Applied Mathematics and Computation 171 (2005) 242–271
www.elsevier.com/locate/amc
Bifurcations of travelling wave solutions ina model of the hydrogen-bonded systems
Jianwei Shen a,c,*, Jibin Li b, Wei Xu a
a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an,
Shaanxi 710072, PR Chinab School of Science, Kunming University of Science and Technology,
Kunming, Yunnan 650093, PR Chinac Department of Mathematics, Xuchang University, Xuchang, Henan 461000, PR China
Abstract
By using the theory of bifucations of dynamical systems to a model of the Hydrogen-
bonded systems, the existence of solitary wave, kink and anti-kink wave solutions and
uncountably infinite many smooth and non-smooth periodic wave solutions is obtained.
Under different parametric conditions, various sufficient conditions to guarantee the
existence of the above solutions are given. In some simple parametric conditions, exact
explicit and implicit solution formulas are listed.
� 2005 Elsevier Inc. All rights reserved.
Keywords: Solitary travelling wave solution; Periodic travelling wave solution; Kink and anti-kink
wave solutions; Smoothness of waves
0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2005.01.058
* Corresponding author. Address: Department of Applied Mathematics, Northwestern Poly-
technical University, Xi�an, Shaanxi 710072, PR China.
E-mail addresses: [email protected] (J. Shen), [email protected] (J. Li).
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 243
1. Introduction
As Mei et al. [10] state that ‘‘Hydrogen-bonded systems in condensed matter
physics and biology have stimulated wide interest in recent years. Understand-
ing the proton properties of such systems provides information on many phys-
ical and biological systems such as ice, solid alcohol, hydrogen halidesimidazote, and carbonhydrates, and on the complicated processes of proton
transfer across cellular membranes, the vision-related molecule rhodopsin’’.
In 1992, Mei et al. [10] studied the soliton model in the anharmonic interaction
approximation in hydrogen-bonded systems, and obtained some analytical
solutions of kink and anti-kink solutions. In the continuum limit approxima-
tion, they give the equation of motion
utt � uxx � buxuxx þ X2 dVdu
¼ 0; ð1:1Þ
where
V ðuÞ ¼2 cos u
2
� �� cos u0
2
� �� �21� cos2 u0
2
� � :
u0 5 0 is a constant, and 0 6 cos u02
� �< 1.
In this paper, we shall consider all travelling wave solutions in the parameterspace of systems. Let u(x, t) = /(x � ct) = /(n), where c is the wave speed.
Substituting the above travelling wave solutions into (1.1), we have
c2 � 1� �
/nn � b/n/nn �2X2
1� cos2 u02
� � sin /2
� �cos
/2
� �� cos
u02
� �� �¼ 0:
ð1:2ÞLet d/
dn ¼ y, (1.2) becomes two dimension system
d/dn
¼ y;dydn
¼ 2X2
1� cos2 u02
� � sin /2
� �cos /
2
� �� cos u0
2
� �� �c2 � 1� by
: ð1:3Þ
which has first integral
Hð/; yÞ ¼ c2 � 1
2y2 � b
3y3 þ 2X2
1� cos2 u02
� � cos/2
� �� cos
u02
� �� �2
¼ h:
ð1:4Þ
From (1.3) and (1.4), we see that we always can assume that b P 0. Other-
wise, making the transformation y ! �y, n ! �n, b ! �b, we have the same
systems.
Obviously, the straight line c2 � 1� by = 0, on which the vector field defined
by (1.3) has no definition, is a singular straight line. It implies that a smooth
244 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
system (1.1) sometimes has non-smooth travelling wave solutions. This phe-
nomenon has been considered by some authors (see [4–8]). In this paper, we
claim that the existence of a singular line for the nonlinear wave equation is
the reason for travelling waves lose their smoothness.
Suppose that u(x, t) = /(x � ct) = /(n) is a continuous solution of Eq. (1.1)
for n 2 (�1,1), and limn!1/(n) = a, limn!�1/(n) = b. It is well known that(i) u(x, t) is called a solitary wave solution if a = b. (ii) u(x, t) is called a kink or
anti-kink solution is a 5 b. Usually, a solitary wave solution of (1.1) corre-
sponds to a homoclinic orbit of (1.2). A kink (or anti-kink) wave solution of
(1.1) corresponds to a heteroclinic orbit (or so called connecting orbit) of
(1.2). Similarly, a periodic orbit of (1.2) corresponds to a periodically travelling
wave solution of (1.1). Thus, to investigate all bifurcations of solitary wave,
kink waves and periodic waves of Eq. (1.1), we should find all periodic annuli,
homoclinic and heteroclinic orbits of (1.2) depending on the parameter space ofthis system.
The paper is organized as follows. In Section 2, we discuss bifurcations of
phase portraits of (1.2) and explicit parametric conditions will be given. In Sec-
tion 3, we discuss smooth travelling wave solutions of (1.1). In Section 4, we
show the existence of breaking solutions and uncountably infinite many non-
smooth periodic travelling wave solutions of (1.1).
2. Bifurcations of phase portraits of (1.1)
In this section, we shall study all possible bifurcations defined by vector
fields of (1.3) when c, b and u0 are varied from (1.3). We see that
dyd/
¼ 2X2
1� cos2 u02
� � sin /2
� �cos /
2
� �� cos u0
2
� �� �yðc2 � 1� byÞ : ð2:1Þ
Let dn = (c2 � 1� by)df, then (1.3) can be become the following integrable
system
d/df
¼ yðc2 � 1� byÞ; dydf
¼2X2 sin /
2
� �cos /
2
� �� cos u0
2
� �� �1� cos2 u0
2
� � : ð2:2Þ
It is easy to see from (2.1) that system (1.3) has the same topological phase por-
traits as (2.2) except on that singular line c2 � 1� by = 0.Let (/e,ye) be an equilibrium of (2.2). At this point, the determinant of the
linearized system of (2.2) has the form
Jð/e; yeÞ ¼X2
1� cos2 u02
� � ð2bye þ 1� c2Þ cosð/eÞ � cos/e
2
� �cos
u02
� �� �:
ð2:3Þ
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 245
By the theory of planar dynamical systems (see [1–3,9]), we know that, if
J(/e,ye) > 0 (or <0) then the equilibrium (/e,ye) is a center (or a saddle point).
if J(/e,ye) = 0 and the Poincare index of the equilibrium point is 0, then (/e,ye)
is a cusp.
By using the above facts to do qualitative analysis, we obtain the following
results.Case I. b = 0 and c2 � 1 5 0.
