KG Bifurcation

7/27/2019 KG Bifurcation

1/13

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]


2/13

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.


3/13

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.


4/13

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


5/13


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)


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)


7/13


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


8/13

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


9/13


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 )


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


11/13


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.


12/13


13/13


18. Zhang, L.H.: Explicit traveling wave solutions of five kinds

of nonlinear evolution equations. J. Math. Anal. Appl.

379(1), 91124 (2011)19. Abdou, M.A.: New solitons and periodic wave solutions

for nonlinear physical models. Nonlinear Dyn. 52, 129136

(2008)20. Ma, W.X., Lee, J.-H.: A transformed rational function

method and exact solutions to the 3

+1 dimensional Jimbo

Miwa equation. Chaos Solitons Fractals 42, 13561363(2009)

21. Ma, W.X., Liu, Y.P.: Invariant subspaces and exact solu-

tions of a class of dispersive evolution equations. Commun.

Nonlinear Sci. Numer. Simul. 17, 37953801 (2012)22. Ma, W.X.: Complexiton solutions to the Kortewegde Vries

equation. Phys. Lett. A 30, 135144 (2002)23. Ma, W.X., Wu, H.Y., He, J.S.: Partial differential equations

possessing Frobenius integrable decompositions. Phys.

Lett. A 364, 2932 (2007)24. Ma, W.X.: A refined invariant subspace method and appli-

cations to evolution equations. Sci. China Math. 55, 1778

1796 (2012)

25. Ma, W.X., Fan, E.G.: Linear superposition principle apply-

ing to Hirota bilinear equations. Comput. Math. Appl. 61,

950959 (2011)

26. Ma, W.X.: Generalized bilinear differential equations. Stud.

Nonlinear Sci. 2, 140144 (2011)

27. Ma, W.X., Gu, X., Gao, L.: A note on exact solutions to

linear differential equations by the matrix exponential. Adv.

Appl. Math. Mech. 1(4), 573580 (2009)

28. Feng, Z.: On travelling wave solutions of the Burgers

Kortewegde Vries equation. Nonlinearity 20, 343356

(2007)

29. Liu, H., Fang, Y., Xu, C.: The bifurcation and exact trav-

elling wave solutions of (1 + 2)-dimensional nonlinearSchrodinger equation with dual-power law nonlinearity.

Nonlinear Dyn. 67, 465473 (2012)

30. Wang, Y., Bi, Q.: Different wave solutions associated with

singular lines on phase plane. Nonlinear Dyn. 69, 1705

1731 (2012)

Documents

KG Bifurcation