Bifurcation method and traveling wave solution to Whitham-Broer-Kaup equation

Applied Mathematics and Computation 171 (2005) 677–702

www.elsevier.com/locate/amc

Bifurcation method and traveling wavesolution to Whitham–Broer–Kaup equation

Jianwei Shen a,b,*, Wei Xu a, Yanfei Jin a

a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an,

Shaanxi 710072, Chinab Department of Mathematics, Xuchang University, Xuchang, Henan 461000, China

Abstract

By using bifurcation method to Whitham–Broer–Kaup shallow water equations,

bifurcation parameter sets are shown. Numbers of solitary waves, kink waves and peri-

odic waves are given. Under various parameter conditions, all explicit formulas of sol-

itary wave solutions, kink wave solutions and periodic wave solutions are listed.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Solitary wave solution; Kink and anti-kink wave solution; Periodic wave solution;

WBK equations; Bifurcation theory; Dynamical systems

1. Introduction

Under Boussinesq approximation, Whitham, Broer and Kaup [8–10] ob-

tained WBK equations

ut þ uux þ vx þ buxx ¼ 0; ð1:1aÞ

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.01.078

* Corresponding author.

E-mail addresses: [email protected] (J. Shen), [email protected] (W. Xu).

mailto:[email protected]

mailto:[email protected]

678 J. Shen et al. / Appl. Math. Comput. 171 (2005) 677–702

vt þ ðuvÞx þ auxxx � bvxx ¼ 0; ð1:1bÞwhere u = u(x, t) is the field of horizontal velocity, v = v(x, t) is the height thatdeviate from equilibrium position of liquid, a, b are constants that represent

different diffusion power, Eq. (1.1) is a very good model to describe dispersive

wave. For the background materials of model equation, we refer to the paper

[8–10] and the references therein. If a = 0 and b 5 0, Eq. (1.1) is classic long-

wave equations that describe shallow water wave with diffusion [12]. If a = 1

and b = 0, Eq. (1.1) is modified Boussinesq equations [13]. Kaup [8] and

Ablowitz [13] studied inverse transformation solution for the special case of

Eq. (1.1), Kupershmidt discussed their symmetries and conservation laws. Byusing of Backlund transformation, Fan [12] found three pairs of solutions of

WBK equation. Xie et al. [14] made use of hyperbolic function method and

Wu elimination method obtained four pairs of solutions of WBK equation.

Chen et al. [15] present a more general transformation and applied it to

WBK equation and obtained some traveling wave solutions. Unfortunately,

these results in [8–15] are not complete since the authors did not consider

the bifurcation behavior of phase portraits for the corresponding traveling

wave equation.In this paper, by using bifurcation theory and methods of dynamical systems

(see [1–4]), we consider bifurcations of Eq. (1.1). Under fixed parameter condi-

tions, we obtain the all explicit formulas of traveling wave solution which

including solitary wave, kink and anti-kink wave, periodic wave solutions.

To find the traveling wave solution of (1.1), we first consider the traveling

wave solutions in the form

uðx; tÞ ¼ uðnÞ; vðx; tÞ ¼ vðnÞ; n ¼ x� ct; ð1:2Þ

where c denotes the wave speed. Therefore (1.1) reduce to be

�cun þ uun þ vn þ bunn ¼ 0; ð1:3aÞ

�cvn þ ðuvÞn þ aunnn � bvnn ¼ 0: ð1:3bÞ

Integrating (1.3) once with respect to n leads to

�cuþ 1

2u2 þ vþ bun ¼ g1; ð1:4aÞ

�cvþ uvþ aunn � bvn ¼ g2: ð1:4bÞwhere g1, g2 2 R are integral constant. Derived from (1.4), we can get

v ¼ cu� 1

2u2 � bun þ g1; ð1:5Þ

vn ¼ cun � uun � bunn: ð1:6Þ

J. Shen et al. / Appl. Math. Comput. 171 (2005) 677–702 679

Substituting (1.5) and (1.6) into (1.4b), we get

unn ¼1

2ðaþ b2Þðu3 � 3cu2 þ 2ðg1 � c2Þuþ 2ðg1cþ g2ÞÞ: ð1:7Þ

Let dudn ¼ y. Then we have the following traveling wave system which is an pla-

nar dynamical system:

dudn

¼ y;dydn

¼ 1

2ðaþ b2Þðu3 � 3cu2 þ 2ðg1 � c2Þuþ 2ðg1cþ g2ÞÞ: ð1:8Þ

Clearly, (1.8) is a Hamiltonian system with Hamiltonian function

Hðu; yÞ ¼ 1

2y2 � 1

2ðaþ b2Þ1

4u4 � cu3 þ ðg1 � c2Þu2 þ 2ðg1cþ g2Þu

� �:

ð1:9ÞFrom (1.8), we can know that we only consider the traveling wave solutions of

(1.8).

