DECOMPOSITION METHOD FOR CAMASSA-HOLM EQUATION

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Jules Sadefo Kamdem, Zhijun Qiao

To cite this version:

Jules Sadefo Kamdem, Zhijun Qiao. DECOMPOSITION METHOD FOR CAMASSA-HOLM EQUA-

TION. Chaos, Solitons and Fractals, Elsevier, 2005, xxxx, �10.1016/j.chaos.2005.09.071�. �hal-

00004557v4�

UNCORRECTED

PROOF

1

2 Decomposition method for the Camassa–Holm equation

3 J. Sadefo Kamdem

^a

, Zhijun Qiao

^b,*

4

^aLaboratoire de Mathe´matiques, CNRS UMR 6056, BP 1039 moulinde la housse, 51687 Reims, France

5

^bDepartment of Mathematics, The University of Texas-Pan American, 1201 W University Drive,

6

Edinburg, TX 78541, United States

7

Accepted 30 September 2005

8

9

Communicated by Prof. Ji-Huan He

10 Abstract

11 The Adomian decomposition method is applied to the Camassa–Holm equation. Approximate solutions are 12 obtained for three smooth initial values. These solutions are weak solutions with some peaks. We plot those approx- 13 imate solutions and find that they are very similar to the peaked soliton solutions. Also, one single and two anti-peakon 14 approximate solutions are presented. Compared with the existing method, our procedure just works with the polyno- 15 mial and algebraic computations for the CH equation.

16

2005 Elsevier Ltd. All rights reserved.

17

18 1. Introduction

19 The generalized shallow water equation—the Camassa–Holm (CH) equation, which was derived physically as a 20 shallow water wave equation by Camassa and Holm in [10], takes the form

21

m

t

þ m

x

u þ 2mu

_x

¼ 0; m ¼ u 1

4 u

xx

ð1:1Þ

23 23

24 where

u

=

u(x,t) represents the horizontal component of the fluid velocity, and

m ¼ u

¹₄

u

xx

is the momentum variable.

25 The subscripts

x,t

of

u

denote the partial derivatives of the function

u

w.r.t.

x,t, for example,u_t

=

ou/ot,u_xxt

=

o³u/

26

o²xot, similar notations will be used frequently later in this paper. This equation was first included in the work of Fuchs-

27 steiner and Fokas [15] on their theory of hereditary symmetries of soliton equations. As it was shown by Camassa and 28 Holm, Eq. (1.1) describes the unidirectional propagation of two dimensional waves in shallow water over a flat bottom.

29 The solitary waves of Eq. (1.1) regain their shape and speed after interacting nonlinearly with other solitary waves. The 30 most feature of this equation is peaked soliton (called peakon) solution, which is a weak solution with non-smooth 31 property at some points.

32 The CH equation possesses the bi-Hamiltonian structure, Lax pair and multi-dimensional peakon solutions, and 33 retains higher order terms of derivatives in a small amplitude expansion of incompressible EulerÕs equations for unidi- 34 rectional motion of waves at the free surface under the influence of gravity. In 1995, Calogero [9] extended the class of

0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2005.09.071

*Corresponding author. Tel.: +33 3268 75 212; fax: +33 3269 18 397.

E-mail addresses:[email protected](J.S. Kamdem),[email protected](Z. Qiao).

Chaos, Solitons and Fractals xxx (2005) xxx–xxx

www.elsevier.com/locate/chaos

UNCORRECTED

PROOF

35 mechanical system of this type. Later, Ragnisco and Bruschi [23] and Suris [24], showed that the CH equation yields the 36 dynamics of the peakons in terms of an

N-dimensional completely integrable Hamiltonian system. Such kind of dynam-

37 ical system has Lax pair and an

N·N r-matrix structure

[23].

