In this section, we study all weak traveling wave solutions of (1.2), i.e. solutions of the form

$u(t,x)=\phi (x-ct),c\in \mathbb{R}$(2.1)

for some $\phi :\mathbb{R}\to \mathbb{R}$ such that $\phi \to 0$ as $|x|\to \mathrm{\infty}$.
Note that if $\phi (x-ct)$ is a traveling wave solution of (1.2), then $-\phi (x+ct)$ is also a traveling wave solution of (1.2). Substituting (2.1) into (1.2) and integrating it, we have

$(c-\phi ){\phi}_{xx}-{\phi}_{x}^{2}=c\phi -\frac{1+3\beta}{2}{\phi}^{2},$(2.2)

We rewrite (2.2) as

$\frac{1}{2}{\left[{(\phi -c)}^{2}\right]}_{xx}=-c\phi +\frac{1+3\beta}{2}{\phi}^{2}.$

Now we give the definition of a traveling wave solution of (1.2).

A function $\phi (x-ct)\in {H}^{1}(\mathbb{R})$ is a nontrivial traveling wave solution of (1.2) with $c\in \mathbb{R}$ and $\phi \to 0$ as $|x|\to \mathrm{\infty}$.

The following lemma deals with the regularity of the traveling wave solutions. The idea is inspired by the study of the traveling waves of the Camassa–Holm equation [5].

*Let $\phi \mathit{}\mathrm{(}x\mathrm{-}c\mathit{}t\mathrm{)}$ be a traveling wave solutions of (1.2). Then*

${(\phi -c)}^{k}\in {C}^{j}(\mathbb{R}\setminus {\phi}^{-1}(c)),k\ge {2}^{j}.$(2.3)

*Therefore*

$\phi \in {C}^{\mathrm{\infty}}(\mathbb{R}\setminus {\phi}^{-1}(c)).$

#### Proof.

Let $\nu =\phi -c$ and denote $P(\nu )=-2c(\nu +c)+(1+3\beta ){(\nu +c)}^{2}$. So $P(\nu )$ is a polynomial in ν.
Then ν satisfies

${({\nu}^{2})}_{xx}=P(\nu ).$

Since $\nu \in {H}^{1}(\mathbb{R})$, we know that ${({\nu}^{2})}_{xx}\in {L}_{\mathrm{loc}}^{1}(\mathbb{R})$. Hence ${({\nu}^{2})}_{x}$ is absolutely continuous and
${\nu}^{2}\in {C}^{1}(\mathbb{R})$. Then $\nu \in {C}^{1}(\mathbb{R}\setminus {\nu}^{-1}(0))$.
Moreover, we have

${({\nu}^{k})}_{xx}={(k{\nu}^{k-1}{\nu}_{x})}_{x}={\displaystyle \frac{k}{2}}{\left({\nu}^{k-2}{({\nu}^{2})}_{x}\right)}_{x}=k(k-2){\nu}^{k-2}{\nu}_{x}^{2}+{\displaystyle \frac{k}{2}}{\nu}^{k-2}{({\nu}^{2})}_{xx}$$=k(k-2){\nu}^{k-2}{\nu}_{x}^{2}+{\displaystyle \frac{k}{2}}{\nu}^{k-2}P(\nu ).$(2.4)

For $k=3$, the right-hand side of (2.4) is in ${L}_{\mathrm{loc}}^{1}(\mathbb{R})$. Thus we conclude that ${\nu}^{3}\in {C}^{1}(\mathbb{R})$.
For $k\ge {2}^{2}=4$ we know that (2.4) implies

${({\nu}^{k})}_{xx}=\frac{k}{4}(k-2){\nu}^{k-4}{\left[{({\nu}^{2})}_{x}\right]}^{2}+\frac{k}{2}{\nu}^{k-2}P(\nu )\in {C}^{1}(\mathbb{R}).$

Therefore ${\nu}^{k}\in {C}^{1}(\mathbb{R})$ for $k\ge {2}^{2}=4$.