The Phase portraits defined by the vector fields of system (1.8) determine all

traveling wave solutions of (1.1). Suppose that u(x, t) = u(x � ct) = u(n) is a

continuous solution of (1.1) for n 2 (�1, 1) and limn!1u(n) = a1 and

limn!�1u(n) = b1. It is well known that (i) u(x, t) is called a solitary wave solu-tion if a1 = b1 and (ii) u(x, t) is called a kink or anti-kink solution if a1 5 b1.

Usually, a solitary wave solution of (1.1) corresponds to a homoclinic orbit

of (1.8). A kink (or anti-kink) wave solution of (1.1) corresponds to a hetero-

clinic orbit (or so-called connecting orbit) of (1.8). Similarly, a periodic orbit of

(1.8) corresponds to a periodically traveling wave solution of (1.1). Thus, to

investigate all bifurcations of solitary wave, kink waves and periodic waves

of (1.1), we should find all periodic annuli, homoclinic and heteroclinic orbits

of (1.8) depending on the parameter space of this system. The bifurcationtheory (see [1–7]) plays an important role in our study.

This paper is organized as follows. In Section 2, we give the bifurcation set

and phase portraits of (1.8). In Section 3, we show all explicit formulae of sol-

itary wave solutions and kink (or anti-kink) wave solutions under given param-

eter conditions. In Section 4, we show all explicit formulae of periodic wave

solutions under given parameter conditions. Our study results give rise to com-

plete description of all traveling waves of (1.1) and contain some results in

[8–15] as special examples.

2. Bifurcations of phase portraits of (1.8)

In this section, we consider bifurcation set and phase portraits of (1.8).

Obviously, on the (u, y)-phase plane, the abscissas of equilibrium points of


system (1.8) are the zeros of f(u) = u3 � 3cu2 + 2(g1 � c2)u + 2(g1c + g2). Notice

that f 0(u) = 3u2 � 6cu + 2(g1 � c2). Let (u0, 0) be an equilibrium of (1.8), at this

point, the determinant of the linearized system (1.8) has the form Jðu0; 0Þ ¼� 1

2ðaþb2Þ f0ðu0Þ. By using the bifurcation theory of planar dynamical system,

we know that if J(u0, 0) > 0 (or < 0), then the equilibrium (u0, 0) is a center

(or saddle point); if J(u0, 0) = 0 and the Poincare index of (u0, 0) is zero, then

the equilibrium (u0, 0) is a cusp. By using the above facts to do qualitative anal-

ysis, we obtain the following results.

Case I. g1 = c2.

Making the transformation 1ffiffiffiffiffiffiffiffiffiffiffiffi2ðaþb2Þ

p n ¼ s, y ! 1ffiffiffiffiffiffiffiffiffiffiffiffi2ðaþb2Þ

p y for a + b2 > 0, or

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2ðaþb2Þ

p n ¼ s, y ! 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2ðaþb2Þ

p y for a + b2 < 0, (1.8) becomes the following

two-dimensional system:

duds

¼ y;dyds

¼ �ðu3 � 3cu2 þ 2aÞ; ð2:1Þ

where a = g1c + g2, and when a + b2 > 0 (or < 0), then the sign in the right-hand side of second equation of (2.1) is ‘‘+’’ (or ‘‘�’’). System (2.1) has Ham-

iltonian function

Hðu; yÞ ¼ 1

2y2 � 1

4u4 � cu3 þ 2au

� �: ð2:2Þ

By using the above discussion to do qualitative analysis, we obtain the follow-

ing results. On the (a, c)-parametric plane, there are three bifurcation curves

(see Fig. 1):

L1 : a ¼ 0; L2 : a ¼ c3; L3 : a ¼ 2c3: ð2:3Þ

–4

–2

0

2

4

a

–4 –2 2 4c

Fig. 1. The partition of the (a, c)-parameter strip of (2.1).


From the above discussion we have the partition in (c, a)-parameter plane

by curves Li, i = 1, 2, 3, shown in Fig. 1 where

ðAÞ 0 < c < L1; ðBÞ L2 < c < L3; ðCÞ c > L3 > 0;

ðDÞ L2 < c < L1; ðEÞ L3 < c < L2; ðF Þ c < L3 < 0:

Corresponding to regions (A)–(F) of the bifurcation set of (a, c), phase por-

traits of (2.1) can be shown in Figs. 2 and 3, respectively.

