* [7] J. Buzzi, _Subshifts of quasi-finite type_ , Inventiones Mathematicae 159 (2005), 369–406.
  * [8] V. Climenhaga, _Specification and towers in shift spaces_ , Comm. Math. Phys. 364 (2018), 441–504.
  * [9] V. Climenhaga, Entropy-bounds-for-equilibrium-states, https://vaughnclimenhaga.wordpress.com/2017/01/26/entropy-bounds-for-equilibrium-states/ (2017)
  * [10] V. Climenhaga and D. Thompson, _Intrinsic ergodicity beyond specification: $\beta$\- shifts, S-gap shifts, and their factors_, Israel J. Math. 192 (2012), no. 2, 785–817.
  * [11] V. Climenhaga and D. Thompson, _Equilibrium states beyond specification and the Bowen property_ , Journal of the London Mathematical Society 87 (2013), 401–427.
  * [12] T. Dillon, _Dynamics and entropy of $\mathcal{S}$-graph shifts_, Discrete and Continuous Dynamical Systems, to appear.
  * [13] D. Fiebig and U. Fiebig, _Covers for coded systems,_ In Peter Walters, editor, Symbolic Dynamics and its Applications, volume 136 of Contemporary Mathematics, American Mathematical Society, 1992.
  * [14] F. García-Ramos and R. Pavlov, _Extender sets and measures of maximal entropy for subshifts_ , Journal of the London Mathematical Society, 100 (2019), 1013-1033.
  * [15] B. M. Gurevič, _Shift entropy and Markov measures in the space of paths of a countable graph_ (Russian). Dokl. Akad. Nauk SSSR 192 (1970), 963–965.
  * [16] N. Haydn, _Phase transitions in one-dimensional subshifts_ , Discrete and Continuous Dynamical Systems 33 (2013) 1965–1973.
  * [17] F. Hofbauer, _$\beta$ -Shifts have unique maximal measure_, Monatshefte für Mathematik 85 (1978), 189–198.
  * [18] W. Krieger, _On the uniqueness of the equilibrium state_ , Math. Systems Theory 8 (1974), 97–104 .
  * [19] D. Lind and B. Marcus, _An Introduction to Symbolic Dynamics and Coding_ , Cambridge University Press, New York, NY, 1989.
  * [20] B. Matson and E. Sattler, _$S$ -limited shifts_, Real Analysis Exchange 43 (2018), 393–416.
  * [21] D. Mauldin and M. Urbanski, _Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets_ , Cambridge Tracts in Mathematics, Series Number 148, Cambridge University Press, (2003), 1st edition.
  * [22] R. Pavlov, _On entropy and intrinsic ergodicity of coded shifts_ , Proceedings of the American Mathematical Society, 148 (2022), 4717–4731.
  * [23] K. Petersen, _Chains, entropy, coding_ , Ergodic Theory and Dynamical Systems, 6 (1986), 415–448.
  * [24] O. Sarig, _Existence of Gibbs measures for countable Markov shifts_ , Proceedings of the American Mathematical Society, 131 (2003), 1751–1758.
  * [25] C. Spandl, _Computing the topological entropy of shifts_ , Mathematical Logic Quarterly, 53 (2007), 493–510.
  * [26] D. Vere-Jones, _Ergodic properties of nonnegative matrices._ I, Pacific J. Math 22 (1967), 361–386.
  * [27] M. S. Shtilman, _On the number of invariant measures with maximal entropy for the shift in a sequence space_ , Mat. Zam. 9 (1971), 291–302 .
  * [28] P. Walters, _An introduction to ergodic theory_ , Graduate Texts in Mathematics 79, Springer, 1981.
  * [29] B. Weiss, _Intrinsically ergodic systems_ , Bulletin American Mathematical Society 76 (1970), 1266–1269.
  * [30] B. Weiss, _Subshifts of finite type and sofic systems_ , Monatsh. Math. 77 (1973), 462–474.


# Periodic and solitary wave solutions to the Fornberg-Whitham equation

Jiangbo Zhou Corresponding author<EMAIL_ADDRESS>Lixin Tian  Nonlinear
Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang,
Jiangsu 212013, China

###### Abstract

In this paper, new travelling wave solutions to the Fornberg-Whitham equation

$u_{t}-u_{xxt}+u_{x}+uu_{x}=uu_{xxx}+3u_{x}u_{xx}$

are investigated. They are characterized by two parameters. The expresssions
of the periodic and solitary wave solutions are obtained.