38 Recently, the algebro-geometric solution of the CH equation and the CH hierarchy arose much more attraction.

39 This kind of solution for most classical integrable PDEs can be obtained by using the inverse spectral transform theory, 40 see Dubrovin [14], Ablowitz and Segur [4], Novikov et al. [19], Newell [18]. This is done usually by adopting the spectral 41 technique associated with the corresponding PDE. Alber and Fedorov [8] studied the stationary and the time-dependent 42 quasi-periodic solution for the CH equation and Dym type equation through using the method of trace formula [7] and 43 Abel mapping and functional analysis on the Riemann surfaces. Constantin and McKean [11] presented the solution of 44 the CH equation on the circle. Later, Alber, Camassa, Fedorov, Holm and Marsden [6] considered the trace formula 45 under the nonstandard Abel-Jacobi equations and by introducing new parameters presented the so-called weak finite- 46 gap piecewise-smooth solutions of the integrable CH equation and Dym type equations. Very recently, Gesztesy and 47 Holden [16], and Qiao [20] discussed the algebro-geometric solutions for the CH hierarchy using polynomial recursion 48 formalism and the trace formula, and constrained method, respectively. Thereafter, Qiao [21] studied an extension ver- 49 sion of the CH equation—the DP equation [13], and presented exact solutions by using the constrained method [22].

50 The present paper provides a different approach to the solutions of the CH equation. The Adomian decomposition 51 method is implemented to solve the Camassa–Holm equation with smooth initial conditions. Numeric algorithm and 52 graphs are analyzed and plotted, respectively. We also compare our solutions with other existing procedures, and find 53 that our approximate solutions are similar to peaked solitons of the CH equation.

54 2. Adomian decomposition method for Camassa–Holm equation 55 The Camassa–Holm equation (1.1) for real

u(x,t)

u

_t

1 4 u

_xxt

þ 3

2 ðu

²

Þ

_x

1

8 ðu

²_x

Þ

_x

1

4 ðuu

xx

Þ

_x

¼ 0 ð2:2Þ

57 57

58 is written as 59

L

t

u 1 4 u

xx

¼ L

x

3

2 ðu

²

Þ

_x

þ 1 8 u

²_x

þ 1

4 uu

xx

ð2:3Þ 61

61

62 where L

t

¼

_ot^o

and L

x

¼

_ox^o

. Then L

¹_x

ðÞ ¼

Rx

0

ðÞ dx and L

¹_t

ðÞ ¼

Rt

0

ðÞ dt. After operating the two sides of Eq. (2.3) with 63 L

¹_t

, we have

64

uðx; tÞ ¼ uðx; 0Þ 1

4 u

xx

ðx; 0Þ þ 1

4 u

xx

þ L

¹_t

L

x

3

2 ðu

²

Þ

_x

þ 1 8 u

²_x

þ 1

4 uu

xx

¼ uðx; 0Þ 1

4 u

xx

ðx; 0Þ þ 1

4 u

xx

þ L

¹_t

ðhðuÞÞ ¼ uðx; 0Þ 1

4 u

xx

ðx; 0Þ þ 1 4 u

xx

þ

Z t 0

hðuðx; sÞÞ ds ð2:4Þ 66

66

67 where

h(u) denote the differential operator

hðuÞ

:¼

L

x

3

2 ðu

²

Þ

_x

þ 1 8 u

²_x

þ 1

4 uu

xx

. ð2:5Þ

69 69

70 The Adomian decomposition method consists of calculating the solution of Eq. (2.4) in a series form u ¼

X¹

n¼0

u

n

ð2:6Þ

72 72

73 and the nonlinear term becomes hðuÞ ¼

X¹

n¼0

A

n

ð2:7Þ

75 75

76 where

An

are polynomials of

u0

,u

1

,. . . ,

un

called AdomianÕs polynomials and are given by A

₀

ðu

0

Þ ¼ hðu

0

Þ n ¼ 0;

A

n

ðu

0;

u

1;. . .;

u

n

Þ ¼

P

b1þþbn¼n

h

^ðb1Þ

ðu

0

Þ

^u

ðb1b2Þ 1 ðb1b2Þ!

^u

ðbn1bnÞ n1 ðb_n1bnÞ!

u^bnⁿ

b_n!

if n 6¼ 0.