For $k\ge {2}^{3}=8$ we see from the above that ${\nu}^{4},{\nu}^{k-4},{\nu}^{k-2},{\nu}^{k-2}P(\nu )\in {C}^{2}(\mathbb{R})$. Moreover, we have

${\nu}^{k-2}{\nu}_{x}^{2}=\frac{1}{4}{({\nu}^{4})}_{x}\frac{1}{k-4}{({\nu}^{k-4})}_{x}\in {C}^{1}(\mathbb{R}).$

Hence from (2.4) we deduce that

${\nu}^{k}\in {C}^{3}(\mathbb{R}\setminus {\nu}^{-1}(0)),k\ge {2}^{3}=8.$

Applying the same argument to higher values of *k* we prove that ${\nu}^{k}\in {C}^{j}(\mathbb{R}\setminus {\nu}^{-1}(0))$ for $k\ge {2}^{j}$, and hence (2.3). This completes the proof of Lemma 2.2.
∎

Let ${\phi}_{x}=G$. Then (2.2) becomes

${({G}^{2})}_{\phi}-\frac{2}{c-\phi}{G}^{2}=\frac{2(c\phi -\frac{1+3\beta}{2}{\phi}^{2})}{c-\phi}.$(2.5)

Solving the first-order ordinary differential equation (2.5), we have

${\phi}_{x}^{2}=\frac{{\phi}^{2}\left[\frac{1+3\beta}{4}{\phi}^{2}-(1+\beta )c\phi +{c}^{2}\right]}{{(\phi -c)}^{2}}:=F(\phi ).$(2.6)

Consider the polynomial

$P(\phi )={\phi}^{2}\left[\frac{1+3\beta}{4}{\phi}^{2}-(1+\beta )c\phi +{c}^{2}\right]$(2.7)

with a double root at $\phi =0$. Then we can classify all traveling wave
solutions of (1.2) depending on the different behaviors of this polynomial.

If $\beta =-\frac{1}{3}$, then we know that $P(\phi )$ is the third-degree polynomial with a double zero at $\phi =0$ and a simple zero at $\phi =\frac{3}{2}c$ such that $P(\phi )=-\frac{2}{3}c{\phi}^{2}(\phi -\frac{3}{2}c)$.

If $\beta \ne -\frac{1}{3}$, then $P(\phi )$ is the fourth-degree polynomial with a double zero at $\phi =0$ and there are the three cases

(i)

$\mathrm{\Delta}<0$,

(ii)

$\mathrm{\Delta}=0$,

(iii)

$\mathrm{\Delta}>0$,

where $\mathrm{\Delta}:={c}^{2}\beta (\beta -1)$ is the determinant of $\frac{1+3\beta}{4}{\phi}^{2}-(1+\beta )c\phi +{c}^{2}$.

(i) $\mathrm{\Delta}<0$: For $0<\beta <1$, we have that $P(\phi )$ is the fourth-degree polynomial with a double zero at $\phi =0$.

(ii) $\mathrm{\Delta}=0$:

•

For $\beta =0$, $P(\phi )={\phi}^{2}{(\frac{1}{2}\phi -c)}^{2}$ is the fourth-degree polynomial with a double zero at $\phi =0$ and $\phi =2c$.

•

For $\beta =1$, $P(\phi )={\phi}^{2}{(\phi -c)}^{2}$ is the fourth-degree polynomial with a double zero at $\phi =0$ and $\phi =c$.

(iii) $\mathrm{\Delta}>0$:
For $\beta <0$ or $\beta >1$, $P(\phi )=\frac{1+3\beta}{4}{\phi}^{2}(c-{l}_{1}-\phi )(c-{l}_{2}-\phi )$ is the fourth-degree polynomial with a double zero at $\phi =0$ and a simple zero at $\phi =c-{l}_{1}$ or $\phi =c-{l}_{2}$, where

${l}_{1}=c\left[\frac{\beta -1}{1+3\beta}+\sqrt{{\left(\frac{\beta -1}{1+3\beta}\right)}^{2}+\frac{\beta -1}{1+3\beta}}\right],{l}_{2}=c\left[\frac{\beta -1}{1+3\beta}-\sqrt{{\left(\frac{\beta -1}{1+3\beta}\right)}^{2}+\frac{\beta -1}{1+3\beta}}\right]$

for $c>0$ and

${l}_{1}=c\left[\frac{\beta -1}{1+3\beta}-\sqrt{{\left(\frac{\beta -1}{1+3\beta}\right)}^{2}+\frac{\beta -1}{1+3\beta}}\right],{l}_{2}=c\left[\frac{\beta -1}{1+3\beta}+\sqrt{{\left(\frac{\beta -1}{1+3\beta}\right)}^{2}+\frac{\beta -1}{1+3\beta}}\right]$

for $c<0$ are two roots of the equation

${y}^{2}-\frac{2(\beta -1)}{1+3\beta}cy-\frac{\beta -1}{1+3\beta}{c}^{2}=0.$

If $\beta <-\frac{1}{3}$ then

${l}_{2}<0<c<{l}_{1}\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{l}_{2}<c<0<{l}_{1}.$(2.8)