In this case, (1.3) becomes
d/dn
¼ y;dydn
¼ A0
sin /2
� �ðcos /
2� cosðu0
2Þ
� �1� cos2 u0
2
� � ; ð2:4Þ
where
A0 ¼2X2
c2 � 1:
Making the transformationffiffiffiffiffiA0
pn ¼ s; y !
ffiffiffiffiffiA0
py for c2 � 1 > 0, orffiffiffiffiffiffiffiffiffi
�A0
pn ¼ s; y !
ffiffiffiffiffiffiffiffiffi�A0
py for c2 � 1 < 0. We obtain the following two-dimen-
sional system
d/ds
¼ y;dyds
¼ �sin /
2
� �ðcos /
2� cos u0
2
� �� �1� cos2 u0
2
� � : ð2:5Þ
which has first integral
Hð/; yÞ ¼ 1
2y2 �
cos /2
� �� cos u0
2
� �� �21� cos2 u0
2
� � ¼ h: ð2:6Þ
System (2.5) is periodic in /. Hence, the state (/,y) can be viewed on a phase
cylinder S1 · R, where S1 = [�2p,2p] with �2p, 2p identified (see Fig. 1 and 2).
Clearly, for / 2 (�2p,2p), there exist five equilibrium points of (2.5) at O(0,0),
A±(±u0,0), B±(±2p, 0). Let M(/i, 0) be the coefficient matrix of the linearized
system of (2.5) at the equilibrium point (/i, 0), we have
Jð0; 0Þ ¼ det Mð0; 0Þ ¼ � 1
2 1þ cos u02
� �� � ;Jð�u0; 0Þ ¼ � 1
2; Jð�2p; 0Þ ¼ � 1
2 1� cos u02
� �� � : ð2:7Þ
By using the signs of these J values, we can determine the types (saddle or cen-
ter) of above equilibrium point. Note that for (2.6), we have h1 = H(±u0,0) = 0,
h0 ¼ Hð0; 0Þ ¼ � tanu04
� �� �2; h2 ¼ Hð�2p; 0Þ ¼ � tan
u04
� �� ��2
: ð2:8Þ
We know from (2.8) that, for every u0 2 (0,2p), h0 = h2 if and only if u0 = p.
Fig. 1. Phase portraits of (2.4) when c2 > 1. (1) u0 2 (0,p), (2) u0 = p, (3) u0 2 (p, 2p).
246 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
From the above discussion, we have the following portraits of phase of (2.5)
(see Figs. 1 and 2).
Case II. b 5 0 and c2 � 1 = 0.
In this case, (1.3) has the form
d/dn
¼ y;dydn
¼ �2X2
b
sin /2
� �cos /
2
� �� cos u0
2
� �� �y 1� cos2 u0
2
� �� � ð2:9Þ
which has first integral
Hð/; yÞ ¼ � b3y3 þ 2X2
1� cos2 u02
� � cos/2
� �� cos
u02
� �� �2
: ð2:10Þ
Fig. 2. Phase portraits of (2.4) when c2 < 1. (1) u0 2 (0,p), (2) u0 = p, (3) u0 2 (p, 2p).
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 247
Obviously, y = 0 is the singular line. It is easy to see that all of equilibrium
points in the /-axis are cusp, which have the following phase portraits for b > 0
(see Fig. 3).
Case III. b 5 0 and c2 � 1 5 0.
Denote that A ¼ 2X2
c2�1; b ¼ b
ffiffiffiffiffi�A
p
c2�1. Making the transformation
ffiffiffiA
pn ¼ s; y !ffiffiffi
Ap
y for c2 � 1 > 0, orffiffiffiffiffiffiffi�A
pn ¼ s; y !
ffiffiffiffiffiffiffi�A
py for c2 � 1 < 0, then (1.3)
becomes
d/ ¼ y;dy ¼ �
sin /2
� �cos /
2
� �� cos u0
2
� �� �u� �� � : ð2:11Þ
ds ds 1� cos2 0
2ð1� byÞ
Fig. 3. Phase portraits of (1.3) for c2 = 1 and b > 0. (1) u0 2 (0,p), (2) u0 = p, (3) u0 2 (p, 2p).
248 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
Thus, (2.2) becomes
d/ds
¼ yð1� byÞ; dyds
¼ �sin /
2
� �cos /
2
� �� cos u0
2
� �� �1� cos2 u0
2
� � : ð2:12Þ
which has first integral
Hð/; yÞ ¼ 1
2y2 � b
3y3 �
cos /2
� �� cos u0
2
� �� �21� cos2 u0
2
� � : ð2:13Þ
We next consider the phase portrait of (2.10). System (2.10) also can beviewed on a phase cylinder S1 · R, where S1 = [�2p,2p] with �2p, 2p identified
(see Figs. 5 and 6). There exist 10 equilibrium points of (2.5) at O(0,0),
A±(±u0,0), B±(±2p, 0), Cð0; 1bÞ, D�ð�u0; 1bÞ, E�ð�2p; 1bÞ. At an equilibrium points
(/i,yi), we have
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 249
Jð0; 0Þ ¼ � 1
2 1þ cos u02
� �� � ; Jð�u0; 0Þ ¼ � 1
2;
Jð�2p; 0Þ ¼ � 1
2 1� cos u02
� �� � ; J 0;1
b
� �¼ � 1
2 1þ cos u02
� �� � ;J �u0;
1
b
� �¼ � 1
2; J �2p;
1
b
� �¼ � 1
2 1� cos u02
� �� � : ð2:14Þ
Notice that for (2.11), we have
h0 ¼ Hð0; 0Þ ¼ � tanu04
� �� �2; h1 ¼ Hð�u0; 0Þ ¼ 0;
h2 ¼ Hð�2p; 0Þ ¼ � tanu04
� �� ��2
; h3 ¼ H 0;1
b
� �¼ 1
6b2� tan
u04
� �� �2;
h4 ¼ H �u0;1
b
� �¼ 1
6b2; h5 ¼ H �2p;
1
b
� �¼ 1
6b2� tan
u04
� �� ��2
:
ð2:15Þ
(1) For c2 > 1 and b > 0, we know from (2.15) that for every u0 2 (p,2p), h4 = h0if and only if the point of (u0,b)-parameter plane lies in the curve
C1 : b ¼ 1ffiffiffi6
ptan u0
4
� � � f1ðu0Þ:
h2 = h4 if and only if the point of (u0,b)-parameter plane lies in the curve
C2 : b ¼tan u0
4
� �ffiffiffi6
p � f2ðu0Þ:
h2 = h0 if and only if the point of (u0,b)-parameter plane lies in the curve
C0 : u0 ¼ p � f0ðu0Þ:From the above discussion, we have the partition of the open strip region
0 < u0 < 2p in (u0,b)-parameter half-plane by the curves Ci, i = 0,1,2, shown
in Fig. 4 where
ðAÞ 0<u0<p; b< f2ðu0Þ; ðBÞ 0<u0<p; b¼ f2ðu0Þ;ðCÞ 0<u0<p; f 2ðu0Þ<b< f1ðu0Þ; ðDÞ 0<u0<p; b¼ f1ðu0Þ;
ðEÞ 0<u0<p; b> f1ðu0Þ; ðFÞ u0¼p; b>1ffiffiffi6
p ;
ðGÞ p<u0<2p; b> f2ðu0Þ; ðHÞ p<u0<2p; b¼ f2ðu0Þ;ðIÞ p<u0<2p; f 1ðu0Þ<b< f2ðu0Þ; ðJÞ p<u0<2p; b¼ f1ðu0Þ;
ðKÞ p<u0<2p; b< f1ðu0Þ; ðLÞ u0¼p; 0<b<1ffiffiffi6
p ; ðMÞ ðu0;bÞ¼ p;1ffiffiffi6
p� �
:
Fig. 4. The partition of the (u0,b)-parameter strip of (2.12) for b > 0.
