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7/27/2019 KG Bifurcation
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Nonlinear Dyn (2013) 72:789801
DOI 10.1007/s11071-013-0753-7
O R I G I N A L P A P E R
The effects of horizontal singular straight line
in a generalized nonlinear KleinGordon model equation
Lijun Zhang Li-Qun Chen Xuwen Huo
Received: 21 December 2012 / Accepted: 2 January 2013 / Published online: 17 January 2013
Springer Science+Business Media Dordrecht 2013
Abstract In this paper, we investigate bounded travel-
ing waves of the generalized nonlinear KleinGordon
model equations by using bifurcation theory of planar
dynamical systems to study the effects of horizontal
singular straight lines in nonlinear wave equations. Be-
sides the well-known smooth traveling wave solutions
and the non-smooth ones, four kinds of new bounded
singular traveling wave solution are found for the first
time. These singular traveling wave solutions are char-
acterized by discontinuous second-order derivatives at
some points, even though their first-order derivativesare continuous. Obviously, they are different from the
singular traveling wave solutions such as compactons,
cuspons, peakons. Their implicit expressions are also
studied in this paper. These new interesting singular
L. Zhang ()
School of Science, Zhejiang Sci-Tech University,
Hangzhou, Zhejiang 310018, P.R. China
e-mail: [email protected]
L.-Q. Chen
Shanghai Institute of Applied Mathematics and Mechanics,
Shanghai University, Shanghai 200072, P.R. China
e-mail: [email protected]
L.-Q. Chen
Department of Mechanics, Shanghai University, Shanghai
200444, P.R. China
X. Huo
School of Information and Electronic, Zhejiang Sci-Tech
University, Hangzhou, Zhejiang, P.R. China
solutions, which are firstly founded, enrich the results
on the traveling wave solutions of nonlinear equations.
It is worth mentioning that the nonlinear equations
with horizontal singular straight lines may have abun-
dant and interesting new kinds of traveling wave solu-
tion.
Keywords Horizontal singular straight lines Singular traveling wave solutions Nonlinear waveequations
Bifurcation theory of dynamical system
1 Introduction
There is an enormous literature on the study of nonlin-
ear wave equations, in which the existence, dynamical
stabilities and the bifurcations of solitary waves, kink
waves, periodic waves and other traveling waves are
discussed. A lot of methods have been developed to
find these exact traveling wave solutions for nonlinear
wave equations, such as the inverse scattering method,Backlund transformation method, Darboux transfor-
mation method, Hirota bilinear method, tanh method,
invariant subspace method and so on. Some special
functions or integrable ODEs are well applied to study
the solutions of nonlinear wave equations [1820, 23].
The general solutions to linear ODEs with variable
coefficients was studied in [27] helps us better un-
derstand about the solutions of nonlinear ODEs and
PDEs. Moreover, the superposition principle has been
mailto:[email protected]:[email protected]:[email protected]:[email protected]7/27/2019 KG Bifurcation
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790 L. Zhang et al.
used to construct subspaces of solutions to Hirota bi-
linear equation [25] and generalized bilinear equa-
tions [26]. More recently, the subspaces of solutions to
linear ordinary differential equations is skillfully taken
as invariant subspaces of nonlinear evolution equation
to study the exact solutions by Ma [21, 24]. Not only
the smooth solutions but also some singular solutionssuch as compacton, peakon [35] and complexiton so-
lutions [22] have attracted much attention. There have
been some excellent works on the explicit solutions of
some nonlinear evolution equations (see [1830] and
references therein).