8<

:

ð2:8Þ

78

78

UNCORRECTED

PROOF

79 where

h

is a real function. (See for instance [5,1,2] for more details about the preceded procedure.)

80 By the use of the relationships shown in the paper of Abbaoui and Cherruault [1], the

An

are determined as follows:

81

A

0

¼ hðu

0

Þ A

1

¼ h

^ð1Þ

ðu

0

Þu

1

A

2

¼ h

^ð1Þ

ðu

0

Þu

2

þ

¹₂

h

^ð2Þ

ðu

0

Þu

²₁

A

3

¼ h

^ð1Þ

ðu

0

Þu

3

þ h

^ð2Þ

ðu

0

Þu

1

u

2

þ

¹₆

h

^ð3Þ

ðu

0

Þu

²₁

A

4

¼ h

^ð1Þ

ðu

0

Þu

4

þ h

^ð2Þ

ðu

0

Þ u

1

u

3

þ

¹₂

u

²₂

þ

¹₂

h

^ð3Þ

ðu

0

Þu

²₁

u

2

þ

₂₄¹

h

^ð4Þ

ðu

0

Þu

⁴₁ ...

8>

>>

>>

>>

>>

><

>>

>>

>>

>>

>>

:

ð2:9Þ

83 83

84 which recursively generates the formula of

u_n

: u

0

¼ uðx;0Þ

¹₄

u

xx

ðx; 0Þ n ¼ 0 u

_nþ1

¼

¹₄

u

nxx

þ

Rt

0

A

n

ds if n 6¼ 0

(

ð2:10Þ 86

86

87 Following Adomian decomposition methods, we consider the following functional equation:

88

u w ¼ NLðuÞ þ LðuÞ ð2:11Þ

90 90

91 where

u

is to be determined approximately in some appropriate functional space

S,w

is a given element of

S,NL

and

L

92 are a nonlinear operator and a linear operator from a subset

X

of the functional space

S

onto itself, respectively. Here, 93 we seek a solution of Eq. (2.11) in the form u ¼

P1

n¼0

u

n

. To do so, we approximate the nonlinear operator

NL

with NLðuÞ ¼ hðuÞ ¼

X¹

n¼0

A

n

fug; ð2:12Þ

95 95

96 where the functions

AnÕs (n

= 0, 1, 2 ,. . .) are the so-called AdomianÕs polynomials and determined by A

n

fug ¼ 1

n!

d dk

ⁿ

hðu

k

Þ

k¼0

¼ 1 n!

d

dk

ⁿ

a

X¹

n¼0

k^j

u

j

!2 x

þ b

X¹

n¼0

k^j

u

jx

!2

þ c

X¹

n¼0

k^j

u

j

! X¹

n¼0

k^j

u

jxx

0 !

@

1 A_x 2

4

3 5 2

4

3 5_k¼0

¼ 1 n!

Xⁿ

j¼0

n j

!

j!ðn jÞ! aðu

jx

u

nj

þ u

ðnjÞx

u

j

Þ

_x

þ bðu

jx

u

ðnjÞx

Þ

_x

þ cðu

j

u

ðnjÞxx

Þ

_x

" #

¼ a

Xⁿ

j¼0

½u

jxx

u

_nj

þ 2u

jx

u

_ðnjÞx

þ u

_ðnjÞxx

u

j

þ b

Xⁿ

j¼0

½u

jxx

u

_ðnjÞx

þ u

jx

u

_ðnjÞxx

þ c

Xⁿ

j¼0

½u

jx

u

ðnjÞxx

þ u

j

u

ðnjÞxxx

. 98

98

99 where

a

= 3/2 ,

b

= 1/8,

c

= 1/4 and u

_k

¼

P1 i¼0kⁱ

u

_i

. 100 The expected solution u ¼

P1

n¼0

u

n

is approximated by the following

m

termÕs sum:

101

/_m

½u ¼

X^m1

n¼0

u

n

ð2:13Þ

103 103

104 which rapidly converges

u. In this sense,m

is able to be chosen as a small number so that this series is convergent to

u.