Case II. g1 > c2.Making the transformation

ffiffiffiffiffiffiffiffiffig1�c2

aþb2

qn ¼ f, 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðg1�c2Þp u ¼ /,

ffiffiffiffiffiffiffiffiffiffiffiffiffi0:5jaþb2j

p

ðg1�c2Þ32

y ! y,

(1.8) becomes the following two-dimensional system:

d/df

¼ �y;dydf

¼ /3 � 3b/2 þ /þ d; ð2:4Þ

where

b ¼ cffiffiffi2

pðg1 � c2Þ

12

; d ¼ g1cþ g2

ð2g1 � 2c2Þ32

;

and when a + b2 > 0 (or < 0), the sign of the term y in the right-hand side of the

first equation of (2.4) is ‘‘+’’ (or ‘‘�’’). This system (2.4) is a Hamiltonian sys-tem with Hamiltonian function

Hð/; yÞ ¼ � 1

2y2 � 1

4/4 � b/3 þ 1

2/2 þ d/

� �: ð2:5Þ

Denote that f1(/) = /3 � 3b/2 + / + d, f 0(/) = 3/2 � 6b/ + 1. Thus f 0(/) has

two zeros at /1;2 ¼ 3b�ffiffiffiffiffiffiffiffiffi9b2�3

p3

. Obviously, /1,2 is real if and only if jbj P 1ffiffi3

p .

Hence, in this situation there exist three bifurcation curves (see Fig. 4):

L4 : d ¼ �2

27ð9b2 � 3Þ

32 þ 2b3 � b; L5 : d ¼ 2b3 � b;

L6 : d ¼ 2

27ð9b2 � 3Þ

32 þ 2b3 � b:

ð2:6Þ

From the above discussion we have the partition in (b, d)-parameter plane by

curves Li, i = 4, 5, 6, shown in Fig. 4 where

ðI1Þ d < L4; ðII1Þ L4 < d < L5; ðIII1Þ L5 < d < L6; ðIV 1Þ d > L6:

Corresponding to regions (I1)–(IV1) of the bifurcation set of (b, d), the bifurca-

tions of phase portraits of (2.4) can be shown in Figs. 5 and 6, respectively.

Case III. g1 < c2.

Making the transformationffiffiffiffiffiffiffiffiffic2�g1jaþb2j

qn ¼ 1, 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðc2�g1Þp u ¼ u,

ffiffiffiffiffiffiffiffiffiffiffiffiffi0:5jaþb2j

p

ðc2�g1Þ32

y ! y,

(1.8) becomes the following two-dimensional system:

—4

—2

0

2

4

y

—2 —1 1 2 3 4x

—4

—2

0

2

4

y

–2 —1 1 2 3 4x

–4

—2

0

2

4

y

—2 —1 1 2 3 4

x

—4

—2

0

2

4

y

—2 —1 1 2 3 4x

—4

—2

0

2

4

y

—2 —1 1 2 3 4x

—4

–2

0

2

4

y

–2.5 —2 —1 .5 —1 —0 .5 0 .5 1x

(1) (2)

(3) (4)

(5) (6)

Fig. 2. Phase portraits of (2.1) for g1 = c2 and a + b2 > 0. (1) (a, c) 2 L1, (2) (a, c) 2 (A), (3) (a, c) 2L2, (4) (a, c) 2 (B), (5) (a, c) 2 L3, (6) (a, c) 2 (C), (7) (a, c) 2 L1, (8) (a, c) 2 (D), (9) (a, c) 2 L2, (10)

(a, c) 2 E, (11) (a, c) 2 L3, (12) (a, c) 2 (F).