To study the traveling wave solutions of a nonlinear
equation
F(u,ut, ux , uxx , uxt, ut t, . . . ) = 0 (1.1)let
=x
ct,u(x,t)
=(), where c is the wave
speed. Substituting them into (1.1), we have
F1
, , , . . .= 0 (1.2)
Here, we consider the case that (1.2) can be reduced to
the following planar dynamical system:
d
d= y, dy
d= F2(,y) (1.3)
by integrals and let = y, that is to say, (1.3) is thecorresponding traveling wave system of the nonlin-
ear equation (1.1). That means that to study the trav-eling wave solutions of the nonlinear equation (1.1)
we only need to study the corresponding traveling
wave system (1.3). Recently, Li and Liu [6], Liu [12],
Zhang [16, 17], Li and Chen [9], Li and Dai [10], Chen
and Huang [2] and Shen [14, 15] studied the travel-
ing wave solutions of some classes of nonlinear equa-
tions, which analysis is based on the bifurcation the-
ory of dynamical systems [1, 13]. However, almost all
nonlinear equations have the same class of traveling
wave systems which can be written in the following
form [11]:
d
d= y = 1
D2()
H
y,
dy
d= 1
D2()
H
= D
()y2 + g()D2()
(1.4)
where H = H(,y) = 12
y2D2() + D()g()dis the first integral. It is easy to see that (1.4) is actually
a special case of (1.3) with F2(,y) = 1D2()H
. If
there is a function = s such that D(s ) = 0, then = s is a vertical straight line solution of the systemd
d= yD(), dy
d= D()y2 + g() (1.5)
where d = D()d for = s . The two systemshave the same topological phase portraits except for
the vertical straight line = s and the directions ofthe time. Consequently, we can obtain bifurcation and
smooth solutions of the nonlinear equation (1.1) by
studying the system (1.5), if the corresponding orbits
are bounded and do not intersect the vertical straight
line = s in its phase portraits. However, the orbitswhich do intersect the vertical straight line = s orthose are unbounded but can go near enough to the
vertical straight line correspond to the non-smooth sin-
gular traveling waves which were proved by many au-
thors [2, 6, 10, 14, 16, 17, 29, 30]. This class of non-
linear equations was classified as the first class of non-
linear wave equations by Li in [11]. And it has been
pointed out that traveling waves sometimes lose their
smoothness during the propagation due to the exis-
tence of singular curves within the solution surfaces
of the wave equation.
However, most of these works are concentrated on
the first class of nonlinear equations, the relationships
between the traveling waves of the nonlinear equa-
tions with a vertical singular straight line and the orbits
of the corresponding traveling wave systems are well
known [10]. But till now there have been few works
on the integrable nonlinear equations with other types
of singular straight line, that is to say, there are only
few primary works [79, 11, 14, 15] on the following
types of wave equation:
d
d= y, dy
d= F2(,y) =
f (,y )
g(,y)(1.6)
where the functions f (,y ) and g(,y) satisfy
y(g(,y)/) + f (, y)/y 0. It is worth topoint out that a more general procedure to study the so-
lution of nonlinear equations which possess specifiedFrobenius integrable decompositions including (1.3)
and (1.6) was presented in [23]. Obviously, g(,y) =0 defines a set of real planar curves such that the right-
hand side of the second equation of system (1.6) is not
well defined on these curves which were named as sin-
gular curves of the corresponding nonlinear equations.
What kinds of traveling wave solution will appear with
the appearance of the singular curves for a given non-
linear wave equation still needs a further study.
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The effects of horizontal singular straight line in a generalized nonlinear KleinGordon model 791
In this paper, we study a generalized nonlinear
KleinGordon model equation as a special concrete
example to investigate the effects of horizontal sin-
gular straight line of integrable nonlinear wave equa-
tions. A nonlinear equation (1.1), which we named
a nonlinear wave equation with horizontal singular
straight line, possesses a singular straight line y = y0which is a horizontal straight line in the phase por-
trait of its corresponding traveling wave system (1.3),
if (1.3) can be reduced to (1.6) and g(,y) in the right-
hand of the second equation satisfies g(,y0) 0 forsome real constant y0. We find that the appearance of
the horizontal singular straight line makes the whole
work more complicated and some new types of sin-
gular traveling wave solution to a nonlinear equation
with a horizontal singular straight line appear, includ-
ing the recently known kink compacton solutions in
the following work.To clarify the characters of the corresponding trav-
eling wave solutions of the orbits near or intersected
with the horizontal singular straight line, we study the
following generalized nonlinear KleinGordon model
equation:
2
t2
C20 + 3Cn1
x
22
x2+ 20
VRP( )
= 0
(1.7)
which is a typical nonlinear equation with a horizontal
singular straight line. In the continuum limit, a one-
dimensional chain model of particles with identical
mass m anharmonically coupled to their nearest neigh-
bors and subjected to a nonlinear periodic double-well
one site potential VRP( ) are governed by the above
Eq. (1.7). (x,t) is the scalar dimensionless displace-
ment of particles, the parameters C20 and 20 are the
characteristic velocity and frequency, respectively. V0represents the amplitude of the substrate potential. For
the background materials of model equation, we refer
to the paper [3] and references therein.