###### keywords:

Fornberg-Whitham equation , solitary wave, periodic wave

,

##  1 Introduction

Recently, Ivanov [1] investigated the integrability of a class of nonlinear
dispersive wave equations

$u_{t}-u_{xxt}+\partial_{x}(\kappa u+\alpha u^{2}+\beta u^{3})=\nu
u_{x}u_{xx}+\gamma uu_{xxx},$ (1.1)

where and $\alpha,\beta,\gamma,\kappa,\nu$ are real constants.

The important cases of Eq.(1.1) are:

The hyperelastic-rod wave equation

$u_{t}-u_{xxt}+3uu_{x}=\gamma(2u_{x}u_{xx}+uu_{xxx}),$ (1.2)

has been recently studied as a model, describing nonlinear dispersive waves in
cylindrical compressible hyperelastic rods [2]-[7]. The physical parameters of
various compressible materials put $\gamma$ in the range from -29.4760 to
3.4174 [2, 4].

The Camassa-Holm equation

$u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx},$ (1.3)

describes the unidirectional propagation of shallow water waves over a flat
bottom [8, 9]. It is completely integrable [1] and admits, in addition to
smooth waves, a multitude of travelling wave solutions with singularities:
peakons, cuspons, stumpons and composite waves [9]-[12]. The solitary waves of
Eq.(1.2) are smooth if $\kappa>0$ and peaked if $\kappa=0$ [9, 10]. Its
solitary waves are stable solitons [13, 14], retaining their shape and form
after interactions [15]. It models wave breaking [16]-[18].

The Degasperis-Procesi equation

$u_{t}-u_{xxt}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx},$ (1.4)

models nonlinear shallow water dynamics. It is completely integrable [1] and
has a variety of travelling wave solutions including solitary wave solutions,
peakon solutions and shock waves solutions [20]-[27].

The Fornberg-Whitham equation

$u_{t}-u_{xxt}+u_{x}+uu_{x}=uu_{xxx}+3u_{x}u_{xx},$ (1.5)

appeared in the study qualitative behaviors of wave-breaking [28]. It admits a
wave of greatest height, as a peaked limiting form of the travelling wave
solution [29], $u(x,t)=A\exp(-\frac{1}{2}\left|{x-\frac{4}{3}t}\right|)$,
where $A$ is an arbitrary constant. It is not completely integrable [1].

The regularized long-wave or BBM equation

$u_{t}-u_{xxt}+u_{x}+uu_{x}=0,$ (1.6)

and the modified BBM equation

$u_{t}-u_{xxt}+u_{x}+3u^{2}u_{x}=0,$ (1.7)

have also been investigated by many authors [30]-[38].

Many efforts have been devoted to study Eq.(1.2)-(1.4),(1.6) and (1.7),
however, little attention was paid to study Eq.(1.5). In [39], we constructed
two types of bounded travelling wave solutions $u(\xi)(\xi=x-ct)$ to Eq.(1.5),
which are defined on semifinal bounded domains and called kink-like and
antikink-like wave solutions. In this paper, we continue to study the
travelling wave solutions to Eq.(1.5). Following Vakhnenko and Parkes’s
strategy in [19], we obtain some periodic and solitary wave solutions $u(\xi)$
to Eq.(1.5) which are defined on $(-\infty,+\infty)$. The travelling wave
solutions obtained in this paper are obviously different from those obtained
in our previous work [39]. To the best of our knowledge, these solutions are
new for Eq.(1.5). Our work may help people to know deeply the described
physical process and possible applications of the Fornberg-Whitham equation.

The remainder of the paper is organized as follows. In Section 2, for
completeness and readability, we repeat Appendix A in [19], which discuss the
solutions to a first-order ordinary differential equaion. In Section 3, we
show that, for travelling wave solutions, Eq.(1.5) may be reduced to a first-
order ordinary differential equation involving two arbitrary integration
constants $a$ and $b$. We show that there are four distinct periodic solutions
corresponding to four different ranges of values of $a$ and restricted ranges
of values of $b$. A short conclusion is given in Section 4.

##  2 Solutions to a first-order ordinary differential equaion

This section is due to Vakhnenko and Parkes (see Appendix A in [19]). For
completeness and readability, we repeat it in the following.