250 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
Corresponding to the regions (A)–(M) of the point set (u0,b), the bifurca-
tions of phase portraits of (2.12) can be showed in Fig. 5 for b > 0.
(2) For c2 < 1, b > 0, we know from (2.15) that for every u0 2 (0,2p), h1 = h3if and only if the point of (u0,b)-parameter plane lies in the curve
C1 : b ¼ f1ðu0Þ:h0 = h2 if and only if the point of (u0,b)-parameter plane lies in the curve
C0 : u0 ¼ p ¼ f0ðu0Þ:h1 = h5 if and only if the point of (u0,b)-parameter plane lies in the curve
C2 : b ¼ f2ðu0Þ:From the above discussion, we have the partition of the open strip region
0 < u0 < 2p in (u0,b)-parameter half-plane by the curves Ci, i = 0,1,2, shown
in Fig. 4.
Corresponding to the regions (A)–(M) of the point set (u0,b), phase portraits
of (2.12) can be showed in Fig. 6 for b > 0.
3. The existence of smooth travelling wave solutions
In this section, we discuss the travelling wave solutions of (1.1) by using the
results of Section 2. We notice that since (2.1) is defined in phase cylinder
S1 · R with / = �2p, / = 2p identified. In other words, (�2p, 0) and (2p, 0)are the same point in the cylinder. The heteroclinic orbits connecting from
�2p to 2p should view as two homoclinic orbits. Therefore, we have the follow-ing conclusions.
Fig. 5. Phase portraits of (2.12) and the curve y ¼ 1b for c
2 > 1. (1) (u0,b) 2 (A), (2) (u0,b) 2 (B), (3)
(u0,b) 2 (C), (4) (u0,b) 2 (D), (5) (u0,b) 2 (E), (6) (u0,b) 2 (F), (7) (u0,b) 2 (G), (8) (u0,b) 2 (H), (9)
(u0,b) 2 (I), (10) (u0,b) 2 (J), (11) (u0,b) 2 (K), (12) (u0,b) 2 (L), (13) (u0,b) 2 (M).
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 251
Case I. Suppose that b = 0, c2 5 1. We notice that for b = 0, the right hand
of (1.3) has no singularity. So that, all solutions are smooth, we have the fol-
lowing conclusions.
Theorem 3.1
1. For c2 > 1, 0 < u0 < p, then,(1) corresponding to H(/, y) = h, h 2 (h1,h0) defined by (2.6), Eq. (1.1) has
two families of smooth periodic wave solutions;
(2) corresponding to H(/, y) = h0, defined by (2.6), Eq. (1.1) has a pair of
smooth solitary wave solutions with peak type and valley type, respectively;
Fig. 5 (continued)
252 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
(3) corresponding to H(/, y) = h, h 2 (h0,h2) defined by (2.6), Eq. (1.1) has a
family of smooth periodic wave solutions;
(4) corresponding to H(/, y) = h2 defined by (2.6), Eq. (1.1) has a pair of
smooth solitary wave solutions with peak type and valley type, respectively;
(5) corresponding to H(/, y) = h, h 2 (h2,1) defined by (2.6), Eq. (1.1) hastwo families of smooth periodic wave solutions, which correspond to the
two families of the rotation orbits in the phase cylinder.
2. For c2 > 1, u0 = p, then,(1) corresponding to H(/, y) = h, h 2 (h1 = 0,h0 = h2) defined by (2.6), Eq.
(1.1) has two families of smooth periodic wave solutions;
(2) corresponding to H(/, y) = h0 = h2 defined by (2.6), Eq. (1.1) has two kinkand two anti-kink wave solutions;
Fig. 6. Phase portraits of (2.12) and the curve y ¼ 1b for c
2 < 1. (1) (u0,b) 2 (A), (2) (u0,b) 2 (B), (3)
(u0,b) 2 (C), (4) (u0,b) 2 (D), (5) (u0,b) 2 (E), (6) (u0,b) 2 (F), (7) (u0,b) 2 (G), (8) (u0,b) 2 (H), (9)
(u0,b) 2 (I), (10)(u0,b) 2 (J), (11) (u0,b) 2 (K), (12) (u0,b) 2 (L), (13) (u0,b) 2 (M).
254 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
(3) corresponding to H(/, y) = h, h 2 (h0 = h2,1) defined by (2.6), Eq. (1.1)
has two families of smooth periodic wave solutions, which correspond to
the two families of rotation orbits in the phase cylinder.
3. For c2 > 1, p < u0 < 2p, then,(1) corresponding to H(/, y) = h, h 2 (h1,h2) defined by (2.6), Eq. (1.1) hastwo families of smooth periodic wave solutions;
(2) corresponding to H(/, y) = h2 defined by (2.6), Eq. (1.1) has a pairs of
smooth solitary wave solutions with peak type and valley type, respectively;
(3) corresponding to H(/, y) = h, h 2 (h2,h0) defined by (2.6), Eq. (1.1) has
one family of smooth periodic wave solutions;
Fig. 6 (continued)
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 255
(4) corresponding to H(/, y) = h0 defined by (2.6), Eq. (1.1) has a pair of sol-
itary wave solution with peak type and valley type, respectively;
(5) corresponding to H(/, y) = h, h 2 (h0,1) defined by (2.6), Eq. (1.1) has
two families of smooth periodic wave solutions, which correspond to the
two families of rotation orbits in the phase cylinder.