105 This method has been investigated in several authorsÕ work (see [12,1,2] for more details).

106 As we see, it is not hard to write a program for generating the Adomian polynomials. We summarize the entire pro- 107 cedure in the following algorithm:

108 Algorithm

109

•

Input: J(x)—initial conditions, i.e: uðx; 0Þ

¹₄

u

xx

ðx;0Þ ¼ J ðxÞ.k—number of terms in the approximation 110

•

Output: u

approx

(x,t) : the approximate solution

111 – Step 1: Set u

₀

= J(x) and u

_approx

(x,t) = u

₀

.

112 – Step 2: For k = 0 to n 1, do Step 3, Step 4, and Step 5.

113 – Step 3: Compute

UNCORRECTED

PROOF

A

k

¼ a

X^k

j¼0

½u

jxx

u

_kj

þ 2u

jx

u

_ðkjÞx

þ u

_ðkjÞxx

u

j

þ b

X^k

j¼0

½u

jxx

u

_ðkjÞx

þ u

jx

u

_ðkjÞxx

þ c

X^k

j¼0

½u

jx

u

_ðkjÞxx

þ u

j

u

_ðkjÞxxx

. 115

115

116 – Step 4: Compute u

_kþ1

¼ 1

4 u

kxx

þ

Z t

0

A

k

ds if k 6¼ 0.

118 118

119 – Step 5: Compute u

approx

= u

approx

+ u

k+1

.

120 – Stop

121 122 123 124

125 Remark 2.1. It is not hard to see that the above procedure also works for the following general equation:

u

_t

þ au

_xxt

þ bðu

²

Þ

_x

þ cðu

²_x

Þ

_x

þ dðuu

xx

Þ

_x

¼

cðx;

tÞ ð2:14Þ 127

127

128 where a, b, c, d are real constants and the function

c

is sufficiently smooth.

129 3. Convergence analysis

130 In this section, we discuss the convergence property of the approximated solution for the CH equation.

131 Let us consider the CH equation in the Hilbert space

H

=

L²

((a,b)

·

[0,

T]):

H ¼ v

:

ða;bÞ ½0; T with

Z

ða;bÞ½0;T

v

²

ðx; sÞ ds ds

<

þ1

( )

ð3:15Þ 133

133

134 Then the operator is of the form

T ðuÞ ¼ L

_t

ðu þ au

_xx

Þ ¼ bðu

²

Þ

_x

cðu

²_x

Þ

_x

d ðuu

xx

Þ

_x

þ

cðx;

tÞ ð3:16Þ 136

136

137 The Adomian decomposition method is convergent if the following two hypotheses are satisfied:

¹

138

•

(Hyp1): There exists a constant

k

> 0 such that the following inner product holds in

H:

ðT ðuÞ T ðvÞ;u vÞ

P

kku vk; 8u; v 2 H; ð3:17Þ

140 140 141

142

•

(Hyp2): As long as both

u

2

H

and

v

2

H

are bounded (i.e. there is a positive number

M

such that kuk

6M,

143 kvk

6M), there exists a constanth(M) > 0 such that

ðT ðuÞ T ðvÞ;u vÞ

6hðM

Þku vkkwk; 8w 2 H. ð3:18Þ 145

145 146

147 Theorem 3.1 (Sufficient conditions of convergence for the CH equation). Let

T ðuÞ ¼ L

_t

ðu þ au

_xx

Þ ¼ bðu

²

Þ

_x

cðu

²_x

Þ

_x

d ðuu

xx

Þ

_x

þ

cðx;

tÞ; with d c

>

0; L

_t

¼

o

149

ot

149

150 and consider the free initial and boundary conditions for the CH equation. Then the Adomian decomposition method leads 151 to a special solution of the CH equation.

