We consider the hyperelastic rod equation describing nonlinear dispersive waves in compressible hyperelastic rods. We investigate the existence of certain traveling wave solutions to this equation. We also determine whether two other equations(the
b
family equation and the modified CamassaHolm equation) have our solution type.
AMS Mathematics Subject Classification : 35Q35, 37K45.
1. Introduction
This paper is concerned with the hyperelastic rod equation
which was obtained as a model for nonlinear waves in cylindrical hyperelastic rods with
u
(
t, x
) representing the radial stretch relative to a prestressed state, and the physical parameter
γ
ranging from −29.4760 to 3.1474
[6
,
7
,
8]
. From the mathematical view point, we regard
γ
as a real number. Among (1), there are two other important equations.
When
γ
= 1, it recovers the standard CamassaHolm(CH) equation
[3]
arises as a model for the unidirectional propagation of shallow water waves over a flat bottom
[3]
, as well as water waves moving over an underlying shear flow
[12]
. The CH equation is completely integrable systems with a corresponding Lax pair formulation, a biHamiltonian structure, and an infinite sequence of conservation laws
[3]
. The CH equation also admits peaked solitary waves or ”peakons”
[3]
:
u
(
t, x
) =
ce
^{−x−ct}
,
c
≠ 0, which are smooth except at the crests, where they are continuous, but have a jump discontinuity in the first derivative. These peakons are shown to be stable
[5]
. Moreover, it admits the multipeakon solutions (cf.
[3]
).
If
γ
= 0, then (1) becomes the regularized long wave equation, a wellknown equation for surface wave in a channel
[1]
. All solutions are global and the solitary waves are smooth. Despite having a Hamiltonian structure, the equation is not integrable and its solitary waves are not solitons
[9]
.
For general
γ
∈
, mathematical properties of (1) have been studied further in many works. Dai and Huo
[8]
observed that (1) was formally shown to admit smooth, peaked, and cusped traveling waves. Subsequently, Lenells
[13]
used a suitable framework for weak solutions to classify all weak traveling waves of the hyperelastic rod equation (1). Constantin and Strauss
[6]
also investigated the stability of a class of solitary waves for the rod equation (1) on the line.
Note that if
p
(
x
) :=
,
x
∈
, then
u
=
=
p
∗
m
, where
m
:=
u
−
u_{xx}
and ∗ denotes the convolution product on
, given by
This formulation allows us to define a weak form of (1) as follows :
In this paper, we are interested in a special traveling wave (i.e. function of
x
−
ct
for some
c
> 0) and in fact that solution will also be a solitary wave solution (i.e. function of
x
−
ct
, decays to zero as 
x
 → ∞). Our solution will be bounded in
x
∈
for each fixed
t
> 0. In fact, it will be bounded by
. The solution will be valid for any fixed value of
c
> 0. This will be a weak solution, namely
u
(
t, x
) will not be differentiable at the ”peak” point. At any other points the solution will have all the derivatives.
We are motivated by the paper
[4]
, where they exhibited a similar solution to the following equation
where 0 ≤
β
< 1, but their result is also applicable to
for any α
^{2}
> 0 and
β
∈ [0, 1). By using transformation
x
⟼
cx
, we have
Choose
c
such that
c
=
. Then we obtain
Our case is then obtained by choosing
α
> 0 such that
3). Our case is
with
β
∈ [0, 1). In our case, we can use result in ]undefined
[4]
to get a solution for our equation
when
γ
∈ (1, 3]. Solution in
[4]
covers our equation when
γ
∈ (1, 3], but not when
γ
> 3. In our case, we also cover
γ
> 3. In fact, our solutionwill be valid when
γ
> 1.
This paper is organized as follows. In Section 2, we give the needed results to pursue our goal. In Section 3, we shall show the existence of traveling wave solutions for (2) under the condition
γ
> 1, by using an analogous analysis in
[4]
. Finally, in Section 4 we determine whether the
b
family equation
[11]
and the modified Camassaholm equation
[10]
have our solution type.
2. Preliminaries
In this section, we present the following basic technical Lemmas which play a key role to obtain our main results.
Lemma 2.1.
Suppose γ
> 1
and
be a sequence of positive real numbers defined
Define
Then, there exist R
> 0
such that φ
(
x
)
is welldefined on

x
 <
R, where R is the radius of the convergence for φ
(
x
).
Proof
. Let
be a sequence of positive real numbers defined
Set
It is obviously that
α_{n}
≤
ξ_{n}
, for
n
= 1, 2, .... Note that
Solving the equation (5), we have
This implies
ϱ
(
x
) is welldefined on
Hence,
φ
(
x
) is welldefined, at least, on
which means there is
such that
φ
(
x
) is welldefined on 
x
 <
R
. This completes the proof of Lemma2.1.
Lemma 2.2.
Consider the following initial value probem
(
IVP
)
The solution of (6) can be implicitly expressed as (11).
Proof
. Notice that (6) is independent of
z
. By assuming that
:= ᴪ(
φ
), we reach
Solving the firstorder ordinary differential equation (7), we get
By (8) and the transformation
z
→
x
with
x
:=
e^{z}
used in (15), we obtain
Using the transformation
φ
⟼
η
with
φ
=
we have
Solving (10) to obtain an implicit analytic formula for function
φ
:
for
x
> 0. This completes the proof of Lemma 2.2.
Under the same assumption of Lemma 2.1, we have the following important result to get (39).
Lemma 2.3.
Assume γ
> 1.
Let
and φ
(
x
)
be defined in (3) and (4). Then φ
(
x
)
has a continuous extension to x
=
R such that φ
(
R
) =
,
or
lim
_{x}
_{→}
_{R}
−
φ
(
x
) =
.
