c satisfies <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\lim \limits _{\xi \rightarrow -\infty }f(\xi )=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>ξ</mi> <mo stretchy="false">→</mo> <mo>-</mo> <mi>∞</mi> </mrow> </munder> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\lim \limits _{\xi \rightarrow \infty }f(\xi )=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>ξ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, while for <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$c>\max \{c^,kn\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <msup> <mi>c</mi> <mo>∗</mo> </msup> <mo>,</mo> <mi>k</mi> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> any traveling wave with speed c satisfies <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\lim \limits _{\xi \rightarrow -\infty }f(\xi )=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>ξ</mi> <mo stretchy="false">→</mo> <mo>-</mo> <mi>∞</mi> </mrow> </munder> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\lim \limits _{\xi \rightarrow \infty }f(\xi )=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>ξ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, for any speed <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$c\in (0,c^)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>c</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, there are traveling wave solutions u with speed c such that <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$u(x,t)\rightarrow 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, in contrast to the non-convective case <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$k=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

Traveling Wave Solutions for the Generalized Burgers-Fisher Equation

期刊论文
发表日期：

2026年5月21日

查看全文

Razvan Gabriel Iagar,
Ariel Sánchez

摘要

Traveling wave solutions, in the form $u(x,t)=f(x+ct)$ $u (x, t) = f (x + c t)$ , 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 ), $ $\partial_{t} u = u_{xx} + k {(u^{n})}_{x} + u^{p} - u^{q}, (x, t) \in R \times (0, \infty),$ with $n\ge 2$ $n \geq 2$ , $p>q\ge 1$ $p > q \geq 1$ and $k>0$ $k > 0$ , are classified with respect to their speed $c\in (-\infty ,\infty )$ $c \in (- \infty, \infty)$ and the behavior at $\pm \infty $ $\pm \infty$ . The existence and uniqueness of traveling waves with any speed $c\in \mathbb {R}$ $c \in R$ is established and their behavior as $x\rightarrow \pm \infty $ $x \to \pm \infty$ is described. In particular, it is shown that there exists a unique $c^*\in (0,\infty )$ $c^{*} \in (0, \infty)$ such that there exists a unique soliton $f^*$ $f^{*}$ with speed $c^*$ $c^{*}$ and such that $ \lim \limits _{\xi \rightarrow -\infty }f^*(\xi )=\lim \limits _{\xi \rightarrow \infty }f^*(\xi )=0, \quad \xi =x+ct. $ $lim_{ξ \to - \infty} f^{*} (ξ) = lim_{ξ \to \infty} f^{*} (ξ) = 0, ξ = x + c t .$ Moreover, if $n<p+q+1$ $n p+q+1$ $n > p + q + 1$ then $c^*>kn$ $c^{*} > k n$ . For $c<\min \{c^*,kn\}$ $c < min {c^{*}, k n}$ , any traveling wave with speed c satisfies $\lim \limits _{\xi \rightarrow -\infty }f(\xi )=0$ $lim_{ξ \to - \infty} f (ξ) = 0$ and $\lim \limits _{\xi \rightarrow \infty }f(\xi )=1$ $lim_{ξ \to \infty} f (ξ) = 1$ , while for $c>\max \{c^*,kn\}$ $c > max {c^{*}, k n}$ any traveling wave with speed c satisfies $\lim \limits _{\xi \rightarrow -\infty }f(\xi )=1$ $lim_{ξ \to - \infty} f (ξ) = 1$ and $\lim \limits _{\xi \rightarrow \infty }f(\xi )=0$ $lim_{ξ \to \infty} f (ξ) = 0$ . In particular, for any speed $c\in (0,c^*)$ $c \in (0, c^{*})$ , there are traveling wave solutions u with speed c such that $u(x,t)\rightarrow 1$ $u (x, t) \to 1$ as $t\rightarrow \infty $ $t \to \infty$ , in contrast to the non-convective case $k=0$ $k = 0$ .