1 See Abbaoui and Cherruault[1,2]and some references therein for more details.

UNCORRECTED

PROOF

152 Proof. To prove the theorem, we just verify the conditions (Hyp1) and (Hyp2). For

"u,v

2 H, let us calculate:

TðuÞ T ðvÞ ¼ bðu

²

v

²

Þ

_x

cðu

²_x

v

²_x

Þ

_x

dðuu

xx

vv

xx

Þ

_x

¼ bðu

²

v

²

Þ

_x

ð2c þ dÞðu

x

u

xx

v

x

v

xx

Þ dðuu

xxx

vv

xxx

Þ

¼ b

o

ox

ðu

²

v

²

Þ ð2c þ d Þðu

x

u

xx

v

x

v

xx

Þ d 2

o³

ox³

ðu

²

v

²

Þ 3

o

ox

ðu

²_x

v

²_x

Þ

¼ b

o

ox

ðu

²

v

²

Þ ðc dÞ

o

ox

ðu

²_x

v

²_x

Þ d 2

o³

ox³

ðu

²

v

²

Þ

154 154

155 Therefore, we have the inner product 156

T ðuÞ T ðvÞ; u v

ð Þ ¼ b

o

ox

ðu

²

v

²

Þ; u v

þ ðc d Þ

o

ox

ðu

²_x

v

²_x

Þ; u v

þ d 2

o³

ox³

ðu

²

v

²

Þ; u v

ð3:19Þ 158

158

159 Let us assume that u, v are bounded and there is a constant M > 0 such that (u, u), (v,v) < M

²

. By using Schwartz 160 inequality

o

ox

ðu

²

v

²

Þ; u v

6

kðu

²

v

²

Þ

_x

kku vk ð3:20Þ

162 162

163 and since there exist

h1

and

h2

such that k(u v)

x

k

6h1

ku vk, k(u + v)

x

k

6h2

ku vk and ku + vk

6

2M, we have 164

o

ox

ðu

²

v

²

Þ; u v

6

2M

h1h2

ku vk

²

. ()

o

ox

ðu

²

v

²

Þ; u v

P

2M

h₁h₂

ku vk

²

.

ð3:21Þ 166

166

167 Following the preceding procedure, we can calculate:

168

o

ox

ðu

²_x

v

²_x

Þ; u v

6

kðu

²_x

v

²_x

Þ

_x

kku vk

6h3

ku

x

þ v

x

kku

x

v

x

kku vk

6

2M

h₃h₄h₅

ku vk

²;

()

o

ox

ðu

²_x

v

²_x

Þ; u v

P

2M

h3h4h5

ku vk

²;

ð3:22Þ

170 170

171 where

hi

(i = 3, 4, 5) are positive constants.

172 Moreover, the Cauchy–Schwartz–Buniakowski inequality yields

o³

ox³

ðu

²

v

²

Þ; u v

6

kðu

²

v

²

Þ

_xxx

kku vk ð3:23Þ

174 174

175 then by using the mean value theorem, we have 176

o³

ox³

ðu

²

v

²

Þ; u v

6h6h7h8

ku

²

v

²

kku vk

6

2M

h6h7h8

ku vk

²

()

o³

ox³

ðu

²

v

²

Þ; u v

P

2M

h₆h₇h₈

ku vk

²

ð3:24Þ

178 178

179 where

h_j

(j = 6, 7, 8) are three positive constants, and k(u

²

v

²

)

_xxx

k

6h₆

k(u

²

v

²

)

_x

xk, k(u + v)

_xx

k

6h₇

k(u + v)

_x

k and

180 k(u + v)

x

k

6h8

ku + vk.

UNCORRECTED

PROOF

181 Substituting (3.21), (3.22), (3.24) into (3.19) generates the following inner product:

ðT ðuÞ T ðvÞ;u vÞ ¼ b

o

ox

ðu

²

v

²

Þ; u v

ðc dÞ

o

ox

ðu

²_x

v

²_x

Þ; u v

d 2

o³

ox³

ðu

²

v

²

Þ;u v

P

kku vk

²;

183

183

184 where k = (2bh

₁h₂

+ 2(c d)h

₃h₄h₅

+ dh

₆h₇h₈

)M. So, (Hyp1) is true for the CH equation.