—4

—2

0

2

4

y

—2 —1 1 2 3 4x

—4

—2

0

2

4

y

–2 —1 1 2 3 4

x

—4

—2

0

2

4

y

—4 —3 —2 —1 1 2x

—4

—2

0

2

4

y

—4 —3 —2 —1 1 2x

—4

—2

0

2

4

y

—4 —3 —2 —1 1 2x

—4

—2

0

2

4

y

—1 1 2x

(7) (8)

(9) (10)

(11) (12)

Fig. 2 (continued)


–4

–2

2

4

y

–1 1 2 3 4

x

–3

–2

–1

0

1

2

3

y

–1 1 2 3

x

–3

–2

–1

0

1

2

3

y

–1 1 2 3

x

–4

–2

0

2

4

y

–2 –1 1 2 3

x

–2

–1

0

1

2

y

–1.5 –1 –0.5

x

–3

–2

–1

0

1

2

3

y

–1 1 2 3

x

(1) (2)

(3) (4)

(5) (6)

Fig. 3. Phase portraits of (2.1) for g1 = c2 and a + b2 < 0. (1) (a, c) 2 L1, (2) (a, c) 2 (A), (3) (a, c) 2L2, (4) (a, c) 2 (B), (5) (a, c) 2 L3, (6) (a, c) 2 (C), (7) (a, c) 2 L1, (8) (a, c) 2 (D), (9) (a, c) 2 L2, (10)

(a, c) 2 E, (11) (a, c) 2 L3, (12) (a, c) 2 (F).


–4

–2

2

4

y

–4 –3 –2 –1 1

x

–3

–2

–1

1

2

3

y

–3 –2 –1 1

x

–2

–1

1

2

y

–3 –2 –1 1

x

–3

–2

–1

1

2

3

y

–3 –2 –1 1

x

–4

–2

0

2

4

y

–3 –2 –1 1 2

x

–2

–1

0

1

2

y

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x

(7) (8)

(9) (10)

(11) (12)

Fig. 3 (continued)


–4

–2

0

2

4

d

±4 ±2 2 4

b

Fig. 4. The partition of the (b, d)-parameter strip of (2.4) for g1 > c2.


dud1

¼ �y;dyd1

¼ u3 � 3eu2 � uþ g; ð2:7Þ

where

e ¼ cffiffiffi2

pðc2 � g1Þ

12

; g ¼ g1cþ g2ffiffiffi2

pðc2 � g1Þ

32

;

and when a + b2 > 0 (or < 0), the sign of the term y in the right-hand side of the

first equation of (2.7) is ‘‘+’’ (or ‘‘�’’). This system (2.7) is a Hamiltonian sys-

tem with Hamiltonian function

Hðu; yÞ ¼ � 1

2y2 � 1

4u4 � eu3 � 1

2u2 þ gu

� �: ð2:8Þ

On the (e, g)-parameter plane there exist three bifurcation curves (see Fig. 7):

L7 : g ¼ � 2

27ð9e2 þ 3Þ

32 þ eþ 2e3; L8 : g ¼ eþ 2e3;

L9 : g ¼ 2

27ð9e2 þ 3Þ

32 þ eþ 2e3:

ð2:9Þ

From the above discussion we have the partition in (e, g)-parameter plane

by curves Li, i = 7, 8, 9, shown in Fig. 7 where

ðI2Þ g < L7; ðII2Þ L7 < g < L8; ðIII2Þ L8 < g < L9; ðIV 2Þ g > L9:

Corresponding to regions (I2)–(IV2) of the bifurcation set of (e, g), phase por-

traits of (2.4) can be shown in Figs. 8 and 9.

From the above discussion, we have

Theorem 1. Corresponding to (1.1), all of the solitary wave, kink (or anti-kink)

wave and periodic wave solutions are smooth.

–4

–2

0

2

4

y

–1 1 2 3 4

x

–4

–2

0

2

4

y

–2 –1 1 2 3

x

–4

–2

0

2

4

y

–2 –1 1 2 3 4

x

–4

–2

0

2

4

y

–2 –1 1 2 3 4

x

–4

–2

0

2

4

y

–2 –1 1 2 3 4

x

–4

–2

0

2

4

y

–2 –1 1 2 3

x

(1) (2)

(3) (4)

(5) (6)

Fig. 5. Phase portraits of (2.4) for g1 > c2 and a + b2 > 0. (1) (b, d) 2 I1, (2) (b, d) 2 (L4), (3) (b, d) 2(II1), (4) (b, d) 2 (L5), (5) (b, d) 2 (III1), (6) (b, d) 2 L6, (7) (b, d) 2 (IV1).


–4

–2

0

2

4

y

–2 –1.5 –1 –0.5 0.5

x

(7)

Fig. 5 (continued)


3. Solitary wave and kink wave solutions determined by (1.8)

In this section we will give all explicit solitary wave solutions and kink (or

anti-kink) wave solutions of (1.8) under given parameter conditions. By using

the traveling wave system (1.8) and the Hamiltonian function (1.9) to do the

calculations and table of integral [16], we have the following results:

Case I. g1 = c2.