We consider this model equation with the external
potential VRP( ) = V02 (1 2)2, that is, we have theequation
2
t2
Cl + 3Cn1
x
22
x2 220V0
3
= 0 (1.8)We also study the dynamical behaviors of its trav-
eling wave solutions under various parameter condi-
tions. Here, we are interested in the singular travel-
ing wave solutions such as kink compactons and try to
point out the relationship between these singular trav-
eling wave solutions and the appearance of horizontal
singular straight lines in their corresponding traveling
wave system.
As we discussed above, the corresponding traveling
wave system of (1.8) isc2 Cl 3Cn12
220V0
3= 0 (1.9)
where the prime denotes differentiation with respect
to , and = x ct, c is the wave speed. When c2 Cl 3Cn12 = 0, i.e. 2 = c
2Cl3Cn1
, (1.9) is equivalent
to the following two-dimensional planar system:
d
d= y, dy
d= 2
20V0( 3)
c2 Cl 3Cn1y2(1.10)
When c
2
Cl 3Cn1y2
= 0, making the time scaletransformation = (c2Cl3Cn1y2)2V0 , (1.10) becomes
d
d= a by2y, dy
d= 3 (1.11)
where a = c2Cl220 V0
, b = 3Cn1220 V0
. Obviously, when
c2Cl3Cn1
< 0, system (1.10) and (1.11) have the same
topological phase portraits (with opposite directions
when c2 Cl < 0), then we can obtain various kinds oftraveling wave solution of (1.8) by the study of the bi-
furcations of (1.11) and its portraits under various pa-
rameter conditions. However, when c2Cl3Cn1
0, (1.10)has the same topological phase portraits with (1.11)
except the horizontal singular straight lines y =
ab
,
i.e. y =
c2Cl3Cn1
. The corresponding traveling wave
solutions of those orbits which intersect the horizontal
singular straight lines of (1.11) have gigantic differ-
ences with others and have many complicated proper-
ties and thus need further careful research.
This paper is organized as follows. In Sect. 2, westudy the bifurcation sets and phase portraits of (1.11).
In Sect. 3, we consider the existence of the smooth
solitary and periodic wave solutions of (1.8). We
prove the existence of the non-smooth wave solutions
of (1.8) not only including the kink compactons and
breaking wave but also some new types of non-smooth
traveling wave in Sect. 4. The numerical simulation re-
sults show the consistence with the theoretical analysis
given at the same time.
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792 L. Zhang et al.
2 Bifurcation sets and phase portraits of (1.11)
It is easy to see that (1.11) is a Hamiltonian system
which has the following Hamilton:
H(,y) = 14
2ay2 by4 22 + 4
= h (2.1)
O(0, 0), A(1, 0) are three equilibrium pointsof (1.11) and B(1,
ab
), O(0,
ab
)D(1,
ab
) are also six other equilibrium points of (1.11)
which lie on the horizontal singular straight line
y =
ab
or y =
ab
when the parameters a and b
satisfy ab > 0. By the symmetry of (1.11), we can
easily know that the equivalent points B(1,
ab
)
and D(1,
ab
) have same properties. A(1, 0)and O
(0,
a
b
) also do. Therefore, we only need
to know the types of these four equivalent points
O(0, 0), A+(1, 0), B+(1,
ab
) and O+(0,
ab
). The
determinate of coefficient matrix of the linearized sys-
tem (1.11) at the equilibrium point is
J = 0 a 3by21 32 0
= a 3by232 1Consequently, O(0, 0), B(1,
ab
) and D(1,
ab
) are saddles and O(0,
ab
) and A(1, 0)
are centers when a > 0 and b > 0; B(1,ab )and D(1,
ab
) are centers and O(0,
ab
) are
A(1, 0) saddles when a < 0 and b < 0; when a = 0and b = 0, O(0, 0) and A(1, 0) are three higherorder equilibrium points. O(0, 0) is a center, while
A(1, 0) are saddles for b > 0 and O(0, 0) is a sad-dle, while A(1, 0) are centers for b < 0. However,when a, b have opposite signs or when a = 0, b = 0,the system has only three simple hyperbolic equiv-
alent points. O(0, 0) is a saddle and A(1, 0) arecenters when a > 0, b
0; while O(0, 0) is a center
and A(1, 0) are saddles when a < 0, b 0.By careful calculations, we obtain the Hamilton of
each point as H(O) = 0, H (A) = 14 , H (O) = a2
4b
and H (B) = a2
4b 14 from which we get the bifurca-
tion curves as follows:
L1: b = a2, a > 0; L2: a = 0, b > 0;L3: a = 0, b < 0; L4: a > 0, b = 0;L5: a < 0, b = 0; L6: b = a2, a < 0.