185 Let us now verify the hypotheses (Hyp2) for the operator T(u). We directly compute:

ðT ðuÞ T ðvÞ;wÞ ¼ b

o

ox

½u

²

v

²

; w

ðc d Þ

o

ox

½u

²_x

v

²_x

; w

d 2

o³

ox³

ðu

²

v

²

Þ

;

w

6hðMÞku

vkkwk 187

187

188 where

h(M) = (2b

+ d 2c)M. Therefore, (Hyp2) is correct as well.

h

.

189 Remark 3.2. Choice of b = 3/2 , c = 1/8, d = 1/4,

c(x,t)

0 corresponds to the CH equation. So, the Adomian 190 decomposition method works for the CH equation.

191 4. Implementation of the method and approximate solutions

192 In this section, we take some examples to show the procedure and present some approximate solutions for the CH 193 equation.

194 Example 4.1

u

t

¹₄

u

xxt

þ

³₂

ðu

²

Þ

_x

¹₈

ðu

²_x

Þ

_x

¹₄

ðuu

xx

Þ

_x

¼ 0 u

0

¼ uðx; 0Þ

¹₄

u

xx

ðx; 0Þ ¼ c sinhðxÞ

(

ð4:25Þ 196

196

197 In this case, one straightforwardly gets u

0xx

¼ u

0;

u

0x

¼ c coshðxÞ; u

²_0x

u

²₀

¼ c

²

and h

^ðnþ1Þ

ðu

0

Þ ¼ ðh

^ðnÞ

ðu

0

ÞÞ

_x=u0x

198 where

u0x5

0,

h⁽⁰⁾

=

h

and

h⁽ⁿ⁾

denotes the

nth derivative ofh. Since the formula

(2.13) implies the formula (2.9), 199 we need the explicit expression of the

nth derivative ofh. Through direct calculations, we obtain the following formulas:

A

0

¼ hðu

0

Þ ¼ 3 u

²_0x

u

²₀

1 4 u

0

u

0x

x

¼ 3c

²

4 ðcosh

²

ðxÞ þ sinh

²

ðxÞÞ 201

201

u

1

ðx; tÞ ¼ 1 4 u

0xx

þ

Z t 0

hðu

0

Þ ds ¼ c

4 sinhðxÞ 3c

²

4 ðcosh

²

ðxÞ þ sinh

²

ðxÞÞt 203

203

A

1

¼ u

1

h

^ð1Þ

ðu

0

Þ ¼ 1 4 u

0

þ A

0

t

3 u

²_0x

u

²₀

¹₄

u

0

u

0x

x

x

u

0x

¼ 1 4 u

0

þ A

0

t

3 2u

0xx

u

0x

2u

0x

u

0

¹₄

ðu

0x

u

0x

þ u

0

u

0xx

Þ

x

u

0x

¼ 3c c

4 sinhðxÞ 3c

²

4 cosh

²

ðxÞ þ sinh

²

ðxÞ t

sinhðxÞ.