Denote that u1 < u2 < u3 are the roots of u3 � 3cu2 + 2a = 0,

a1i = 6ui(ui � 2c), b1i = 4(ui � c), q1i ¼ 8ðu2i � 2uic� 2c2Þ, i = 1, 2, 3, n = x � ct.

Suppose that a + b2 > 0.

(1) Corresponding to Fig. 2(2) and (10), (1.1) has one smooth solitary wave

solutionwithpeak form,whichhas the followingparametric representation:

u ¼ u1 �2a11

b11 þffiffiffiffiffiffiq11

psinhð

ffiffiffiffiffiffiffiffiffiffiffiffi0:5a11

psÞ;

s ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðaþ b2Þ

q n; v ¼ cu� 1

2u2 � bun þ g1:

ð3:1Þ

(2) Corresponding to Fig. 2(3) and (9), (1.1) has one kink wave and one anti-

kink wave solutions, which has the following parametric representation:

u ¼ c�ffiffiffi3

pc tanhð

ffiffiffiffiffiffiffi1:5

pcsÞ;


q n; v ¼ cu� 1

2u2 � bun þ g1:

ð3:2Þ

–2

–1

0

1

2

y

0.5 1 1.5 2 2.5 3

x

–1.5

–1

–0.5

0

0.5

1

1.5

y

–0.5 0.5 1 1.5 2 2.5

x

–3

–2

–1

1

2

3

y

–1 1 2 3 4

x

–3

–2

–1

0

1

2

3

y

–1 1 2 3 4

x

–4

–2

2

4

y

–2 –1 1 2 3 4

x

–1.5

–1

–0.5

0

0.5

1

1.5

y

–1 –0.5 0.5 1 1.5 2

x

(1) (2)

(3) (4)

(5) (6)

Fig. 6. Phase portraits of (2.4) for g1 > c2 and a + b2 < 0. (1) (b, d) 2 I1, (2) (b, d) 2 (L4), (3) (b, d) 2(II1), (4) (b, d) 2 (L5), (5) (b, d) 2 (III1), (6) (b, d) 2 L6, (7) (b, d) 2 (IV1).


–2

–1

0

1

2

y

–3 –2.5 –2 –1.5 –1 –0.5

x

(7)

Fig. 6 (continued)

–4

–2

0

2

4

g

±4 ±2 2 4

e

Fig. 7. The partition of the (e, g)-parameter strip of (2.7) for g1 < c2.



solution with valley form, which has the following parametric

representation:

u ¼ u3 �2a13


psinhð


psÞ;


q n; v ¼ cu� 1

2u2 � bun þ g1:

ð3:3Þ

Fig. 8. Phase portraits of (2.7) for g1 < c2 and a + b2 > 0. (1) (e, g) 2 I2, (2) (e, g) 2 (L7), (3) (e,

g) 2 (II2), (4) (e, g) 2 (L8), (5) (e, g) 2 (III2), (6) (e, g) 2 L9, (7) (e, g) 2 (IV2).


Fig. 8 (continued)


Suppose that a + b2 < 0.

(4) Corresponding to Fig. 3(1) and (7), (1.1) has one smooth solitary wavesolution with peak form, which has the following parametric

representation:

u ¼ 4c1þ 2c2s2

;

s ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2ðaþ b2Þ

q n; v ¼ cu� 1

2u2 � baun þ g1:

ð3:4Þ

(5) Corresponding to Fig. 3(2), (4), (8) and (10), (1.1) has a pair of smooth

solitary wave solutions, which have the following parametricrepresentation:

u ¼ u2 þ2a12

�b12 þffiffiffiffiffiffiq12

psinhð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�0:5a12

psÞ;


q n; v ¼ cu� 1

2u2 � bun þ g1:

ð3:5Þ

(6) Corresponding to Fig. 3(3) and (9), (1.1) has a pair of smooth solitary

wave solutions, which have the following parametric representation:

u ¼ cþffiffiffi6

pc sechð

ffiffiffi3

pcsÞ;


q n; v ¼ cu� 1

2u2 � bun þ g1:

ð3:6Þ

Fig. 9. Phase portraits of (2.7) for g1 < c2 and a + b2 < 0. (1) (e, g) 2 I2, (2) (e, g) 2 (L7), (3) (e, g) 2(II2), (4) (e, g) 2 (L8), (5) (e, g) 2 (III2), (6) (e, g) 2 L9, (7) (e, g) 2 (IV2).


Fig. 9 (continued)



solution with valley form, which has the following parametric

representation:

u ¼ 2c� 4c1þ 2c2s2

;


q n; v ¼ cu� 1

2u2 � bun þ g1:

ð3:7Þ

Case II. g1�c2 > 0.