4. For c2 < 1, then,(1) corresponding to H(/, y) = h, h 2 (h0,h1) and h 2 (h2,h1) defined by (2.6),
Eq. (1.1) has two families of smooth periodic wave solutions;
(2) corresponding to H(/, y) = 0 defined by (2.6), Eq. (1.1) has an kink wave
solutions and an anti-kink wave solutions and two solitary wave solutions
with peak type and valley type;
Fig. 7. The variation of smoothness of periodic travelling wave as h varies. (1) Smooth wave, (2)
Periodic cusp wave, (3) Breaking wave.
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 257
(3) corresponding to H(/, y) = h, h 2 (0,1) defined by (2.6), Eq. (1.1) has two
families of smooth periodic wave solutions, which correspond to the two
families of rotation orbits in the phase cylinder.
To understand the above results, we shall give some exact explicit solutions for
b = 0 and c2 � 15 0. In this case h1 = H(±u0,0) = 0, h0 ¼Hð0; 0Þ¼� tan u04
� �� �2,
h2 ¼Hð�2p;0Þ¼� 1
tanu04ð Þð Þ2
. Denote that sn(u,k), cn(u,k), dn(u,k), tn(u,k) are the
Jacobian elliptic functions with the modulo k. K(k) is the complete elliptic integral
of the first kind. E(k) is the complete elliptic integral of the second kind.
258 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
(I). The case c2 > 1.
1. u0 2 (0,p).(1) For h 2 (h1 = 0,h0), two families of periodic wave solutions have the
parametric representation
/ ¼ �4 arctan a20 � ða20 � b20ÞðsnðX0
ffiffiffiffiffiA0
pðx� ctÞ; k0ÞÞ2
� �12
: ð3:1Þ
where
a20 ¼ffiffiffiffiffih0
pþ
ffiffiffih
pffiffiffiffiffih2
p�
ffiffiffih
p ; b20 ¼ffiffiffiffiffih0
p�
ffiffiffih
pffiffiffiffiffih2
pþ
ffiffiffih
p ;
k20 ¼2ffiffiffih
pðffiffiffiffiffih0
pþ
ffiffiffiffiffih2
pÞ
ðffiffiffih
pþ
ffiffiffiffiffih2
pÞð
ffiffiffiffiffih0
pþ
ffiffiffih
pÞ; X0 ¼
ð2ðffiffiffiffiffih2
pþ
ffiffiffih
pÞð
ffiffiffiffiffih0
pþ
ffiffiffih
pÞÞ
12
4
(2) For h = h0, two solitary wave solutions of (1.1) have the parametric
representation
/ ¼ � arctanð4h0Þ
14
ðffiffiffiffiffih2
p�
ffiffiffiffiffih0
pÞ12
coshh
14
0ðffiffiffiffiffih2
pþ
ffiffiffiffiffih0
pÞ12
2
ffiffiffiffiffiA0
pðx� ctÞ
! :
ð3:2Þ(3) For h 2 (h0,h2), periodic wave solutions have the parametric
representation
/ ¼ �4 arctan
ffiffiffih
pþ
ffiffiffiffiffih0
pffiffiffiffiffih2
p�
ffiffiffih
p !1
2
cnðX1
ffiffiffiffiffiA0
pðx� ctÞ; k1Þ
0@
1A: ð3:3Þ
where
X1 ¼1
2h
14ð
ffiffiffiffiffih0
pþ
ffiffiffiffiffih2
pÞ12; k21 ¼
ðffiffiffih
pþ
ffiffiffiffiffih0
pÞð
ffiffiffih
pþ
ffiffiffiffiffih2
pÞ
2ffiffiffih
pðffiffiffiffiffih0
pþ
ffiffiffiffiffih2
pÞ
(4) For h = h2, solitary wave solution have the parametric representation
/ ¼ �4 arctanðffiffiffiffiffih2
p�
ffiffiffiffiffih0
pÞ12
ð4h2Þ14
sinhðh2 þ
ffiffiffiffiffiffiffiffiffih0h2
pÞ12
2
ffiffiffiffiffiA0
pðx� ctÞ
! !:
ð3:4Þ(5) For h 2 (h2,1), the family of periodic wave solutions, which correspond
to the rotation orbits in the phase cylinder, have the parametric
representation
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 259
/ ¼ �4 arctan
ffiffiffih
p�
ffiffiffiffiffih0
pffiffiffih
pþ
ffiffiffiffiffih2
p !1
2
tnðX0
ffiffiffiffiffiA0
pðx� ctÞ; k0Þ
0@
1A: ð3:5Þ
2. u0 = p.(1) For h 2 (h1 = 0,h0 = h2) = 1, two families of periodic wave solutions
have the parametric representation
/ ¼ �4 arctan1þ
ffiffiffih
p
1�ffiffiffih
p � 4ffiffiffih
p
1� hðsnðX2
ffiffiffiffiffiA0
pðx� ctÞ; k2ÞÞ2
!12
0@
1A: ð3:6Þ
where
X2 ¼ffiffiffi2
pð1þ
ffiffiffih
pÞ
4; k2 ¼
2h14
1þffiffiffih
p :
(2) For h = h0 = h2 = 1, two kink and two anti-kink wave solutions have the
parametric representation
/ ¼ �4 arctan exp
ffiffiffiffiffiffiffiffi2A0
p
2ðx� ctÞ
� �: ð3:7Þ
(3) For h 2 (h0,1), the family of periodic wave solutions, which correspondto the rotation orbits in the phase cylinder, have the parametric
representation
/ ¼ �4 arctan
ffiffiffih
p� 1ffiffiffi
hp
þ 1
!12
tnðX2
ffiffiffiffiffiA0
pðx� ctÞ; k2Þ
0@
1A: ð3:8Þ
3. u0 2 (p,2p),0 < h2 < h0.(1) For h 2 (h2,h0), periodic wave solutions are same as (3.1).