Consider solutions to the following ordinary differential equation

$(\varphi\varphi_{\xi})^{2}=\varepsilon^{2}f(\varphi),$ (2.1)

where

$f(\varphi)=(\varphi-\varphi_{1})(\varphi-\varphi_{2})(\varphi_{3}-\varphi)(\varphi_{4}-\varphi),$
(2.2)

and $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$, $\varphi_{4}$ are chosen to
be real constants with
$\varphi_{1}\leq\varphi_{2}\leq\varphi\leq\varphi_{3}\leq\varphi_{4}$.

Following [40] we introduce $\zeta$ defined by

$\frac{d\xi}{d\zeta}=\frac{\varphi}{\varepsilon},$ (2.3)

so that Eq.(2.1) becomes

$(\varphi_{\zeta})^{2}=f(\varphi).$ (2.4)

Eq.(2.4) has two possible forms of solution. The first form is found using
result 254.00 in [41]. Its parametric form is

$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{2}-\varphi_{1}n\mathrm{sn}^{2}(w|m)}{\textstyle
1-n\mathrm{sn}^{2}(w|m)},\\\ \xi=\frac{\displaystyle
1}{\displaystyle\varepsilon
p}(w\varphi_{1}+(\varphi_{2}-\varphi_{1})\Pi(n;w|m)),\\\ \end{array}}\right.$
(2.5)

with $w$ as the parameter, where

$m=\frac{(\varphi_{3}-\varphi_{2})(\varphi_{4}-\varphi_{1})}{(\varphi_{4}-\varphi_{2})(\varphi_{3}-\varphi_{1})},p=\frac{1}{2}\sqrt{(\varphi_{4}-\varphi_{2})(\varphi_{3}-\varphi_{1})},w=p\zeta,$
(2.6)

and

$n=\frac{\varphi_{3}-\varphi_{2}}{\varphi_{3}-\varphi_{1}}.$ (2.7)

In (2.5) $\mathrm{sn}(w|m)$ is a Jacobian elliptic function, where the
notation is as used in Chapter 16 of [42]. $\Pi(n;w|m)$ is the elliptic
integral of the third kind and the notation is as used in Section 17.2.15 of
[42].

The solution to (2.1) is given in parametric form by (2.5) with $w$ as the
parameter. With respect to $w$, $\varphi$ in (2.5) is periodic with period
$2K(m)$, where $K(m)$ is the complete elliptic integral of the first kind. It
follows from (2.5) that the wavelength $\lambda$ of the solution to (2.1) is

$\lambda=\frac{\displaystyle 2}{\displaystyle\varepsilon
p}|\varphi_{1}K(m)+(\varphi_{2}-\varphi_{1})\Pi(n|m)|,$ (2.8)

where $\Pi(n|m)$ is the complete elliptic integral of the third kind.

When $\varphi_{3}=\varphi_{4}$, $m=1$, (2.5) becomes

$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{2}-\varphi_{1}n\tanh^{2}w}{\textstyle
1-n\tanh^{2}w},\\\ \xi=\frac{\textstyle
1}{\textstyle\varepsilon}(\frac{\textstyle w\varphi_{3}}{\textstyle
p}-2\tanh^{-1}(\sqrt{n}\tanh w)).\\\ \end{array}}\right.$ (2.9)

The second form of solution of (2.5) is found using result 255.00 in [41]. Its
parametric form is

$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{3}-\varphi_{4}n\mathrm{sn}^{2}(w|m)}{\textstyle
1-n\mathrm{sn}^{2}(w|m)},\\\ \xi=\frac{\displaystyle
1}{\displaystyle\varepsilon
p}(w\varphi_{4}-(\varphi_{4}-\varphi_{3})\Pi(n;w|m)),\\\ \end{array}}\right.$
(2.10)

where $m,p,w$ are as in (2.6), and

$n=\frac{\varphi_{3}-\varphi_{2}}{\varphi_{4}-\varphi_{2}}.$ (2.11)

The solution to (2.1) is given in parametric form by (2.10) with $w$ as the
parameter. The wavelength $\lambda$ of the solution to (2.1) is

$\lambda=\frac{\displaystyle 2}{\displaystyle\varepsilon
p}|\varphi_{4}K(m)-(\varphi_{4}-\varphi_{3})\Pi(n|m)|.$ (2.12)

