Bifurcations of travelling wave solutions in a model of the hydrogen-bonded systems

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).

mailto:[email protected]

mailto:[email protected]

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


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


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).


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


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).


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).


Thus, (2.2) becomes

d/ds

¼ yð1� byÞ; dyds

¼ �sin /

2

� �cos /

2

� �� cos u0

2

� �� 1� cos2 u0

2

� � : ð2:12Þ


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


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.


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).


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)


(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


(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. 5 (continued)


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).


(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


(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)


(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. 6 (continued)


Fig. 7. The variation of smoothness of periodic travelling wave as h varies. (1) Smooth wave, (2)

Periodic cusp wave, (3) Breaking wave.


(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.


(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


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


/ 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


/ 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


where

X3 ¼1

2h

14

� ffiffiffiffiffih0

pþ

ffiffiffiffiffih2

p �12; k23 ¼

ðffiffiffiffiffih0

p�

ffiffiffih

pÞð

ffiffiffih

p�

ffiffiffiffiffih2

pÞ

2ffiffiffih


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


/ ¼ �4 arctanðb4snðX4


pðx� ctÞ; k4ÞÞ: ð3:13Þ

where

a24 ¼ffiffiffiffiffiffiffiffi�h0

pþ

ffiffiffiffiffiffiffi�h

pffiffiffiffiffiffiffiffi�h2

p�


p ; b24 ¼ffiffiffiffiffiffiffiffi�h0

p�



pþ


p :

k4 ¼b4a4

; X4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð

ffiffiffiffiffiffiffiffi�h0

pþ


pÞð


pþ


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


pþ



p�


p !1

2

ðcnðX5


pðx� ctÞ; k5ÞÞ�1

0@

1A;

ð3:15Þ


where

X5 ¼1

2ð�hÞ

14


pþ


p� �12

; k25 ¼ðffiffiffiffiffiffiffi�h

p�


pÞð


p�


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


pðx� ctÞ; k6Þ

1� cnðX6


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�


p

1þffiffiffiffiffiffiffi�h

p !1

2

snðX7


pðx� ctÞ; k7Þ

0@

1A: ð3:17Þ

where

X7 ¼ffiffiffi2

p

4ð1þ


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


spond to the rotation orbits in the phase cylinder, the family of periodic

wave solutions have the parametric representation

/ ¼ �4 arctan1þ cnðX8


pðx� ctÞ; k8Þ

1� cnðX8


pðx� ctÞ; k8Þ

� �12

!: ð3:19Þ


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


p�



pþ


p !1

2

cnðX9


pðx� ctÞ; k9Þ

0@

1A: ð3:20Þ

where

X9 ¼ffiffiffi2

p

4


pþ


p� �12

; k29 ¼ðffiffiffiffiffiffiffiffi�h0

p�


pÞð


p�


pÞ

2ðffiffiffiffiffiffiffih0h

pþ


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).


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;



has one family of smooth periodic wave solution;


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,


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,


has two families of smooth periodic wave solutions;


smooth solitary wave solution with peak type and valley type,

respectively;


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,

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


5. For (u0,b) 2 (E). Then,



(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.


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,






8. For (u0,b) 2 (H), then,






9. For (u0,b) 2 (I), then,



(2) corresponding to h 2 (h1,h2) defined by (2.13), Eq. (1.1) has two families

of smooth periodic wave solutions;


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,


smooth solitary solution with valley type;


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;


(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,

smooth solitary solutions with peak type and valley type, respectively;



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,


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


(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,


respectively.

2. For (u0,b) 2 (B), then,




(2) corresponding to H(/, y) = h1 defined by (2.13), Eq. (1.1) has a smooth

kink and two anti-kink wave solutions;


(2.13), Eq. (1.1) has one family of periodic wave solutions, which corre-


3. For (u0,b) 2 (C), then,








4. For (u0,b) 2 (D), then,


smooth anti-kink wave solutions;


(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,





6. For (u0,b) 2 (F), then,


(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;




8. For (u0,b) 2 (H), then,


smooth anti-kink wave solutions;(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h3,1) defined by




9. For (u0,b) 2 (I), then,




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,


respectively.

10. For (u0,b) 2 (J), then,



(2) corresponding to h 2 (h2,h1) defined by (2.13), Eq. (1.1) has one familyof periodic wave solutions;




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


(3) corresponding to H(/, y) = h, h 2 (h1,1), h 2 (h1,h3) and h 2 (h3,1)



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;



(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,



(2) corresponding to H(/, y) = h, h 2 (h1,1) and h 2 (h1 = h3 = h5,1)



respectively.


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-



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


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


family of uncountably infinite many periodic travelling wave solutions with

varying smoothness, respectively. When h varies from h1 to h4, periodic travel-



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



gradually lose their smoothness, and evolve from smooth periodic travelling

wave to periodic cusp travelling wave.


6. For (u0,b) 2 (I), then,





wave to periodic cusp travelling wave.7. For (u0,b) 2 (J), then,






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


(ii) Suppose that b > 0, c2 < 1.

Theorem 4.2

1. For (u0,b) 2 (B), then,




gradually lose their smoothness, and evolve from smooth periodic travellingwave to periodic cusp travelling wave.

2. For (u0,b) 2 (C), then,






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.


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.


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).

References

[1] S.N. Chow, J.K. Hale, Method of Bifurcation Theory, Springer-Verlag, New York, 1981.

[2] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhauser,

Boston, 1997.

[3] J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations

of Vector Fields, Springer-Verlag, New York, 1983.

[4] J. Li, J. Shen, Travelling wave solutions in a model of the Helix Polypeptide chains, Chaos,

Solitons and Fractals 20 (2004) 827–841.

[5] J. Li, Z. Liu, Smooth and non-smooth traveling waves in a nonlinearly dispersive equation,

Appl. Math. Modell. 25 (2000) 41–56.

[6] J. Li, Z. Liu, Travelling wave solutions for a class of nonlinear dispersive equations, Chin.

Ann. Math. Ser. B 23 (3) (2002) 397–418.

[7] Y.A. Li, P.J. Olver, Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian

dynamical system I: Compactons and peakons, Discr. Contin. Dyn. Syst. 3 (1997) 419–432.

[8] Y.A. Li, P.J. Olver, Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian

dynamical system II: Complex analytic behaviour and convergence to non-analytic solutions,

Discr. Contin. Dyn. Syst. 4 (1998) 159–191.

[9] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.

[10] Y.P. Mei, J.R. Yan, X.H. Yan, J.Q. You, Proton soliton in anharmonic interaction

approximation, Phys. Lett. A 170 (1992) 289–292.

Bifurcations of travelling wave solutions in a model of the hydrogen-bonded systems

Documents

Transcript of Bifurcations of travelling wave solutions in a model of the hydrogen-bonded systems