(2) For h = h2, a pair of solitary wave solutions have the parametric
representation
/ ¼ �4 arctanðffiffiffiffiffih0
p�
ffiffiffiffiffih2
pÞ12
ð4h2Þ14
cosðh0 � h2Þ
12
2ffiffiffi2
pffiffiffiffiffiA0
pðx� ctÞ
! !�10@
1A:
ð3:9Þ(3) For h 2 (h2,h0), periodic wave solutions have the parametric
representation
/ ¼ �4 arctan
ffiffiffiffiffih0
p�
ffiffiffih
pffiffiffih
pþ
ffiffiffiffiffih2
p !1
2
ðcnðX3
ffiffiffiffiffiA0
pðx� ctÞ; k3ÞÞ�1
0@
1A: ð3:10Þ
260 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
where
X3 ¼1
2h
14
� ffiffiffiffiffih0
pþ
ffiffiffiffiffih2
p �12; k23 ¼
ðffiffiffiffiffih0
p�
ffiffiffih
pÞð
ffiffiffih
p�
ffiffiffiffiffih2
pÞ
2ffiffiffih
pðffiffiffiffiffih0
pþ
ffiffiffiffiffih2
pÞ
:
(4) For h = h0, solitary wave solution have the parametric representation
/ ¼ �4 arctan2ffiffiffiffiffih0
pffiffiffiffiffih0
p�
ffiffiffiffiffih2
p� �1
2
sinh1
2ðffiffiffiffiffih0
pþ
ffiffiffiffiffih2
pÞ12h
14
0
ffiffiffiffiffiA0
pðx� ctÞ
� � !:
ð3:11Þ(5) For h 2 (h0,1), the family of periodic wave solutions, which correspond
to the rotation orbits in the phase cylinder, have the parametric
representation
/ ¼ �4 arctan
ffiffiffih
p�
ffiffiffiffiffih0
pffiffiffih
pþ
ffiffiffiffiffih2
p !1
2
tnðX0
ffiffiffiffiffiA0
pðx� ctÞ; k0Þ
0@
1A: ð3:12Þ
(II). The case c2 < 1.
1. u0 2 (0,p).(1) For h 2 (h0,h1 = 0), the family of periodic wave solutions have the
parametric representation
/ ¼ �4 arctanðb4snðX4
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k4ÞÞ: ð3:13Þ
where
a24 ¼ffiffiffiffiffiffiffiffi�h0
pþ
ffiffiffiffiffiffiffi�h
pffiffiffiffiffiffiffiffi�h2
p�
ffiffiffiffiffiffiffi�h
p ; b24 ¼ffiffiffiffiffiffiffiffi�h0
p�
ffiffiffiffiffiffiffi�h
pffiffiffiffiffiffiffiffi�h2
pþ
ffiffiffiffiffiffiffi�h
p :
k4 ¼b4a4
; X4 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð
ffiffiffiffiffiffiffiffi�h0
pþ
ffiffiffiffiffiffiffi�h
pÞð
ffiffiffiffiffiffiffiffi�h2
pþ
ffiffiffiffiffiffiffi�h
pÞ
q4
(2) For h = h1 = 0, kink and anti-kink wave solutions have the parametric
representation
/ ¼ �4 arctan tanu04
� �tanh
ffiffiffiffiffiffiffiffiffiffiffi�2A0
p
4ðx� ctÞ
� �� �: ð3:14Þ
(3) For h 2 (h2,h1 = 0), then
(i) when h 2 (h2,h0), the family of periodic wave solutions have the para-
metric representation
/ ¼ �4 arctan
ffiffiffiffiffiffiffiffi�h0
pþ
ffiffiffiffiffiffiffi�h
pffiffiffiffiffiffiffiffi�h2
p�
ffiffiffiffiffiffiffi�h
p !1
2
ðcnðX5
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k5ÞÞ�1
0@
1A;
ð3:15Þ
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 261
where
X5 ¼1
2ð�hÞ
14
ffiffiffiffiffiffiffiffi�h0
pþ
ffiffiffiffiffiffiffiffi�h2
p� �12
; k25 ¼ðffiffiffiffiffiffiffi�h
p�
ffiffiffiffiffiffiffiffi�h0
pÞð
ffiffiffiffiffiffiffiffi�h2
p�
ffiffiffiffiffiffiffi�h
pÞ
2ðffiffiffiffiffiffiffih0h
pþ
ffiffiffiffiffiffiffih2h
pÞ
(ii) when h 2 (h0,0 = h1), the family of periodic wave solutions are same
as (3.13)
(4) For h 2 (h1 = 0,1), the family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, have the parametric
representation
/ ¼ � arctanh� h0h� h2
� �14 1þ cnðX6
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k6Þ
1� cnðX6
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k6Þ
� �12
!: ð3:16Þ
where
X6 ¼ffiffiffi2
p
2ððh� h2Þðh� h0ÞÞ
14; k6 ¼
1
2� ðh� 1Þ
4X26
:
2. u0 = p.(1) For h 2 (h0 = h2,h1), the family of periodic wave solutions have the para-
metric representation
/ ¼ �4 arctan1�
ffiffiffiffiffiffiffi�h
p
1þffiffiffiffiffiffiffi�h
p !1
2
snðX7
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k7Þ
0@
1A: ð3:17Þ
where
X7 ¼ffiffiffi2
p
4ð1þ
ffiffiffiffiffiffiffi�h
pÞ; k7 ¼
1�ffiffiffiffiffiffiffi�h
p
1þffiffiffiffiffiffiffi�h
p :
(2) For h = h1 = 0, kink and anti-kink wave solutions have the parametric
representation
/ ¼ �4 arctan exp
ffiffiffiffiffiffiffiffiffiffiffi�2A0
p
2ðx� ctÞ
� �� p: ð3:18Þ
(3) For h 2 (h1 = 0,1), the family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, the family of periodic
wave solutions have the parametric representation
/ ¼ �4 arctan1þ cnðX8
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k8Þ
1� cnðX8
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k8Þ
� �12
!: ð3:19Þ
262 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
where
X8 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1þ hÞ
p2
; k8 ¼1ffiffiffiffiffiffiffiffiffiffiffi1þ h
p :
3. u0 2 (p,2p).(1) For For h 2 (h0,h1)
(i) when h 2 (h0,h2), the periodic wave solutions have the parametric
representation
/ ¼ �4 arctan
ffiffiffiffiffiffiffiffi�h0
p�
ffiffiffiffiffiffiffi�h
pffiffiffiffiffiffiffiffi�h2
pþ
ffiffiffiffiffiffiffi�h
p !1
2
cnðX9
ffiffiffiffiffiffiffiffiffi�A0
pðx� ctÞ; k9Þ
0@
1A: ð3:20Þ
where
X9 ¼ffiffiffi2
p
4
ffiffiffiffiffiffiffih0h
pþ
ffiffiffiffiffiffiffih2h
p� �12
; k29 ¼ðffiffiffiffiffiffiffiffi�h0
p�
ffiffiffiffiffiffiffi�h
pÞð
ffiffiffiffiffiffiffiffi�h2
p�
ffiffiffiffiffiffiffi�h
pÞ
2ðffiffiffiffiffiffiffih0h
pþ
ffiffiffiffiffiffiffih2h
pÞ
:
(ii) when h 2 (h2,h1), the periodic wave solutions are same as (3.13).
(2) For h = h1 = 0, kink and anti-kink wave solutions are same as (3.14).(3) For h 2 (h2,h1), the periodic wave solutions are same as (3.13).