When $\varphi_{1}=\varphi_{2}$, $m=1$, (2.10) becomes

$\left\\{{\begin{array}[]{l}\varphi=\frac{\textstyle\varphi_{3}-\varphi_{4}n\tanh^{2}w}{\textstyle
1-n\tanh^{2}w},\\\ \xi=\frac{\textstyle
1}{\textstyle\varepsilon}(\frac{\textstyle w\varphi_{2}}{\textstyle
p}+2\tanh^{-1}(\sqrt{n}\tanh w)).\\\ \end{array}}\right.$ (2.13)

##  3 Periodic and solitary wave solutions to Eq.(1.5)

Eq.(1.5) can also be written in the form

$(u_{t}+uu_{x})_{xx}=u_{t}+uu_{x}+u_{x}.$ (3.1)

Let $u=\varphi(\xi)+c$ with $\xi=x-ct$ be a travelling wave solution to
Eq.(3.1), then it follows that

$(\varphi\varphi_{\xi})_{\xi\xi}=\varphi\varphi_{\xi}+\varphi_{\xi}.$ (3.2)

Integrating (3.2) twice with respect to $\xi$, we have

$(\varphi\varphi_{\xi})^{2}=\frac{1}{4}(\varphi^{4}+\frac{8}{3}\varphi^{3}+a\varphi^{2}+b),$
(3.3)

where $a$ and $b$ are two arbitrary integration constants.

Eq.(3.3) is in the form of Eq.(2.1) with $\varepsilon=\frac{\textstyle
1}{\textstyle 2}$ and $f(\varphi)=(\varphi^{4}+\frac{\textstyle 8}{\textstyle
3}\varphi^{3}+a\varphi^{2}+b)$. For convenience we define $g(\varphi)$ and
$h(\varphi)$ by

$f(\varphi)=\varphi^{2}g(\varphi)+b,\ \mbox{where}\
g(\varphi)=\varphi^{2}+\frac{8}{3}\varphi+a,$ (3.4)
$f^{\prime}(\varphi)=2\varphi h(\varphi),\ \mbox{where}\
h(\varphi)=2\varphi^{2}+4\varphi+a,$ (3.5)

and define $\varphi_{L}$, $\varphi_{R}$, $b_{L}$, and $b_{R}$ by

$\varphi_{L}=-\frac{\textstyle 1}{\textstyle
2}(2+\sqrt{4-2a}),\varphi_{R}=-\frac{\textstyle 1}{\textstyle
2}(2-\sqrt{4-2a}),$ (3.6)
$b_{L}=-\varphi_{L}^{2}g(\varphi_{L})=\frac{\textstyle a^{2}}{\textstyle
4}-2a+\frac{\textstyle 8}{\textstyle 3}+\frac{\textstyle 2}{\textstyle
3}(2-a)\sqrt{4-2a},$ (3.7)
$b_{R}=-\varphi_{R}^{2}g(\varphi_{R})=\frac{\textstyle a^{2}}{\textstyle
4}-2a+\frac{\textstyle 8}{\textstyle 3}-\frac{\textstyle 2}{\textstyle
3}(2-a)\sqrt{4-2a}.$ (3.8)

Obviously, $\varphi_{L}$, $\varphi_{R}$ are the roots of $h(\varphi)=0$.

In the following, suppose that $a<2$ and $a\neq 0$ such that $f(\varphi)$ has
three distinct stationary points: $\varphi_{L}$, $\varphi_{R}$, $0$ and
comprise two minimums separated by a maximum. Under this assumption, Eq.(3.3)
has periodic and solitary wave solutions that have different analytical forms
depending on the values of $a$ and $b$ as follows:

(1) $a<0$

In this case $\varphi_{L}<0<\varphi_{R}$ and $f(\varphi_{L})<f(\varphi_{R})$.
For each value $a<0$ and $0<b<b_{R}$ (a corresponding curve of $f(\varphi)$ is
shown in Fig.1(a)), there are periodic inverted loop-like solutions to
Eq.(3.3) given by (2.5) so that $0<m<1$, and with wavelength given by (2.8);
see Fig.2(a) for an example.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1: The curve of $f(\varphi)$. (a) $a=-50$, $b=233$; (b) $a=-50$,
$b=374.1346$; (c) $a=1.5$, $b=-0.05$; (d) $a=1.5$, $b=0$; (e)
$a=\frac{\textstyle 16}{\textstyle 9}$, $b=-0.1$; (f) $a=\frac{\textstyle
16}{\textstyle 9}$, $b=0$; (g) $a=1.9$, $b=-0.24$; (h) $a=1.9$, $b=-0.2010$.