These curves partition the (a,b) parameter plane
into six regions and the phase portraits in each region
and on the bifurcation curses are shown in Fig. 1.
3 Dynamical behaviors and smooth traveling
wave solutions of (1.8)
Suppose that (x,t) = (x ct ) = () is a smoothsolution of a traveling wave equation with smoothness
for (, ),()+ = and () =. It is well-known that it is a smooth solitary wave if
= and it is a smooth kink (or anti-kink) if = .Usually, a solitary wave solution of (1.8) corresponds
to a homoclinic orbit of (1.11). A kink (or anti-kink)
wave solution (1.8) corresponds to a heteroclinic or-
bit of (1.11). Thus, to investigate all bifurcations of
solitary waves and kink waves of (1.8), we shall findall periodic annuli and their boundary curves of (1.11)
depending on the parameter space of this system. The
bifurcation theory of dynamical systems [1, 13] plays
an important role in our study.
By the bifurcations and phase portraits obtained in
Sect. 2 and above analysis, we can get the smooth soli-
tary wave, kink and periodic wave solutions of (1.8)
under various parameters conditions which are de-
scribed in the following theorem.
Theorem 3.1 Under different parameter conditions,(1.8) has the smooth traveling wave solutions as fol-
lows:
Case (1). When b = a2, a > 0, b > 0, i.e. Cn1 =(c2Cl )2
6V0and c2 > Cl (Fig. 1(1)), for h ( 14 , 0)
in (2.1), there are two families of periodic orbits
of (1.11) which corresponds to two families of un-
countable infinite many smooth periodic traveling
wave solutions of (1.8). And as h increases continu-
ously from
14 to 0, the amplitudes of these periodic
traveling waves increase from 0 and tend to 2.
Case (2). When b > a2, a > 0, b > 0, i.e. Cn1 >(c2Cl )2
6V0and c2 > Cl (Fig. 1(2)), for h ( 14 , a
2
4b
14 ) in (2.1), there are two families of periodic or-
bits of (1.11) which corresponds to two families of
uncountable infinite many smooth periodic traveling
wave solutions of (1.8). And as h increases contin-
uously from 14
to a2
4b 1
4, the amplitudes of these
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The effects of horizontal singular straight line in a generalized nonlinear KleinGordon model 793
Fig. 1 Phase portraits of
(1.11) in each bifurcation
region and on the
bifurcation curves
periodic traveling waves increase from 0 and tend to
( 4a2
b)1/4.
Case (3). When 0 < b < a2
, a > 0, i.e. 0 < Cn1 Cl (Fig. 1(3)), for h ( 14 , 0)
in (2.1), there are two families of periodic orbits
of(1.8) which corresponds to two families of uncount-
able infinite many smooth periodic traveling wave so-
lutions of(1.11). And as h increases continuously from
14 to 0, the amplitudes of these periodic travelingwaves increase from 0 and tend to
2; for h = 0,
there are two homoclinic orbits of (1.11) which cor-
responds to a peak-form solitary wave solution and a
valley-form solitary wave solution of (1.8). For 0 0, b 0, i.e. Cn1 0 andc2 > Cl (Fig. 1(5)), forh ( 14 , 0) in (2.1), there aretwo families of periodic orbits of (1.11) which corre-
sponds to two families of uncountable infinite many
smooth periodic traveling wave solutions of (1.8). And
as h increases continuously from 14 to 0, the ampli-tudes of these periodic traveling waves increase from 0
and tend to
2; for h = 0, there are two homoclinicorbits of(1.11) which correspond to a peak-form soli-
tary wave solution and a valley-form solitary wave so-
lution of (1.8); for h > 0, and as h increases continu-
ously from 0, the amplitudes of these periodic traveling
waves increase from 22 and tend to infinity finally.
Case (6). When 0 > b > a2, a < 0, i.e. 0 >Cn1 > (c
2Cl )26V0
and c2 < Cl (Fig. 1(6)), for h ( 1
4 , 0) in (2.1), there are two families of periodic
orbits of (1.11) which corresponds to two families of
uncountable infinite many smooth periodic traveling
wave solutions of (1.8). And as h decreases contin-
uously from 0 to 14 , the amplitudes of these peri-odic traveling waves increase from 0 and tend to 2;
forh=
1
4, there are two heteroclinic orbits of (1.11)
which corresponds to a kink wave solution and an anti-
kink wave solution of(1.8).