(4) For h 2 (h1 = 0,1), the family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, the family of periodic
wave solutions are same as (3.16).
Case II. b 5 0 and c2 � 1 = 0. In this case, y = 0 is a singular straight line,
and we have the following
Theorem 3.2. Suppose that c2 � 1 = 0,b > 0, Eq. (1.1) has two families of
smooth periodic solutions, which correspond to the rotation orbits in the phase
cylinder.
Case III. b 5 0 and c2 � 1 5 0.
Similar to case I and II, we discuss the travelling wave solutions (2.2) by
using the results of Section 2 for b 5 0 and c2 � 1 5 0.
(i) Suppose that b > 0 and c2 � 1 > 0.
Theorem 3.3
1. For (u0,b) 2 (A), then,
(1) corresponding to H(/, y) = h, h 2 (h1,h0) defined by (2.13), Eq. (1.1)
has two families of smooth periodic wave solution;
(2) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has a pair
of smooth solitary wave solutions with peak type and valley type,
respectively;
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 263
(3) corresponding to H(/, y) = h, h 2 (h0,h2) defined by (2.13), Eq. (1.1)
has one family of smooth periodic wave solution;
(4) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has a pair of
smooth solitary wave solutions with peak type and valley type,
respectively;
(5) corresponding to H(/, y) = h, h 2 (h2,1), h 2 (h2,h4) and h 2 (h4,1)defined by (2.13), Eq. (1.1) has one family of smooth periodic wave solu-
tions, which correspond to the rotation orbits in the phase cylinder,
respectively.
2. For (u0,b) 2 (B), then,
(1) corresponding to H(/, y) = h, h 2 (h1,h0) defined by (2.13), Eq. (1.1)
has two families of smooth periodic wave solutions;
(2) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has a pair ofsmooth solitary wave solutions with peak type and valley type,
respectively;
(3) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has one
smooth solitary wave solutions with valley type;
(4) corresponding to h 2 (h2,1) and h 2 (h4,1) defined by (2.13), Eq. (1.1)
has one family of periodic wave solutions, which correspond to the rota-
tion orbits in the phase cylinder, respectively.
3. For (u0,b) 2 (C), then,
(1) corresponding to H(/, y) = h, h 2 (h1,h0) defined by (2.13), Eq. (1.1)
has two families of smooth periodic wave solutions;
(2) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has a pair of
smooth solitary wave solution with peak type and valley type,
respectively;
(3) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has one
smooth solitary wave solution with valley type;(4) corresponding to H(/, y) = h, h 2 (h2,1) and h 2 (h4,1) defined by
(2.13), Eq. (1.1) has one family of smooth periodic wave solutions, which
correspond to the rotation orbits in the phase cylinder, respectively.
4. For (u0,b) 2 (D), then,(1) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has one
smooth solitary wave solution with valley type;
(2) corresponding to H(/, y) = h, h 2 (h2,1) and h 2 (h4,1) defined by(2.13), Eq. (1.1) has one family of smooth periodic wave solutions, which
correspond to the rotation orbits in the phase cylinder, respectively.
5. For (u0,b) 2 (E). Then,
(1) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has one
smooth solitary wave solution with valley type;
(2) corresponding to H(/, y) = h, h 2 (h2,1) and h 2 (h4,1) defined by
(2.13), Eq. (1.1) has one family of smooth periodic wave solutions, whichcorrespond to the rotation orbits in the phase cylinder, respectively.
264 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
6. For (u0,b) 2 (F), then,
(1) corresponding to H(/, y) = h0 = h2 defined by (2.13), Eq. (1.1) has a
smooth kink and an anti-kink wave solutions;
(2) corresponding to H(/, y) = h, h 2 (h0 = h2,1) and h 2 (h4,1) defined
by (2.13), Eq. (1.1) has one family of smooth periodic wave solutions,
which correspond to the rotation orbits in the phase cylinder,respectively.
7. For (u0,b) 2 (G), then,
(1) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has one
smooth solitary wave solution with valley type;
(2) corresponding to H(/, y) = h, h 2 (h0,1) and h 2 (h4,1) defined by
(2.13), Eq. (1.1) has one family of smooth periodic wave solutions, which
correspond to the rotation orbits in the phase cylinder, respectively.
8. For (u0,b) 2 (H), then,
(1) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has one
smooth solitary wave solution with valley type;
(2) corresponding to H(/, y) = h, h 2 (h0,1) and h 2 (h4,1) defined by
(2.13), Eq. (1.1) has one family of smooth periodic wave solutions, which
correspond to the rotation orbits in the phase cylinder, respectively.
9. For (u0,b) 2 (I), then,
(1) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has one
smooth solitary wave solution with valley type;
(2) corresponding to h 2 (h1,h2) defined by (2.13), Eq. (1.1) has two families
of smooth periodic wave solutions;
(3) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has a pair of
solitary solutions with peak type and valley type, respectively;
(4) corresponding to H(/, y) = h, h 2 (h0,1) and h 2 (h4,1) defined by(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, respectively.
10. For (u0,b) 2 (J), then,
(1) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has one
smooth solitary solution with valley type;
(2) corresponding to H(/, y) = h, h 2 (h1,h2) defined by (2.13), Eq. (1.1)
has two families of smooth periodic wave solutions;(3) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has a pair of
solitary solutions with peak type and valley type, respectively;
(4) corresponding to H(/, y) = h, h 2 (h0,1) and h 2 (h4,1) defined by
(2.13), Eq. (1.1) has one family of smooth periodic wave solutions,
which correspond to the rotation orbits in the phase cylinder,
respectively.
11. For (u0,b) 2 (K), then,(1) corresponding to H(/, y) = h0 defined by (2.13), Eq. (1.1) has a pair of
smooth solitary solutions with peak type and valley type, respectively;
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 265
(2) corresponding to H(/, y) = h, h 2 (h0,h2) defined by (2.13), Eq. (1.1)
has one family of smooth periodic wave solutions;
(3) corresponding to H(/, y) = h2 defined by (2.13), Eq. (1.1) has a pair
of smooth solitary solutions with peak type and valley type, respec-
tively;
(4) corresponding to H(/, y) = h, h 2 (h1,h2) defined by (2.13), Eq. (1.1)has two families of smooth periodic wave solutions;
(5) corresponding to H(/, y) = h, h 2 (h0,1),h 2 (h0,h4) and h 2 (h4,1)
defined by (2.13), Eq. (1.1) has one families of periodic wave solutions,
which correspond to the rotation orbits in the phase cylinder,
respectively.