The case $a<0$ and $b=b_{R}$ (a corresponding curve of $f(\varphi)$ is shown
in Fig.1(b)) corresponds to the limit $\varphi_{3}=\varphi_{4}=\varphi_{R}$ so
that $m=1$, and then the solution is an inverted loop-like solitary wave given
by (2.9) with $\varphi_{2}\leq\varphi<\varphi_{R}$ and

$\varphi_{1}=-\frac{\textstyle 1}{\textstyle
6}(2+3\sqrt{4-2a}+2\sqrt{4+6\sqrt{4-2a}}),$ (3.9)
$\varphi_{2}=-\frac{\textstyle 1}{\textstyle
6}(2+3\sqrt{4-2a}-2\sqrt{4+6\sqrt{4-2a}}),$ (3.10)

see Fig.3(a) for an example.

(2) $0<a<\frac{\textstyle 16}{\textstyle 9}$

In this case $\varphi_{L}<\varphi_{R}<0$ and $f(\varphi_{L})<f(0)$. For each
value $0<a<\frac{\textstyle 16}{\textstyle 9}$ and $b_{R}<b<0$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(c)), there are periodic
hump-like solutions to Eq.(3.3) given by (2.5) so that $0<m<1$, and with
wavelength given by (2.8); see Fig.2(b) for an example.

The case $0<a<\frac{\textstyle 16}{\textstyle 9}$ and $b=0$ (a corresponding
curve of $f(\varphi)$ is shown in Fig.1(d)) corresponds to the limit
$\varphi_{3}=\varphi_{4}=0$ so that $m=1$, and then the solution can be given
by (2.9) with $\varphi_{1}$ and $\varphi_{2}$ given by the roots of
$g(\varphi)=0$, namely

$\varphi_{1}=-\frac{\textstyle 4}{\textstyle 3}-\frac{\textstyle 1}{\textstyle
3}\sqrt{16-9a},\varphi_{2}=-\frac{\textstyle 4}{\textstyle 3}+\frac{\textstyle
1}{\textstyle 3}\sqrt{16-9a}.$ (3.11)

In this case we obtain a weak solution, namely the periodic upward-cusp wave

$\varphi=\varphi(\xi-2j\xi_{m}),(2j-1)\xi_{m}<\xi<(2j+1)\xi_{m},\ j=0,\pm
1,\pm 2,\cdots,$ (3.12)

where

$\varphi(\xi)=(\varphi_{2}-\varphi_{1}\tanh^{2}(\xi/4))\cosh^{2}(\xi/4),$
(3.13)

and

$\xi_{m}=4\tanh^{-1}\sqrt{\frac{\varphi_{2}}{\varphi_{1}}},$ (3.14)

see Fig.3(b) for an example.

(a)

(b)

(c)

(d)

Figure 2: Periodic solutions to Eq.(3.3) with $0<m<1$. (a) $a=-50$, $b=233$ so
$m=0.7885$; (b) $a=1.5$, $b=-0.05$ so $m=0.6893$; (c) $a=\frac{\textstyle
16}{\textstyle 9}$, $b=-0.1$ so $m=0.8254$; (d) $a=1.9$, $b=-0.24$ so
$m=0.6121$.

(a)

(b)

(c)

(d)

Figure 3: Solutions to Eq.(3.3) with $m=1$. (a) $a=-50$, $b=374.1346$; (b)
$a=1.5$, $b=0$; (c) $a=\frac{\textstyle 16}{\textstyle 9}$, $b=0$; (d)
$a=1.9$, $b=-0.2010$.

(3) $a=\frac{\textstyle 16}{\textstyle 9}$

In this case $\varphi_{L}<\varphi_{R}<0$ and $f(\varphi_{L})=f(0)$. For
$a=\frac{\textstyle 16}{\textstyle 9}$ and each value $b_{R}<b<0$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(e)), there are periodic
hump-like solutions to Eq.(3.3) given by (2.10) so that $0<m<1$, and with
wavelength given by (2.12); see Fig.2(c) for an example.