Case (7). When b < a2, a < 0, i.e. Cn1 < (c2Cl )2
6V0and c2 > Cl (Fig. 1(7)), for h ( a
2
4b, 0)
in (2.1), there are a family of periodic orbits of(1.11)
which correspond to a family of uncountable infi-
nite many smooth periodic traveling wave solutions
of (1.8). And as h decreases continuously from 0 toa2
4b, the amplitudes of these periodic traveling waves
increase from 0 and tend to 2(1 ba2b )(1/2).Case (8). When b = a2, a < 0, i.e. Cn1 =
(c2Cl )26V0
and c2 > Cl (Fig. 1(8)), for h ( 14 , 0)in (2.1), there are a family of periodic orbits of(1.11)
which correspond to a family of uncountable infi-
nite many smooth periodic traveling wave solutions
of (1.8). And as h decreases continuously from 0 to
14
, the amplitudes of these periodic traveling waves
increase from 0 and tend to 2.
We calculate the expressions of these smooth trav-
eling wave solutions as follows.
1. When a > 0 and b > 0, for every h ( 14 , 0),corresponding to H(,y) = h i.e. 4 22 = by4 2ay2 + 4h, there is an orbit of (1.11) which inter-sects with the x-axis at four points with x-coordinates0 =
1 + 1 + 4h, 1 =
1 1 + 4h. The
expressions of these two orbits are
y = 1b
a
b(4 22 4h) + a2,
0 < < 1 or
+1 < <
+0 (3.1)
Inserting (3.1) into the first equation of (1.10), we
obtain the parametric representation for two families
of periodic orbits as
0
da
b(4 22 4h) + a2
= 1b| 2nT1|, | 2nT1| T1 (3.2)
where
T1 =+0
+1
b d
a
b(4 22 4h) + a2and
h 1
4, 0
Specially, when b = a2, two families of periodicorbits can be expressed as
0
d1
(2 1)2 4h
= a| 2nT1|,
| 2nT1| T1 (3.3)When 0 < b < a2, the two homoclinic orbits of (1.11)
corresponding to h = 0 can be expressed as
2
da
b(4 22) + a2 =
1
b (3.4)
and a family of periodic orbits of (1.11) corresponding
to h (0, a24b 14 ) can be expressed as0
da
b(4 22 4h) + a2
= 1b
2nT1 , 2nT1 T1 (3.5)
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where
T1 =+0
0
bd
a
b(4 22 4h) + a22. When a < 0, b < 0, the periodic orbits surrounding
the center O(0, 0) can be expressed as
y = 1b
a
b(4 22 4h) + a2
0 < h < max
1
4,
a2
4b
(3.6)
which intersect the x-axis at two points with the
x-coordinates 1 =
1 1 + 4h respectively.From (3.6) and the first equation of (1.10), we ob-
tain the parametric representation for the correspond-
ing periodic orbits as
1
da
b(4 22 4h) + a2
= 1b | 2nT2|, | 2nT2| T2 (3.7)
where
T2 =+1
1
b da
b(4 22 4h) + a2
3. When a < 0 and b > 0, we have a family of peri-
odic orbits surrounding the center O(0, 0) that can be
expressed as
y = 1b
a +
b(4 22 4h) + a2
h
14
, 0
(3.8)
which intersect the x-axis at two points with the
x-coordinates 1 =
1 1 + 4h respectively.From (3.8) and the first equation of (1.10), we ob-
tain the parametric representation for the correspond-
ing family of periodic orbits as1
da +
b(4 22 4h) + a2
= 1b| 2nT3|, | 2nT3| T3 (3.9)
where
T3 =+1
1
b d
a +
b(4 22 4h) + a2
The kink wave solution and the anti-kink wave solu-
tion corresponding to h = 14
can be expressed, re-
spectively, as0
d
a +b(2
1)2
+a2
= 1b
(3.10)
4. When a > 0, b < 0, the two families of periodic or-
bits corresponding to h ( 14
, 0) can be expressed,
respectively, as follows:
y = 1b
a +
b(4 22 4h) + a2 (3.11)
which intersect the x-axis at four points with x-
coordinates 0 =
1 + 1 + 4h and 1 =
1 1 + 4h, respectively. From (3.11) and thefirst equation of (1.10), we obtain the parametric rep-
resentation for the corresponding two families of peri-
odic orbits as0
da +
b(4 22 4h) + a2
= 1b | 2nT4|, | 2nT4| T4 (3.12)
where
T4 = 1
+1
b d
a +
b(4 22 4h) + a2
The two solitary wave solutions of (1.8) corresponding
to h = 0 can be expressed, respectively, as
2
da +
b(4 22) + a2
= 1b (3.13)
and a family of periodic orbits corresponding to h > 0
can be expressed as
0
da +
b(4 22 4h) + a2
= 1b 2nT4 , 2nT4 T4 (3.14)
where
T4 =+0
0
b da +
b(4 22 4h) + a2
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796 L. Zhang et al.