Proof
. By Lemma 2.1, we see tht
φ
(
x
) is welldefined on 
x
 <
R
. In order to show that
φ
(
x
) has a continuous extension such that
φ
(
R
) =
, or lim
_{x}
_{→}
_{R}
−
φ
(
x
) =
, let us define
Since
we obtain
Notice that for 
x
 <
R
By (12) and (13), we have
Since our goal is to show lim
_{x}
_{→}
_{R}
−
φ
(
x
) =
, we may assume that
x
> 0. Using the transformation
z
⟼
x
with
x
:=
, we have
Substituting (15) into (14), we obtain the ordinary differential equation
which is independent of
z
. By Lemma 2.2, we obtain implicitly expression of solution for
φ
for
x
> 0. Let
By direct calculation we have
By the Implicit Function Theorem, we conclude that
φ
(
x
) is welldefined in 
x
 <
R
, where
and
Since
α_{n}
≥ 0,
n
= 1, 2, ..., it is easy to check that
This completes the proof of Lemma 2.3.
3. Traveling wave solutions
In this section, we discuss the existence of traveling wave solutions to the hyperelastic rod equation (2) with
γ
> 1.
Theorem 3.1.
Let γ
> 1.
For every c
> 0,
the functions of the form
where α n and R are defined in (3) and (20), is a solution to (2) in the weak sense
.
Proof
. We need to show that
u
(
t, x
) satisfies (2). Notice that
Using (23), (24), and
p
(
x
) =
for
x
∈
, we calculate from (2) that
where
with
B_{n}
=
When
x
>
ct
, by using
we write
K
_{2}
defined in (25) as
K
_{2}
=
I
_{1}
+
I
_{2}
+
I
_{3}
, where
We directly compute
I
_{1}
as follows:
In a similar manner,
and
Plugging (27)(29) into (26), we deduce that for
x
>
ct
Using (25) and (30), we obtain for
x
>
ct
While for the case
x
≤
ct
, we split second term(
K
_{2}
) of (25) into the following three parts:
For
J
_{1}
, a direct computation gives rise to
Similarly, one obtains
and
Using (25) and (33)(35), we have for
x
≤
ct
The traveling wave profile u(0.x) for c=2.
The zoom in profile for the traveling wave u(0.x) for c=2.
On the other hand, using (23), we get
As a result, (2) is equivalent to the recursive sequence of equation given by
With the help of (3) and (22), we conclude that (38) holds if
Now, we apply Lemma 2.1 and 2.3 to show that (39) holds. Let us define
From the definition of
θ
(
x
) in Lemma 2.3, we see that Φ(
x
) =
θ′
(
x
) for 
x
 <
R
. By (12), we have
θ′
(
x
) = 1 + (
γφ
(
x
) − 1)
φ′
(
x
) for 
x
 <
R
. By Lemma 2.3, we have lim
_{x}
_{→}
_{R}
−
φ
(
x
) =
and hence
Therefore,
Since
α_{n}
≥ 0 for
n
≥ 1, one can easily to check that
This completes the proof of Theorem 3.1.
4. Remarks
In Sections 23 we have found explicit expressions for traveling wave solutions to (2) with
γ
> 1 that travel in the positive
x
direction with speed
c
> 0. In the following, we discuss whether two other equations(the
b
family equation and the modified CamassaHolm equation) have our solution type. We expect our nonsmooth traveling wave solutions satisfy to the above both equations. Unfortunately, their equations does not have our traveling wave solution type.
Consider the following two other equations which are the
b
family equation
[11]
and the modified CamassaHolm equation
[10]
Moreover, (40) and (41) are equivalent to the following nonlocal form:
and
We just apply our solution to the
b
family equation. Let us substitute our solution
u
(
t, x
) =
in (22) to (42). In a similar arguments, substituting (23) and (24) into (42) gives us
Remark 4.1.
Since
c
> 0, (44) implies that
β_{n}
= 0 for
n
≥ 2 and
β
_{1}
=
c
or 0. Therefore, we conclude that (42) have
u
(
t, x
) =
ce
^{−x−ct}
or
u
(
t, x
) = 0. In this case, our solution recovers the single peakon solution.
Next we also apply our solution to the modified CamassHolm equation. In a similar manner, we can plug our solution
u
(
t, x
) =
into (43). Then we obtain
Remark 4.2.
Since
c
> 0,
β
_{1}
=
or 0,
β
_{2}
= 0 and
β_{n}
= 0 for
n
≥ 3. Hence (45) have a solution
u
(
t, x
) =
e
^{−x−ct}
or
u
(
t, x
) = 0. In this case, our solution also recovers single peakon solution which was obtained in
[10]
.
In the following remark, we explain our solutions fit into the classification previously studied traveling wave solutions of (1).
Remark 4.3.
In [13], Lenells study our own equation
and categorized all weak traveling wave solutions. Our solution corresponds to his category (f) of Theorem 1 in
[13]
. He proves the existence of a cuspon solution with decay, which corresponds to our solution such that
where
α_{n}
and
R
are defined in (3) and (20).
BIO
Byungsoo Moon received a M.A. from the State University of New York at Buffalo and Ph.D in the University of Texas at Arlington. He is a postdoctoral fellowship of the Pohang Mathematics Institute (PMI) at POSTECH. His current research interests are the study of a certain class of nonlinear partial differential equations referred to as integrable systems.
Pohang Mathematics Institute (PMI), POSTECH, Pohang, Kyungbuk 790784, Korea
email: bsmoon@postech.ac.kr
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