The case $a=\frac{\textstyle 16}{\textstyle 9}$ and $b=0$ (a corresponding
curve of $f(\varphi)$ is shown in Fig.1(f)) corresponds to the limit
$\varphi_{1}=\varphi_{2}=\varphi_{L}=-\frac{\textstyle 4}{\textstyle 3}$ and
$\varphi_{3}=\varphi_{4}=0$ so that $m=1$. In this case neither (2.5) nor
(2.10) is appropriate. Instead we consider Eq.(3.3) with
$f(\varphi)=\frac{\textstyle 1}{\textstyle
4}\varphi^{2}(\varphi+\frac{\textstyle 4}{\textstyle 3})^{2}$ and note that
the bound solution has $-\frac{\textstyle 4}{\textstyle 3}<\varphi\leq 0$. On
integrating Eq.(3.3) and setting $\varphi=0$ at $\xi=0$ we obtain a weak
solution

$\varphi=\frac{\textstyle 4}{\textstyle 3}\exp{(-\frac{\textstyle
1}{\textstyle 2}|\xi|)}-\frac{\textstyle 4}{\textstyle 3},$ (3.15)

i.e. a single peakon solution with amplitude $\frac{\textstyle 4}{\textstyle
3}$, see Fig.3(c).

(4) $\frac{\textstyle 16}{\textstyle 9}<a<2$

In this case $\varphi_{L}<\varphi_{R}<0$ and $f(\varphi_{L})>f(0)$. For each
value $\frac{\textstyle 16}{\textstyle 9}<a<2$ and $b_{R}<b<b_{L}$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(g)), there are periodic
hump-like solutions to Eq.(3.3) given by (2.10) so that $0<m<1$, and with
wavelength given by (2.12); see Fig.2(d) for an example.

The case $\frac{\textstyle 16}{\textstyle 9}<a<2$ and $b=b_{L}$ (a
corresponding curve of $f(\varphi)$ is shown in Fig.1(h)) corresponds to the
limit $\varphi_{1}=\varphi_{2}=\varphi_{L}$ so that $m=1$, and then the
solution is a hump-like solitary wave given by (2.13) with
$\varphi_{L}<\varphi\leq\varphi_{3}$ and

$\varphi_{3}=\frac{\textstyle 1}{\textstyle
6}(-2+3\sqrt{4-2a}-2\sqrt{4-6\sqrt{4-2a}}),$ (3.16)
$\varphi_{4}=\frac{\textstyle 1}{\textstyle
6}(-2+3\sqrt{4-2a}+2\sqrt{4-6\sqrt{4-2a}}),$ (3.17)

see Fig.3(d) for an example.

On the above, we have obtained expressions of parametric form for periodic and
solitary wave solutions $\varphi(\xi)$ to Eq.(3.3). So in terms of
$u=\varphi(\xi)+c$, we can get expressions for the periodic and solitary wave
solutions $u(\xi)$ to Eq.(1.5).

##  4 Conclusion

In this paper, we have found expressions for new travelling wave solutions to
the Fornberg-Whitham equation. These solutions depend, in effect, on two
parameters $a$ and $m$. For $m=1$, there are inverted loop-like ($a<0$),
single peaked ($a=\frac{\textstyle 16}{\textstyle 9}$) and hump-like
($\frac{\textstyle 16}{\textstyle 9}<a<2$) solitary wave solutions. For
$m=1,0<a<\frac{\textstyle 16}{\textstyle 9}$ or $0<m<1,a<2$ and $a\neq 0$,
there are periodic hump-like wave solutions.