If $-\frac{1}{3}<\beta <0$ then

${l}_{2}<{l}_{1}<0<c\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}c<0<{l}_{2}<{l}_{1}.$(2.9)

If $\beta >1$ then

${l}_{2}<0<{l}_{1}<c\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}c<{l}_{2}<0<{l}_{1}.$

Using the idea as introduced in [5] gives us the following conclusions:

(a)

Assume $F(\phi )$ has a simple zero at $\phi =m$ so that $F(m)=0$ and ${F}^{\prime}(m)\ne 0$. The solution φ of (2.6) satisfies

${\phi}_{x}^{2}=(\phi -m){F}^{\prime}(m)+O({(\phi -m)}^{2})\mathit{\hspace{1em}}\text{as}x\to m,$

where $f=O(g)$ as $x\to a$ means that $\left|\frac{f(x)}{g(x)}\right|$ is bounded in some interval $[a-\u03f5,a+\u03f5]$ with $\u03f5>0$. Therefore

$\phi (x)=m+\frac{1}{4}{(x-\xi )}^{2}{F}^{\prime}(m)+O({(x-\xi )}^{4})\mathit{\hspace{1em}}\text{as}x\to \xi ,$(2.10)

where $\phi (\xi )=m$.

(b)

If $F(\phi )$ has a double zero at *m*, so that ${F}^{\prime}(m)=0,{F}^{\mathrm{\prime \prime}}(m)>0$, then

${\phi}_{x}^{2}={(\phi -m)}^{2}{F}^{\mathrm{\prime \prime}}(m)+O({(\phi -m)}^{3})\mathit{\hspace{1em}}\text{as}\phi \to m.$

We obtain

$\phi (x)-m\sim \alpha \mathrm{exp}\left(-x\sqrt{{F}^{\mathrm{\prime \prime}}(m)}\right)\mathit{\hspace{1em}}\text{as}x\to \mathrm{\infty}$(2.11)

for some constant α. Thus $\phi \to m$ exponentially as $x\to \mathrm{\infty}$.

(c)

If φ approaches a double pole $\phi ({x}_{0})=c$ of $F(\phi )$.
Then

$\phi (x)-c=\alpha {|x-{x}_{0}|}^{1/2}+O({(x-{x}_{0})}^{3/2})\hspace{1em}\text{as}x\to {x}_{0},$(2.12)${\phi}_{x}=\{\begin{array}{cc}\frac{1}{2}\alpha {|x-{x}_{0}|}^{-1/2}+O({(x-{x}_{0})}^{1/2})\hfill & \text{as}x\downarrow {x}_{0},\hfill \\ -\frac{1}{2}\alpha {|x-{x}_{0}|}^{-1/2}+O({(x-{x}_{0})}^{1/2})\hfill & \text{as}x\uparrow {x}_{0}\hfill \end{array}$(2.13)

for some constant α. In particular, when *F* has a double pole, the solution φ has a cusp.

(d)

If the evolution of φ according to (2.6) suddenly
changes direction ${\phi}_{x}\mapsto -{\phi}_{x}$, then peaked solitary waves occur.

In view of (a)–(d), we give the following theorem on all bounded traveling wave
solutions of (1.2) with decay.

*Any bounded traveling wave of (1.2) with decay belongs to one of the following categories.*

(1)

*For *
$\beta <-\frac{1}{3}$
*:*

–

*If *
$c>0$
*, then there is a smooth traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c-{l}_{1}$
* and a cusped traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$.

–

*If *
$c<0$
*, then there is a smooth traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c-{l}_{2}$
* and an anticusped traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c$.

(2)

*For *
$\beta =-\frac{1}{3}$
*:*

–

*If *
$c>0$
*, then there is a cusped traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$.

–

*If *
$c<0$
*, then there is an anticusped traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c$.

(3)

*For *
$-\frac{1}{3}<\beta <0$
*:*

–

*If *
$c>0$
*, then there is a cusped traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$.

–

*If *
$c<0$
*, then there is an anticusped traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c$.

(4)

*For *
$\beta =0$
*:*

–

*If *
$c>0$
*, then there is a cusped traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$.