Denote that /1 < /2 < /3 are the roots of /3 � 3b/2 + / + d = 0, n = x � ct,

a2i ¼ 6/2i � 12/ibþ 2, b2i = 4(/i � b), q2i ¼ 8ð/2

i � 2/ibþ 1� 2b2Þ, i = 1, 2, 3.


(1) Corresponding to Fig. 5(3), (1.1) has one smooth solitary wave solution

with peak form, which has the following parametric representation:

/ ¼ /1 �2a21


psinhð


pfÞ:

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1 � c2

jaþ b2j

sn; / ¼ ð2g1 � 2c2Þ�

12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:8Þ(2) Corresponding to Fig. 5(4), (1.1) has one kink and one anti-kink wave

solutions, which has the following parametric representation:


/ ¼ b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3b2 � 1

ptanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:5ð3b2 � 1Þ

qf

� �


jaþ b2j

sn; / ¼ ð2g1 � 2c2Þ�

12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:9Þ(3) Corresponding to Fig. 5(5), (1.1) has one smooth solitary wave solution

with valley form, which has the following parametric representation:

/ ¼ /3 �2a23


psinhð


pfÞ:


jaþ b2j

sn; / ¼ ð2g1 � 2c2Þ�

12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:10ÞSuppose that a + b2 < 0.



/ ¼ b� 1

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9b2 � 3

pþ 4


p

6þ ð3b2 � 1Þf2;


jaþ b2j

sn; / ¼ ð2g1 � 2c2Þ�

12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:11Þ(5) Corresponding to Fig. 6(3) and (5), (1.1) has a pair of smooth solitary

wave solutions with peak form and valley form, which have the following

parametric representation:

/ ¼ /2 þ2a22


psinhð


pfÞ;


jaþ b2j

sn; / ¼ ð2g1 � 2c2Þ�

12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:12Þ(6) Corresponding to Fig. 6(4), (1.1) has a pair of smooth solitary wave solu-

tions with peak form and valley form, which have the following paramet-

ric representation:


/ ¼ b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6b2 � 2

psech


pf

� �;


jaþ b2j

sn; / ¼ ð2g1 � 2c2Þ�

12u; v ¼ cu� 1

2u2 � bun þ g1:



/ ¼ 1

3


pþ b� 4


p

3þ 2ð3b2 � 1Þf2;


jaþ b2j

sn; / ¼ ð2g1 � 2c2Þ�

12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:14ÞCase III. g1 � c2 < 0.

Denote that u1 < u2 < u3 are the roots of u3 � 3eu2 � u + g = 0, n = x � ct,

a3i ¼ 2ð3u2i � 6uie� 1Þ, b3i = 4(ui � e), q3i ¼ 8ðu2

i � 2uie� 2e2 � 1Þ, i = 1,

2, 3.




u ¼ u1 �2a31


psinhð


p1Þ;

1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � g1jaþ b2j

sn; u ¼ ð2c2 � 2g1Þ

�12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:15Þ(2) Corresponding to Fig. 8(4), (1.1) has one kink and one anti-kink wave

solution, which has the following parametric representation:

u ¼ e�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3e2 þ 1

ptanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6e2 þ 2

p

21

� �;


sn; u ¼ ð2c2 � 2g1Þ

�12u; v ¼ cu� 1

2u2 � bun þ g1:




u ¼ u3 �2a33


psinhð


p1Þ;


sn; u ¼ ð2c2 � 2g1Þ

�12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:17Þ


(4) Corresponding to Fig. 9(2), (1.1) has one smooth solitary wave solutionwith peak form, which has the following parametric representation:

u ¼ e� 1

3


pþ 4


p

3þ 2ð3e2 þ 1Þ12 ;


sn; u ¼ ð2c2 � 2g1Þ

�12u; v ¼ cu� 1

2u2 � bun þ g1:


tion with peak form and valley form, which have the following paramet-

ric representation:

u ¼ u2 þ2a32


psinhð


p1Þ;


sn; u ¼ ð2c2 � 2g1Þ

�12u; v ¼ cu� 1

2u2 � bun þ g1:


tion with peak form and valley form, which have the following paramet-

ric representation:

u ¼ eþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6e2 þ 2

psech


p1

� �;


sn; u ¼ ð2c2 � 2g1Þ

�12u; v ¼ cu� 1

2u2 � bun þ g1:




u ¼ eþ 1

3


p� 4


p

3þ 2ð3e2 þ 1Þ12 ;


sn; u ¼ ð2c2 � 2g1Þ

�12u; v ¼ cu� 1

2u2 � bun þ g1:

ð3:21Þ

Theorem 2. There are 21 parametric conditions of (1.1) such that (1.1) has 21

formulae of smooth solitary wave solutions, kink and anti-kink wave solutions

given by (3.1)–(3.21).