## References

  * [1] R. Ivanov, On the integrability of a class of nonlinear dispersive wave equations, J. Nonlinear Math. Phys. 1294 (2005) 462-468.
  * [2] H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech. 127 (1998) 193-207.
  * [3] A. Constantin, W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A 270 (2000) 140-148.
  * [4] H. H. Dai, Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, Proc. R. Soc. A 456 (2000) 331-363.
  * [5] Z. Y. Yin, On the cauchy problem for a nonlinearly dispersive wave equation, J. Nonlinear Math. Phys. 10 (2003) 10-15.
  * [6] Y. Zhou, Stability of solitary waves for a rod equation, Chaos Solitons and Fractals 21 (2004) 977-981.
  * [7] Y. Zhou, Blow-up of solutions to a nonlinear dispersive rod equation, Calc. Var. 25 (2005) 63-77.
  * [8] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech. 457 (2002) 63-82.
  * [9] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661-1664.
  * [10] Y. A. Li, P. J. Olver, P. Rosenau, Non-analytic solutions of nonlinear wave models, in nonlinear theory of generalised functions: Vienna 1997, Chapman & Hall/CRC Res. Notes Math. 401, Chapman & Hall/CRC, Boca Raton, FL, 1999, 129-145.
  * [11] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differ. Equ. 217 (2005) 393-430.
  * [12] Y. A. Li, P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ. 162 (2000) 27-63.
  * [13] A. Constantin, W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci. 12 (2002) 415-422.
  * [14] A. Constantin, W. A. Strauss, Stability of peakons, Commun. Pure Appl. Math. 53 (2000) 603-610.
  * [15] R. S. Johnson, On the solutions of the Camassa-Holm equation, Proc. R. Soc. London A 459 (2003) 1687-1708.
  * [16] A . Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000) 321-362.
  * [17] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998) 229-243.
  * [18] A. Constantin, J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233 (2000) 75-91.
  * [19] V. O. Vakhnenko, E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos, Solitons and Fractals 20 (2004) 1059-1073.
  * [20] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci.17 (2007) 169-198.
  * [21] J. Escher, Y. Liu, Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Ind. Univ. Math. J. 56 (2007) 187-117.
  * [22] G. M. Coclite, K. H. Karlsen, N. H. Risebro, Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal. 28 (2008) 80-105.
  * [23] G. M. Coclite, K. H. Karlsen, N. H. Risebro, On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation, J. Diff. Equ. 234 (2007) 142-160.
  * [24] J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl. 306 (2005) 72-82.
  * [25] Y. Matsuno, Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit, Inverse Problems 21 (2005) 1553-1570.
  * [26] H. Lundmark, J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems 19 (2003) 1241-1245.
  * [27] Y. Matsuno, Cusp and loop soliton solutions of short-wave models, for the Camassa-Holm and Degasperis-Procesi equations, Phys. Lett. A 359 (2006) 451-457.
  * [28] G. B. Whitham, Variational methods and applications to water wave, Proc. R. Soc. Lond. Ser. A 299 (1967) 6-25.
  * [29] B. Fornberg, G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. R. Soc. Lond. Ser. A 289 (1978) 373-404.
  * [30] A. C. Bryan, A. E. G. Stuart, Solitons and the regularized long wave equation: A nonexistence theorem , Chaos, Solitons and Fractals 7 (1996) 1881-1886.
  * [31] D. Bhardwaj, R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl. 40 (2000) 1397-1404.
  * [32] S. I. Zaki, Solitary waves of the splitted RLW equation, Comput. Phys. Commun. 138 (2001) 80-91.
  * [33] J. I. Ramos, Solitary waves of the EW and RLW equations, Chaos, Solitons and Fractals 34 (2007) 1498-1518.
  * [34] J. I. Ramos, Solitary wave interactions of the GRLW equation, Chaos, Solitons and Fractals 33 (2007) 479-491.
  * [35] J. Nickel, Elliptic solutions to a generalized BBM equation, Phys. Lett. A 364 (2007) 221-226.
  * [36] A. H. A. Ali, A. A. Soliman, K. R. Raslanc, Soliton solution for nonlinear partial differential equations by cosine-function method, Phys. Lett. A 368 (2007) 299-304.
  * [37] D. H. Feng, J. B. Li, J. L. Lu, T. L. He, New explicit and exact solutions for a system of variant RLW equations, Appl. Math. Comput. 198 (2008) 715-720.
  * [38] Z. W. Lin, Instability of nonlinear dispersive solitary waves, J. Funct. Anal. 255 (2008) 1191-1224.
  * [39] J. B. Zhou, L. X. Tian, A type of bounded traveling wave solutions for the Fornberg-Whitham equation, J. Math. Anal. Appl. 346 (2008) 255-261.
  * [40] E. J. Parkes, The stability of solutions of Vakhnenko’s equation, J. Phys. A Math. Gen. 26 (1993) 6469-75.
  * [41] P. F. Byrd, M. D. Friedman, Handbook of elliptic integrals for engineers and scientists. Berlin: Springer; 1971.
  * [42] M. Abramowitz, I. A. Stegun. Handbook of mathematical functions. New York: Dover; 1972.