5. When a < 0, b = 0, i.e. Cn1 = 0, a family of peri-odic orbits surrounding the center O(0, 0) and corre-
sponding to h (14 , 0) can be expressed as
y = 12a
4 22 4h, 1 < < +1 (3.15)
which intersect the x-axis at two points with x-coordinates 1 =
1 1 + 4h respectively.
From (3.15) and the first equation of (1.10), we ob-
tain the parametric representation for two families of
corresponding periodic orbits as1
d4 22 4h
= 12a | 2nT5|,
| 2nT5| T5 (3.16)where
T5 = +1
1
2a d
4 22 4hThe kink wave solution and the anti-kink wave solu-
tion corresponding to h = 14 can be expressed, re-spectively, as
= tanh
V0
Cl c2
(3.17)
6. When a > 0, b = 0, i.e. Cn1 = 0, the two familiesof periodic orbits surrounding the center A(1, 0)respectively and corresponding to h
(
1
4, 0) can be
expressed as
y = 12a
4 + 22 + 4h,
0 < < 1
or +1 < <
+0
(3.18)
which intersect the x-axis at four points with x-co-
ordinates 0 =
1 + 1 + 4h and 1 =
1 1 + 4h respectively. From (3.15) and thefirst equation of (1.10), we obtain the parametric rep-
resentation for two families of corresponding periodic
orbits as0
d4 + 22 + 4h
= 12a
| 2nT6|, | 2nT6| T6 (3.19)
where
T6 =+0
+1
2a d
4 + 22 + 4h.
The two solitary wave solutions corresponding to h =0 can be expressed, respectively, as
=
2sec h
2V0
c2 Cl
(3.20)
A family of periodic wave solutions corresponding to
h > 0 can be expressed as0
d4 + 22 + 4h
= 12a
2nT6 , 2nT6 T6 (3.21)where
T6 =+0
0
2a d
4 + 22 + 4h
4 Non-smooth traveling wave solutions of (1.8)
In this section, we will study the traveling wave so-
lutions which correspond to the orbits of (1.8) which
intersect the singular straight lines.
Case (1). When a = 0, b > 0, i.e. Cn1 > 0 and c2 = Cl ,for h ( 14 , 0) in (2.1), there are a family of pe-riodic orbits of equation (1.8) which are separated
up into two parts by the singular straight line y = 0(Fig. 1(10)); for h
= 1
4, there are two heteroclinic
orbits of (1.11) which intersect the singular straight
line at two equilibrium points of (1.11), and thus their
corresponding traveling wave solutions are distinctly
different with others.
When h ( 14
, 0), the orbits satisfy
by4 22 + 4 4h = 0 (4.1)with 1 < < 1 and d2
d2= 3by2 =
3b(4224h)
.
Consequently, along this orbit when
1
1
+4h, d
d
0, d
2
d2
, and thus d
dap-
proach to 0 rapidly. The corresponding traveling wavesbreak and come into being breaking waves. By same
calculations, we can prove that the traveling waves
which correspond to the orbits intersecting with hor-
izontal singular straight lines at regular points break,
that is to say, the orbits intersecting with horizontal
singular straight lines at regular points correspond to
the breaking waves.
For h = 14 , (1.11) has two heteroclinic orbitswhich intersect the horizontal singular straight lines
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only at the two equilibrium pointsA(1, 0). Thesetwo orbits satisfy ( d
d)2 = 1
b(1 2), d2
d2=
b.