4. Periodic solutions determined by (1.8)

In this section, all the periodic wave solutions of (1.1) will be given. Denote

that sn(u, k), cn(u, k), tn(u, k) are the Jacobian elliptic functions with themodulus k, K(k) is the complete elliptic integral of the first kind and E(k) is

the complete elliptic integral of the second kind [17].

Case I. g1 = c2.Suppose that a + b2 > 0.

Denote that u11 < u12 < u13 < u14 are the roots of 2h + 0.5u4 � 2cu3 +

4au = 0, h is Hamiltonian value, n = x � ct.

Corresponding to Fig. 2(2)–(4) and (8)–(10), Eq. (1.1) has one family of

smooth periodic wave solutions, which have the following parametric

representation:

u ¼ u14 � u13w1 sn2ðX1s; k1Þ

1� w1 sn2ðX1s; k1Þ; s ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2jaþ b2jq n;

v ¼ cu� 1

2u2 � bun þ g1;

ð4:1Þ

where

g11 ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðu14� u12Þðu13 � u11Þp ; k21 ¼

ðu13� u12Þðu14 � u11Þðu14� u12Þðu13 � u11Þ

; w1 ¼u14 � u11u13 � u11

;

X1 ¼ffiffiffi2

p

2g11:

ð4:2Þ



Denote that u21 < u22 < u23 < u24 are the roots of 2h � 0.5u4 + 2cu3 � 4au =

0, h is Hamiltonian value, n = x � ct.

(1) Corresponding to Fig. 3(6) and (12), Eq. (1.1) has one family of smoothperiodic wave solutions.

(2) Corresponding to Fig. 3(1), (5), (7) and (11), Eq. (1.1) has two families of

smooth periodic wave solutions.

(3) Corresponding to Fig. 3(2)–(5) and (8)–(10), Eq. (1.1) has three families

of smooth periodic wave solutions.

These periodic wave solutions have the following parametric representation:

u ¼ u24 � w2u21 sn2ðX2s; k2Þ1þ w2 sn2ðX2s; k2Þ

; s ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2jaþ b2j

q n;

v ¼ cu� 1

2u2 � bun þ g1;

ð4:3Þ

where

g2 ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðu24 � u22Þðu23 � u21Þp ; k22 ¼

ðu24 � u23Þðu22 � u21Þðu24 � u22Þðu23 � u21Þ

;

w2 ¼u24 � u23u23 � u21

; X2 ¼ffiffiffi2

p

2g2:

Case II. g1 > c2.Suppose that a + b2 > 0.

Denote that /11 < /12 < /13 < /14 are the roots of 2hþ 12/4 � 2b/3 þ /2 þ

2d/ ¼ 0, h is Hamiltonian value, n = x � ct.Corresponding to Fig. 5(3)–(5), Eq. (1.1) has one family of smooth periodic


/ ¼ /14 � /13w3 sn2ðX3f; k3Þ

1� w3 sn2ðX3f; k3Þ; f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1 � c2

aþ b2

sn;

/ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðg1 � c2Þ

p u; v ¼ cu� 1

2u2 � bun þ g1;

ð4:4Þ

where

g3 ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð/14 � /12Þð/13 � /11Þp ; k23 ¼

ð/13 � /12Þð/14 � /11Þð/14 � /12Þð/13 � /11Þ

;

w3 ¼/14 � /11

/13 � /11

; X3 ¼ffiffiffi2

p

2g3:



Denote that /21 < /22 < /23 < /24 are the roots of �2h� 12/4 þ 2b/3 � /2�

2d/ ¼ 0, h is Hamiltonian value, n = x � ct.


(2) Corresponding to Fig. 6(2) and (6), Eq. (1.1) has two families of smooth

periodic wave solutions.