12. For (u0,b) 2 (L), then,
(1) corresponding to H(/, y) = h, h 2 (h0 = h2,h1) defined by (2.13), Eq.(1.1) has two families of smooth periodic wave solutions;
(2) corresponding to H(/, y) = h2 = h0 defined by (2.13), Eq. (1.1) has two
smooth kink and two anti-kink wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h0 = h2,1), h 2 (h0 = h2,h4) and
h 2 (h4,1) defined by (2.13), Eq. (1.1) has one families of smooth peri-
odic wave solutions, which correspond to the rotation orbits in the phase
cylinder, respectively.
13. For (u0,b) 2 (M), then,
(1) corresponding to H(/, y) = h0 = h2 = h4 defined by (2.13), Eq. (1.1) has
a smooth kink and an anti-kink wave solutions;
(2) corresponding to h 2 (h0 = h2,1) and h 2 (h4,1) defined by (2.13), Eq.
(1.1) has one family of periodic wave solutions, which correspond to the
rotation orbits in the phase cylinder, respectively.
(ii) Suppose that b > 0 and c2 � 1 < 0.
Theorem 3.4
1. For (u0,b) 2 (A), then,
(1) corresponding to H(/, y) = h, h 2 (h0,h1) and h 2 (h2,h1) defined by(2.13), Eq. (1.1) has one family of smooth periodic wave solutions,
respectively;
(2) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth kink and two anti-kink wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h1,1), h 2 (h1,h5) and h 2 (h5,1)
defined by (2.13), Eq. (1.1) has one family of periodic wave solutions,
which correspond to the rotation orbits in the phase cylinder,
respectively.
2. For (u0,b) 2 (B), then,
(1) corresponding to H(/, y) = h, h 2 (h0,h1) defined by (2.13), Eq. (1.1)
has one family of smooth periodic wave solutions;
266 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
(2) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has a smooth
kink and two anti-kink wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h5,1) defined by
(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, respectively.
3. For (u0,b) 2 (C), then,
(1) corresponding to H(/, y) = h, h 2 (h0,h1) defined by (2.13), Eq. (1.1)
has one family of smooth periodic wave solutions;
(2) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has a smooth
kink and two anti-kink wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h5,1) defined by
(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, respectively.
4. For (u0,b) 2 (D), then,
(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth anti-kink wave solutions;
(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h5,1) defined by
(2.13), Eq. (1.1) has one families of periodic wave solutions, which cor-
respond to the rotation orbits in the phase cylinder, respectively.
5. For (u0,b) 2 (E), then,
(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth anti-kink wave solutions;
(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h5,1) defined by
(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, respectively.
6. For (u0,b) 2 (F), then,(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth anti-kink wave solutions;
(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h3 = h5,1) defined
by (2.13), Eq. (1.1) has one family of periodic wave solutions, which cor-
respond to the rotation orbits in the phase cylinder, respectively.
7. For (u0,b) 2 (G), then,
(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has twosmooth anti-kink wave solutions;
(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h3,1) defined by
(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, respectively.
8. For (u0,b) 2 (H), then,
(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth anti-kink wave solutions;(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h3,1) defined by
(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, respectively.
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 267
9. For (u0,b) 2 (I), then,
(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has a smooth
kink and two anti-kink wave solutions;
(2) corresponding to H(/, y) = h, h 2 (h2,h1) defined by (2.13), Eq. (1.1)
has one family of periodic wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h3,1) definedby (2.13), Eq. (1.1) has one family of periodic wave solutions,
which correspond to the rotation orbits in the phase cylinder,
respectively.
10. For (u0,b) 2 (J), then,
(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has a smooth
kink and two anti-kink wave solutions;
(2) corresponding to h 2 (h2,h1) defined by (2.13), Eq. (1.1) has one familyof periodic wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h3,1) defined by
(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-
spond to the rotation orbits in the phase cylinder, respectively.
11. For (u0,b) 2 (K), then,
(1) corresponding to h 2 (h0,h1) and h 2 (h2,h1) defined by (2.13), Eq. (1.1)
has one family of periodic wave solutions, respectively;(2) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth kink and two anti-kink wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h1,1), h 2 (h1,h3) and h 2 (h3,1)
defined by (2.13), Eq. (1.1) has one family of periodic wave solutions,
which correspond to the rotation orbits in the phase cylinder,
respectively.
12. For (u0,b) 2 (L), then,(1) corresponding to h 2 (h0,h1) and h 2 (h2,h1) defined by (2.13), Eq. (1.1)
has one family of periodic wave solutions, respectively;
(2) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth kink and two anti-kink wave solutions;
(3) corresponding to H(/, y) = h, h 2 (h1,1), h 2 (h1,h3 = h5) and
h 2 (h3 = h5,1) defined by (2.13), Eq. (1.1) has one family of periodic
wave solutions, which correspond to the rotation orbits in the phase cyl-
inder, respectively.
13. For (u0,b) 2 (M), then,
(1) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has two
smooth anti-kink wave solutions;
(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h1 = h3 = h5,1)
defined by (2.13), Eq. (1.1) has one family of periodic wave solutions,
which correspond to the rotation orbits in the phase cylinder,
respectively.
268 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
4. The existence of non-smooth periodic travelling waves solutions
In this section, we shall point out that the existence of the singular line
c2 � 1� by = 0 is the original reason for the appearance of non-smooth travelling
wave solutions in our travelling wave models (see [4–8]). In fact, for these periodic
families defined byH(/,y) = h in (1.4), when h varies (say from hi to hj), periodictravelling wave will gradually lose their smoothness, and evolve from smooth
periodic travelling wave to periodic cusp travelling wave (see Fig. 7(1)–7(2)).
(i) Suppose that b > 0, c2 > 1.
Theorem 4.1
1. For (u0,b) 2 (B), then,corresponding to H(/, y) = h, h 2 (h0,h4 = h2) defined by (2.13), Eq. (1.1) has
one family of uncountably infinite many periodic travelling wave solutions with
varying smoothness, respectively. When h varies from h0 to h4, periodic travel-
ling wave will gradually lose their smoothness, and evolve from smooth periodic
travelling wave to periodic cusp travelling wave.
2. For (u0,b) 2 (C), then,
corresponding to H(/, y) = h, h 2 (h0,h4) defined by (2.13), Eq. (1.1) has one
family of uncountably infinite many periodic travelling wave solutions withvarying smoothness, respectively. When h varies from h0 to h4, periodic travel-
ling wave will gradually lose their smoothness, and evolve from smooth periodic
travelling wave to periodic cusp travelling wave.
3. For (u0,b) 2 (D), then,
corresponding to H(/, y) = h, h 2 (h1,h0 = h4) defined by (2.13), Eq. (1.1) has
one family of uncountably infinite many periodic travelling wave solutions with
varying smoothness. When h varies from h1 to h4, periodic travelling wave will
gradually lose their smoothness, and evolve from smooth periodic travellingwave to periodic cusp travelling wave.