Consequently, along these two orbits when 1(1 < < 1), d
d 0, d2
d2 1
b. And thus,
along these orbits reaches 1 in a finite time (timevariable ). Since A
(
1, 0) are equilibrium points
of (1.11), (1.8) has a kink and an anti-kink wave so-
lution. While unlike the well-known kink waves, they
achieve equilibrium state in a finite time. Actually, the
expressions of these two special solutions can be writ-
ten as
=
sin[b1/4( 0)] | 0|
2b1/4
1 0
2b1/4
1
0
2b1/4
(4.2)
And the corresponding traveling waves of (1.8) which
were named kink compactons [3] can be expressed as
=
sin
2V0
3Cn1
1/4(x
Cl t 0)
| 0|
2
3Cn1
2V0
1/4
1 0
2 3Cn1
2V0
1/4
1 0
2
3Cn1
2V0
1/4
(4.3)
Remark These two kink compactons (4.3) are dif-
ferent from the well-known smooth kink that they
are weak solutions which have one-order continuous
derivative, but the second-order derivatives on the two
points 0 + 2 ( 3Cn12V0 ) and 0 2
(3Cn12V0
) do not exist.
That is to say that these kink compactons are non-
smooth singular traveling wave solutions of (1.8).
Case (2). When a = 0, b < 0, i.e. Cn1 < 0 and c2 = Cl ,(1.8) has six families of breaking wave.
Case (3). When b = a2, a > 0, b > 0, i.e. Cn1 =(c2Cl )2
6V0and c2 > Cl (Fig. 1(1)), for h (0, 14 ) in (2.1),
there are two families of periodic orbits of (1.11)
which intersect the singular straight lines, respectively,
and thus correspond to four families of breaking wave
solutions of (1.8); while for h = 0, the correspondingorbits of (1.11) satisfy
2 + ay2 2ay2 2= 0 (4.4)and the two corresponding heteroclinic orbits which join to two equilibrium points on the horizontal
singular straight lines, respectively, satisfy
=
2 ay2 (4.5)We can also prove that approach to 1 (along) in a finite long time as above. The straight lines
y = 1a
, 0 < < 1 and the arch y =
22a
,
1 < 0 (a = 1, b = 1),as h tends to 0, the smooth
periodic wave evolves into
a singular solitary wave
In addition, there exists a smooth curve 2 +ay2 2 = 0 corresponding to the energy curve h = 0which intersects with both the two horizontal singu-
lar straight lines. Without the effect of the horizontal
singular straight lines, there should be a periodic wave
=
2cos(
a) corresponding to this closed curve.
Obviously, =
2cos(
a) is a smooth periodic solu-
tion of (1.8) in mathematic and thus is a classical peri-odic solution. It is well known that a smooth periodic
wave solution is always lies in a family of periodic
wave solutions, that is to say, it is always continuous
respect to the initial conditions. However, this periodic
solution is not so. What is its physical meaning? This
strange phenomenon needs more attention and further
research.
Case (4). When b > a2, a > 0, b > 0, i.e. Cn1 >(c2Cl )2
6V0and c2 > C
l(Fig. 1(2)), for h
=a2
4b 1
4in (2.1), the corresponding orbits of (1.11) satisfy
2 +
by2 1 a
b
by2 2 + 1 a
b
= 0 (4.8)
Along the heteroclinic orbits
by2 2 + 1 ab
=0,
1 ab
|| 1, when 2 1, 2 ab
and
|| 1. Consequently, approaches 1 in a finitelong time. Actually, integrating along their orbits, re-
spectively, we can obtain the time from one end of the
orbit to another 2T1 and 2T2, where
T1 = b1/4 arcch
1 ab
(1/2),
T2 = b1/4 arccos
1 + ab
(1/2) (4.9)
Therefore, two periodic solutions are obtained as fol-
lows:
=
1 a
bcosh
b1/4
2n(T1 + T2)
2n(T1 + T2) T11 + a
bcos
b1/4
(2n + 1)(T1 + T2)
(2n + 1)(T1 + T2)< T2
(4.10)
where n is any real integer. Obviously, these two pe-
riodic solutions are non-smooth weak solutions which
have a continuous first-order derivative. Actually, we
can get these solutions by letting h a24b 14 in thetwo families of periodic orbits of (1.11) with h ( 1
4, a
2
4b 1
4) as above.
In addition, one can find that there exists a smooth
curve 2 +
by2 1 ab
= 0 corresponding to theenergy curve h = a24b 14 which intersects with boththe two horizontal singular straight lines. Without the
effect of the horizontal singular straight lines, there
should be a periodic wave =
1 + ab
cos(b1/4 )corresponding to this closed curve. Obviously, =
2cos(
a) is a smooth periodic solution of (1.8) in
mathematic and thus is a classical periodic solution.