205 205

u

2

ðx; tÞ ¼ 1 4 u

1xx

þ

Z t 0

A

1

ds ¼ c

16 sinhðxÞ 3c

²

1 þ 2sinh

²

ðxÞ t þ 3 c

²

4 sinhðxÞt þ 3c

³

8 ð1 þ 2sinh

²

ðxÞÞt

²

sinhðxÞ

¼ c

16 sinhðxÞ 3c

²

1 þ 7 4 sinh

²

ðxÞ

t þ 9c

³

8 sinhðxÞ

²

þ 2sinh

³

ðxÞ t

²

207

207

A

2

¼ u

2

h

^ð1Þ

ðu

0

Þ þ u

²₁

h

^ð2Þ

ðu

0

Þ ¼ 3u

2

u

0

3 c

4 sinhðxÞ þ 3c

²

4 ðcosh

²

ðxÞ þ sinh

²

ðxÞÞt

2

209

209

UNCORRECTED

PROOF

u

3

ðx; tÞ ¼ 1 4 u

2xx

þ

Z t 0

A

2

ds

¼ c

64 sinhðxÞ 42c

²

16 ðcosh

²

ðxÞ þ sinh

²

ðxÞÞt þ 9c

³

8 ððsinhðxÞ

²

þ cosh

²

ðxÞÞ þ 2sinh

³

ðxÞ þ 12 sinhðxÞcosh

²

ðxÞÞt

²

3 sinhðxÞ c

16 sinhðxÞt 3c

²

2 1 þ 7

4 sinh

²

ðxÞ

t

²

þ 9c

³

24 ðsinhðxÞ

²

þ 2sinh

³

ðxÞÞt

³

4

c

²

ð1 þ 2sinh

²

ðxÞÞ c

4 sinhðxÞ þ 3c

²

4 ð1 þ 2sinh

²

ðxÞÞt

3

211 211

212 So, the approximate solution, truncated in the second term, is uðx;tÞ u

0

þ u

1

ðx; tÞ þ u

2

ðx; tÞ

¼ c

³

2

585

8 coshðxÞ sinhðxÞ

²

9

2 sinhðxÞ

³

27 coshðxÞ

²

sinhðxÞ

t

²

þ c

³

2

585

8 coshðxÞ

³

9 2

coshðxÞ

⁴

sinhðxÞ

!

t

²

þ c

²

2 93

8 sinhðxÞ

²

93

8 ðcoshðxÞ

²

3 coshðxÞ sinhðxÞÞ

t þ 21c 16 sinhðxÞ 214

214

215 The graph of

u(x,t) is plotted in

Fig. 1. From the figure, we see that the approximate solution is similar to a single pea- 216 kon solution of the CH equation.

217 Example 4.2 218

u

t

¹₄

u

xxt

þ

³₂

ðu

²

Þ

_x

¹₈

ðu

²_x

Þ

_x

¹₄

ðuu

xx

Þ

_x

¼ 0

u

0

¼ uðx;0Þ

¹₄

u

xx

ðx; 0Þ ¼ c

1

coshðxÞ; c

1

¼ constant.

(

ð4:26Þ 220

220 221

222 Let us follow the procedure in Example 4.1 and notice u

²₀

u

²_0x

¼ c

²₁

, we obtain the following formulas.

u

1

ðx; tÞ ¼ 1 4 u

0xx

þ

Z t 0

hðu

0

Þ ds

¼ c

4 coshðxÞ 3c

²

4 ð4ðcoshðxÞ

²

þ sinhðxÞ

²

Þ coshðxÞ sinhðxÞÞt.

224 224

Fig. 1. Approximate solution forc= 1.

UNCORRECTED

PROOF

u

2

ðx; tÞ ¼ 1 4 u

1xx

þ

Z t 0

A

1

ds

¼ c

³

2

585

8 coshðxÞ sinhðxÞ

²

t

²

þ c

³

2 9

2 sinhðxÞ

³

27 coshðxÞ

²

sinhðxÞ þ 585

8 coshðxÞ

³

9

4 sinhðxÞ coshðxÞ

⁴

t

²

þ 3c

²

sinhðxÞ

²

2 coshðxÞ

²

þ 5

16 coshðxÞ sinhðxÞ þ 1

16 sinhðxÞ coshðxÞ

³

t þ c 16 coshðxÞ 226

226

227 So, the approximate solution corresponding to Eq. (4.26) is uðx; tÞ u

0

þ u

1

ðx; tÞ þ u

2

ðx; tÞ

¼ c

³

2

585

8 coshðxÞ sinhðxÞ

²

9

4 sinhðxÞ

³

27 coshðxÞ

²

sinhðxÞ

t

²

þ c

³

2

585

8 coshðxÞ

³

9

2 sinhðxÞ coshðxÞ

⁴

t

²

þ c

²

6 sinhðxÞ

²

9 coshðxÞ

²

þ 27

16 coshðxÞ sinhðxÞ þ 3 16 sinhðxÞ coshðxÞ

³

!

t þ 21c

16 coshðxÞ 229

229

230 The graph of

u(x,t) is plotted in

Fig. 2, which shows that the approximate solution is similar to a single anti-peakon 231 solution of the CH equation.