(3) Corresponding to Fig. 6(3)–(5), Eq. (1.1) has three families of smooth



/ ¼ /24 � w4/21 sn2ðX2f; k4Þ

1þ w4 sn2ðX2f; k4Þ; f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1 � c2

jaþ b2j

sn;

/ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g1 � 2c2

p u; v ¼ cu� 1

2u2 � bun þ g1;

ð4:5Þ

where


ð/24 � /22Þð/23 � /21Þp ; k24 ¼

ð/24 � /23Þð/22 � /21Þð/24 � /22Þð/23 � /21Þ

;

w4 ¼/24 � /23

/23 � /21

; X4 ¼ffiffiffi2

p

2g4:

Case III. g1 < c2.Suppose that a + b2 > 0.

Denote that u11 < u12 < u13 < u14 are the roots of 2hþ 12u4 � 2eu3 � u2þ

2gu ¼ 0, h is Hamiltonian value, n = x � ct.

Corresponding to Fig. 8(3)–(5), Eq. (1.1) has one family of smooth periodic


u ¼ u14 � u13w5 sn2ðX51; k5Þ

1� w5 sn2ðX51; k5Þ; 1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � g1aþ b2

sn;

u ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðc2 � g1Þ

p u; v ¼ cu� 1

2u2 � bun þ g1;

ð4:6Þ

where


ðu14 � u12Þðu13 � u11Þp ; k25 ¼


;

w5 ¼u14 � u11

u13 � u11

; X5 ¼ffiffiffi2

p

2g5:



Denote that u21 < u22 < u23 < u24 are the roots of �2h� 12u4 þ 2e/3 þ u2�

2gu ¼ 0, h is Hamiltonian value, n = x � ct.


(2) Corresponding to Fig. 9(2) and (6), Eq. (1.1) has two families of smooth


(3) Corresponding to Fig. 9(3)–(5), Eq. (1.1) has three families of smooth



u ¼ u24 � w6u21 sn2ðX61; k6Þ

1þ w6 sn2ðX61; k6Þ; 1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � g1jaþ b2j

sn;

u ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðg1 � c2Þ

p u; v ¼ cu� 1

2u2 � bun þ g1;

ð4:7Þ

where


ðu24 � u22Þðu23 � u21Þp ; k26 ¼


;

w6 ¼u24 � u23

u23 � u21

; X6 ¼ffiffiffi2

p

2g6:

From the above discussion, we have

Theorem 3. There are six different kind of periodic wave solutions given by

(4.1)–(4.6) for (1.1). When the modulus ki ! 0 or ki ! 1, the periodic wave

solutions approach to solitary wave solutions or kink (anti-kink) wave solutions.

5. Summary and conclusion

In summary, we have derived all solitary, kink (or anti-kink) and periodic

wave solutions. Obviously, this method is very valid to find explicit solutions

for the integrable system and also applied to other nonlinear evolution equa-tions in mathematical physics.

Acknowledgment

This work was supported by the National Natural Science Foundation of

China (Grant no. 10472091 and 10332030) and Natural Science Foundation


of Shaanxi Province. J. Shen is supported partially by the Doctorate Creation

Foundation and Graduate Starting Seed Found of NWPU (Grant no. CX

200423).

References

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

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

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

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

[4] D. Luo et al., Bifurcation Theory and Methods Dynamical Systems, World Scientific

Publishing, London, 1997.

[5] J.B. Li, J.W. Shen, Chaos, Solitons and Fractals 20 (2004) 827.

[6] J.W. Shen et al., Appl. Math. Comput. 161 (2) (2005) 365.

[7] J.B. Li, Z.R. Liu, Appl. Math. Modell. 25 (2000) 41.

[8] D.J. Kaup, Prog. Theor. Phys. 54 (2) (1975) 396.

[9] G.B Whitham, Pro. R. Soc. A 299 (1967) 6.

[10] L.J. Broer, Appl. Sci. Res. 31 (5) (1975) 377.

[11] B.A. Kupershmidt, Commun. Math. Phys. 99 (1) (1985) 51.

[12] E.G. Fan, H.Q. Zhang, Appl. Math. Mech. 19 (8) (1998) 667 (in Chinese).

[13] M.J Ablowitz, Soliton, Nolinear Evolution Equations and Inverse Scatting, Cambridge

University Press, New York, 1991.

[14] F.D. Xie et al., Phys. Lett. A 285 (2001) 76.

[15] Y. Chen et al., Chaos, Solitons and Fractals 22 (2004) 675.

[16] B.O. Peirce, Table of Integral, Shanghai Academic, 1953 (in Chinese).

[17] P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineer and Scientists,

Springer-Verlag, New York, 1971.

Bifurcation method and traveling wave solution to Whitham-Broer-Kaup equation

Documents

Transcript of Bifurcation method and traveling wave solution to Whitham-Broer-Kaup equation