Case (5). When 0 < b < a2, a > 0, i.e. 0 < Cn1 Cl (Fig. 1(3)), or h (14 , a
2
4b )
in (2.1), there are two families of periodic orbits
of (1.11) which intersect the singular straight lines, re-
spectively, and thus correspond with four families ofbreaking wave solutions of (1.8).
In addition, let h a24b 14 (h (0, a2
4b 1
4 ))
in (3.5), then the corresponding periodic orbits of
(1.11) approach a closed curve which intersects with
each horizontal singular straight line at two points, re-
spectively. That means this closed curve has four non-
smooth points. We can also derive this non-smooth
periodic wave solution of (1.8) by integrating along
this closed curve. In the same way, we also can derive
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it by letting h a24b 14 (0 < h < a2
4b 14 ) in (3.5).
Actually, when h a24b 14 (h (0, a2
4b 1
4 )) in (3.5),a
b(4 22 4h) + a2
a
b|2 1|
and T1 T1 +
T1, where
T1 = 210
b d
a
b(1 2),
T1 = 2
1+ ab
1
b d
a
b(2 1),
and thus the limit of this family of periodic wave solu-
tions is obtained as
1+ ab
d
a
b(2 1)
=1
b
2nT1 + T1
2nT1 + T1T11
da
b(1 2)
= 1b
2nT1 + T1
T1
2nT1 + T1 T1 + T1
(4.11)
It is easy to prove that there are four non-smooth points
| 2n(T1 +
T1)| =
T1 or | 2n(T1 +
T1)| = T1 at
which the first-order derivative of this solution is con-
tinuous, but the second-order derivative breaks. Con-
sequently, this periodic solution is a singular solution.
Case (6). When 0 > b > a2, a < 0, i.e. 0 > Cn1 > (c2Cl )2
6V0and c2 < Cl (Fig. 1(6)), for h ( a
2
4b
14 ,
a2
4b ) in (2.1), there are four families of periodic or-
bits of (1.11) intersecting with the horizontal singu-
lar straight lines which correspond to eight families of
breaking waves of (1.8).
Case (7). When b < a2
, a < 0, i.e. Cn1 < (c2
Cl )
2
6V0
and c2 > Cl (Fig. 1(7)), for h ( a2
4b 1
4, 1
4) in (2.1),
there are seven families of periodic orbits of (1.8) in-
tersecting with the horizontal singular straight lines
which correspond to 14 families of breaking waves
of (1.8); for h = a24b
, there is a boundary curve of
a periodic annulus around the center (0, 0) which
corresponds to a non-smooth periodic wave solution
of (1.8). Integrating along this boundary curve, we
have the implicit integral representation
0
dab
2(22)b
= | 2nT|,
| 2nT| T (4.12)
where
n Z, 0 =
1
1 + a
2
band
T = 20
0
dab
2(22)b
.
This is a non-smooth periodic wave solution of (1.8)
which is 2T-period and has continuous first-order
derivative but has discontinuous second-order deriva-
tive when = 2nT T2 .
Case (8). When b = a2, a < 0, i.e. Cn1 = (c2Cl )26V0
and c2 > Cl (Fig. 1(8)) for h ( a2
4b 1
4 ,a2
4b ) in (2.1),
there are five families of periodic orbits of (1.8) in-
tersecting with the horizontal singular straight lines
which correspond to 10 families of breaking waves
of (1.8); for h = a24b , there is a boundary curve of aperiodic annulus around the center (0, 0) which con-
nects two equilibrium points. Each part of this bound-
ary curve above or under the abscissa axis corresponds
to a non-smooth peak-form solitary wave solution or a
valley-form one of (1.8) which second-order deriva-
tive has a discontinuous point at its peak or valley.
Integrating along this boundary curve, we have the
implicit integral representations. Similarly, integrating
along its orbits, respectively, the implicit integral rep-
resentations are derived as
0
d
1 2(2 2)
= || (4.13)
Actually, this solution can be obtained by letting h a2
4bin (3.7) with b = a2and a < 0.
Remark When , 1, and () hascontinuous first-order derivative, but discontinuous
second-order derivative at = 0 ( = 0). Conse-quently, (4.13) is a non-smooth singular solitary wave
solution which is first referred here.
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