–

*If *
$c<0$
*, then there is an anticusped traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c$.

(5)

*For *
$0<\beta <1$
*:*

–

*If *
$c>0$
*, then there is a cusped traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$.

–

*If *
$c<0$
*, then there is an anticusped traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c$.

(6)

*For *
$\beta =1$
*:*

–

*If *
$c>0$
*, then there is a peaked traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$.

–

*If *
$c<0$
*, then there is an antipeaked traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c$.

(7)

*For *
$\beta >1$
*:*

–

*If *
$c>0$
*, then there is a smooth traveling wave with decay *
$\phi >0$
* with *
${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c-{l}_{1}$.

–

*If *
$c<0$
*, then there is a smooth traveling wave with decay *
$\phi <0$
* with *
${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c-{l}_{2}$.

#### Proof.

First, we consider $c>0$. The other case $c<0$ can be handled in a very similar way.

If $\beta =-\frac{1}{3}$, then (2.6) becomes

${\phi}_{x}^{2}=\frac{-\frac{2}{3}c{\phi}^{2}(\phi -\frac{3}{2}c)}{{(\phi -c)}^{2}}:={F}_{1}(\phi ).$

When $c>0$, $\phi =c$ is a double pole of ${F}_{1}(\phi )$. Hence from (2.12) and (2.13) we see that we obtain a traveling
wave with cusp at $\phi =c$, which decays exponentially.

If $0<\beta <1$ and $\beta =0$, then $\phi =c$ is a double pole of $F(\phi )$ and $\phi =0$ is a double zero of $F(\phi )$ in (2.6).
Therefore, in the same manner as above, we obtain a cusped traveling wave with ${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$ for $c>0$.
If $\beta =1$, then (2.6) becomes

${\phi}_{x}^{2}=\frac{{\phi}^{2}{(\phi -c)}^{2}}{{(\phi -c)}^{2}}:={F}_{2}(\phi ).$

When $c>0$, the smooth solution can be constructed until $\phi =c$. But it can make a sudden turn at $\phi =c$ and so give rise to a peak. Since $\phi =0$ is still a double zero of ${F}_{2}(\phi )$, we still have the exponential decay.

Figure 1 There are three different kinds of traveling wave solutions with decay for $c>0$ in Theorem 2.3.(i) Cusped traveling waves with ${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$ for $\beta <1$.(ii) Peaked traveling waves with ${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$ for $\beta =1$.(iii) Smooth traveling waves with ${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c-{l}_{1}$ for $\beta >1$.

Figure 2 The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.

If $\beta <-\frac{1}{3}$, $-\frac{1}{3}<\beta <0$, and $\beta >1$, then (2.6) becomes

${\phi}_{x}^{2}=\frac{(1+3\beta ){\phi}^{2}(c-\phi -{l}_{1})(c-\phi -{l}_{2})}{4{(\phi -c)}^{2}}:={F}_{3}(\phi ).$

When $\beta <-\frac{1}{3}$, we know that ${l}_{2}<0<c<{l}_{1}$ from (2.8). ${F}_{3}(\phi )$ has a simple zero at $\phi =c-{l}_{1}<0$ and a double zero at $\phi =0$. Therefore from (2.10) and (2.11) we see that in this case we can obtain a smooth traveling wave with ${\mathrm{min}}_{x\in \mathbb{R}}\phi (x)=c-{l}_{1}$ and an exponential decay to zero at infinity.
Moreover, since ${F}_{3}(\phi )$ has a double pole at $\phi =c$, we can also obtain a cusped traveling wave with ${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$, which decays exponentially.

If $-\frac{1}{3}<\beta <0$, we know that ${l}_{2}<{l}_{1}<0<c$ from (2.9). In this case ${F}_{3}(\phi )$ has a double pole at $\phi =c$ and a double zero at $\phi =0$. Hence we obtain a cusped traveling wave with ${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c$, which decays exponentially.

If $\beta >1$, we see that ${l}_{2}<0<{l}_{1}<c$ from (2.9). ${F}_{3}(\phi )$ has a simple zero at $\phi =c-{l}_{1}>0$ and a double zero at $\phi =0$. Therefore from (2.10) and (2.11) we see that in this case we can obtain a smooth traveling wave with ${\mathrm{max}}_{x\in \mathbb{R}}\phi (x)=c-{l}_{1}$ and an exponential decay to zero at infinity. This completes the proof of Theorem 2.3.
∎

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.