4. For (u0,b) 2 (E), (u0,b) 2 (F) and (u0,b) 2 (G), then
corresponding to H(/, y) = h, h 2 (h1,h4) defined by (2.13), Eq. (1.1) has one
family of uncountably infinite many periodic travelling wave solutions with
varying smoothness, respectively. When h varies from h1 to h4, periodic travel-
ling wave will gradually lose their smoothness, and evolve from smooth periodic
travelling wave to periodic cusp travelling wave.
5. For (u0,b) 2 (H), then,corresponding to H(/, y) = h, h 2 (h1,h2 = h4) defined by (2.13), Eq. (1.1) has
one family of uncountably infinite many periodic travelling wave solutions with
varying smoothness. When h varies from h1 to h4, periodic travelling wave will
gradually lose their smoothness, and evolve from smooth periodic travelling
wave to periodic cusp travelling wave.
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 269
6. For (u0,b) 2 (I), then,
corresponding to H(/, y) = h, h 2 (h2,h4) defined by (2.13), Eq. (1.1) has one
family of uncountably infinite many periodic travelling wave solutions with
varying smoothness. When h varies from h2 to h4, periodic travelling wave will
gradually lose their smoothness, and evolve from smooth periodic travelling
wave to periodic cusp travelling wave.7. For (u0,b) 2 (J), then,
corresponding to H(/, y) = h, h 2 (h2,h4 = h0) defined by (2.13), Eq. (1.1) has
one family of uncountably infinite many periodic travelling wave solutions with
varying smoothness. When h varies from h2 to h4, periodic travelling wave will
gradually lose their smoothness, and evolve from smooth periodic travelling
wave to periodic cusp travelling wave.
8. For (u0,b) 2 (M), then,
corresponding to H(/, y) = h, h 2 (h1,h4 = h2 = h0) defined by (2.13), Eq. (1.1)has one family of uncountably infinite many periodic travelling wave solutions
with varying smoothness. When h varies from h1 to h4, periodic travelling wave
will gradually lose their smoothness, and evolve from smooth periodic travelling
wave to periodic cusp travelling wave.
(ii) Suppose that b > 0, c2 < 1.
Theorem 4.2
1. For (u0,b) 2 (B), then,
corresponding to H(/, y) = h, h 2 (h2,h5 = h1) defined by (2.13), Eq. (1.1) has
one family of uncountably infinite many periodic travelling wave solutions with
varying smoothness. When h varies from h2 to h5, periodic travelling wave will
gradually lose their smoothness, and evolve from smooth periodic travellingwave to periodic cusp travelling wave.
2. For (u0,b) 2 (C), then,
corresponding to H(/, y) = h, h 2 (h2,h5) defined by (2.13), Eq. (1.1) has one
family of uncountably infinite many periodic travelling wave solutions with
varying smoothness. When h varies from h2 to h5, periodic travelling wave will
gradually lose their smoothness, and evolve from smooth periodic travelling
wave to periodic cusp travelling wave.
3. For (u0,b) 2 (D), then,corresponding to H(/, y) = h, h 2 (h2,h5) and h 2 (h0,h3 = h1) defined by
(2.13), Eq. (1.1) has one family of uncountably infinite many periodic travelling
wave solutions with varying smoothness, respectively. When h varies from h2(or h0) to h5 (or h3), periodic travelling wave will gradually lose their smooth-
ness, and evolve from smooth periodic travelling wave to periodic cusp travel-
ling wave.
270 J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271
4. For (u0,b) 2 (E), (u0,b) 2 (F), and (u0,b) 2 (G), then,
corresponding to H(/, y) = h, h 2 (h2,h5) and h 2 (h0,h3) defined by (2.13), Eq.
(1.1) has one family of uncountably infinite many periodic travelling wave solu-
tions with varying smoothness, respectively. When h varies from h2 (or h0) to h5(or h3), periodic travelling wave will gradually lose their smoothness, and
evolve from smooth periodic travelling wave to periodic cusp travelling wave.5. For (u0,b) 2 (H), then,
corresponding to H(/, y) = h, h 2 (h2,h5 = h1) and h 2 (h0,h3) defined by (2.13),
Eq. (1.1) has one family of uncountably infinite many periodic travelling wave
solutions with varying smoothness, respectively. When h varies from h2 (or h0)
to h5 (or h3), periodic travelling wave will gradually lose their smoothness,
and evolve from smooth periodic travelling wave to periodic cusp travelling wave.
6. For (u0,b) 2 (I), then,
corresponding to h 2 (h0,h3) defined by (2.13), Eq. (1.1) has one family ofuncountably infinite many periodic travelling wave solutions with varying
smoothness. When h varies from h0 to h3, periodic travelling wave will gradu-
ally lose their smoothness, and evolve from smooth periodic travelling wave to
periodic cusp travelling wave.
7. For (u0,b) 2 (J), then,
corresponding to h 2 (h0,h3 = h1) defined by (2.13), Eq. (1.1) has one family of
uncountably infinite many periodic travelling wave solutions with varying
smoothness. When h varies from h0 to h3, periodic travelling wave will gradu-ally lose their smoothness, and evolve from smooth periodic travelling wave to
periodic cusp travelling wave.
8. For (u0,b) 2 (M), then,
corresponding to H(/, y) = h, h 2 (h2,h5 = h1) and h 2 (h0,h3 = h1) defined by
(2.13), Eq. (1.1) has one family of uncountably infinite many periodic travelling
wave solutions with varying smoothness, respectively. When h varies from h2 (or
h0) to h5 (or h3), periodic travelling wave will gradually lose their smoothness,
and evolve from smooth periodic travelling wave to periodic cusp travelling wave.
Finally, it is easy to see the following conclusion holds.
Theorem 4.3. If a orbit of (1.2) intersects the curve c2 � 1� by = 0 at least one
point, then corresponding to this orbit, the travelling wave solution of (1.1) is a
breaking wave (see Fig. 7(3)).
5. Summary
From the above analysis, we can know the global dynamical behavior of
Hydrogen-Bonded Systems, this method can be also applied to another nonlin-
ear wave equation.
J. Shen et al. / Appl. Math. Comput. 171 (2005) 242–271 271
Acknowledgement
This work was supported by the National Natural Science Foundation of
China (Grant no. 10231020, Grant no. 10472091 and Grant no. 10332030)
and Natural Science Foundation of Shaanxi Province. J. Shen is supported
partially by the Doctorate Creation Foundation of NWPU(CX200423).
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