232 Example 4.3 233

u

t

¹₄

u

xxt

þ

³₂

ðu

²

Þ

_x

¹₈

ðu

²_x

Þ

_x

¹₄

ðuu

xx

Þ

_x

¼ 0 u

0

¼ u

xx

ðx; 0Þ

¹₄

u

xx

ðx; 0Þ ¼ ae

^x

þ be

^x (

ð4:27Þ 235

235

236 In this case, by u

0xx

¼ u

0;

e

^x

¼

^u0þu_2a^0x;

e

^x

¼

^u0u_2b^0x

and u

²₀

u

²_0x

¼ 4ab, we obtain those

ujÕs below

A

0

¼ 1

4 21a

²

e

^2x

27b

²

e

^2x

238

238

u

1

¼ 1 4 u

0xx

þ

Z t 0

A

0

ds ¼ ae

^x

þ be

^x

4 1

4 21a

²

e

^2x

þ 27b

²

e

^2x

240 t

240

Fig. 2. Approximate solution forc= 1.

UNCORRECTED

PROOF

Fig. 3. Approximate solution fora=5,b= 2.

Fig. 4. Approximate solution fora=5,b= 10.

Fig. 5. Approximate solution fora= 5,b= 2.

UNCORRECTED

PROOF

u

₂

ðx; tÞ ¼ ae

^x

þ be

^x

16 21a

²

4 e

^2x

þ 27b

²

4 e

^2x

1 ae

²

x b

27ab

²

8 21a

²

b

8 e

^2x

þ 27b

³

8 e

^2x

t

þ 1 ae

²

x b

441

16 a

⁴

e

^5x

729 16 b

⁴

e

^3x

t

²

242

242

243 So, the approximate solution corresponding to Eq. (4.27) is uðx; tÞ u

0

þ u

1

ðx; tÞ þ u

2

ðx; tÞ

¼ 21

16 ðae

^x

þ be

^x

Þ 21a

²

2 e

^2x

þ 27b

²

2 e

^2x

1 ae

²

x b

27ab

²

8 21a

²

b

8 e

^2x

þ 27b

³

8 e

^2x

t

þ 1 ae

²

x b

441

16 a

⁴

e

^5x

729 16 b

⁴

e

^3x

t

²

. 245

245

246 The graphs of

u(x,t) for differentaÕs andbÕs are plotted in

Figs. 3–6. Those figures reveal that the approximate solutions 247 are describing the interactions of two anti-peakons for the CH equation.

248 5. Conclusions

249 In this paper, we successfully apply the Adomian polynomial decomposition method to solve the CH equation in an 250 explicitly approximate form. The initial values we adopted are smooth, but the most interesting is: the approximate 251 solutions are weak solutions with some peaks (see graphs in Figs. 1–6). The approximate solutions in Figs. 1, 2 show 252 the single peakons of the CH equation, while the approximate solutions in Figs. 3–6 provide the interactions of the two 253 anti-peakons. In comparison with the existing method to obtain two exact anti-peakons, our procedure just works on 254 the polynomial and algebraic computations. In the future, we plan to generalize our method to multi-soliton solutions 255 for the CH equation and other higher order equations. In the recent literatures, there are also other methods to deal 256 with nonlinear partial differential equations [3,17], where smooth solutions were obtained. Our paper presents some 257 peaked (i.e. continuous but non-smooth) explicit solutions for the CH equation (1.1).

258 Acknowledgements

259 The authors thank the referees for mentioning the Refs. [3,17]. Qiao did his partial work during he visited the ICTP, 260 Trieste, Italy, University of Kassel, Germany, and Delaware State University, Dover, Delaware this summer. Qiao spe- 261 cially expresses his sincere thanks to Prof. Strampp (University of Kassel), Prof. Fengshan Liu (DSU), Prof. Xiquan Shi

Fig. 6. Approximate solution fora= 5,b= 10.

UNCORRECTED

PROOF

262 (DSU), and Dr. Guoping Zhang (DSU) for their friendly hospitality. QiaoÕs work was partially supported by the 263 UTPA-FDP grant.

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