Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method

Chaos, Solitons and Fractals 12 (2001) 1549±1556

www.elsevier.nl/locate/chaos

Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modi®ed decomposition method A.M. Wazwaz * Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA Accepted 6 June 2000

Abstract In this paper, we present a reliable algorithm to study the known model of nonlinear dispersive waves proposed by Boussinesq. The modi®ed algorithm of Adomian decomposition method is used with an emphasis on the single soliton solution. New exact periodic solutions and polynomial solutions are obtained. The results of numerical examples are presented and only few terms are required to obtain accurate solutions. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction A well-known model of nonlinear dispersive waves which was proposed by Boussinesq is formulated in the form utt ˆ uxx ‡ 3…u2 †xx ‡ uxxxx ;

L0 6 x 6 L 1

…1†

with u ˆ u…x; t† is a suciently often di€erentiable function. The initial conditions associated with the Boussinesq equation (1) are assumed to have the form u…x; 0† ˆ f …x†;

ut …x; 0† ˆ g…x†:

…2†

The Boussinesq equation (1) describes motions of long waves in shallow water under gravity and in a onedimensional nonlinear lattice [1±3]. This particular form (1) is of special interest [4,5] because it admits inverse scattering formalism. For more details about the formulation of Boussinesq equation, see [4±12]. A great deal of research work has been invested in recent years for the study of the Boussinesq equation. Three di€erent methods have been developed independently by which soliton solutions may be obtained for nonlinear evolution equations in general. Ablowitz and Segur [10] implemented the inverse scattering transform method to handle the nonlinear equations of physical signi®cance where soliton solutions were developed. Hirota [1±3] constructed the N-soliton solutions of the evolution equation by reducing it to the bilinear form. One of the most helpful tools in the study of evolution equations, over the last two decades, has been Hirota [1±3] bilinear formalism. On the other hand, Nimmo and Freeman [13,14] introduced an alternative formulation of the N-soliton solutions in terms of some function of the Wronskian determinant of N functions.

*

Fax: +773-779-9061. E-mail address: [email protected] (A.M. Wazwaz).

0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 1 3 3 - 8

1550

A.M. Wazwaz / Chaos, Solitons and Fractals 12 (2001) 1549±1556

More recently, Kaptsov [15] implemented Hirota's bilinear representation to construct a new set of exact solutions of the Boussinesq equation. Some of the new solutions constructed by Kaptsov [15] are interpreted from a hydrodynamic point of view. In particular, solutions that describe the propagation of wave packets and their interaction are obtained in [15,16]. A very unusual solution of the Boussinesq equation developed by Kaptsov [15] was periodic in x and its amplitude tends exponentially rapidly to zero for t ! 1. The interpretation given in [15] that this results as a wave arises from nothing during a short time interval and then damps rapidly. In addition, a solution that describes the elastic-breather interaction was developed by Kaptsov [15]. The approach of Kaptsov [15] and Andreev et al. [16] introduced an ecient algorithm to handle the Boussinesq equation and to develop multisoliton solutions. Most recently, Bratsos [17] approached the good Boussinesq equation and the bad Boussinesq equation where the method of lines has been used to transform the equation into a ®rst-order, nonlinear initial value problem. In [17], numerical methods were developed by replacing the matrix-exponential term in a recurrence relation by rational approximants. The basic motivation of this work is to approach the Boussinesq equation di€erently, but e€ectively, by using an alternative technique. In this work, we will use the modi®ed form of Adomian decomposition method introduced by Wazwaz [18±22] in a direct manner without any need to transformation formulae, bilinear forms or Lax pair. The introduction of this algorithm not only provides the solution in a rapidly convergent series, but it also guarantees considerable savings of the calculation size. The implementation of this modi®ed technique of Adomian method [23,24] has shown reliable results and this gives it a wider applicability in handling evolution models. This work will complement existing works in the literature. 2. Analysis To begin with, Eq. (1) can be written in an operator form Lu ˆ uxx ‡ 3…u2 †xx ‡ uxxxx ;

…3†

where the di€erential operator L is Lˆ

o2 : ot2

…4†

It is assumed that L 1 is a twofold integral operator given by Z tZ t L 1 …
† ˆ …
†dt dt: 0

0

…5†

The Adomian decomposition method [23,24] assumes a series solution for the unknown function u…x; t† given by u…x; t† ˆ

1 X

un …x; t†

…6†

nˆ0

and the nonlinear operator F …u† ˆ …u2 †xx can be decomposed into an in®nite series of polynomials given by F …u† ˆ

1 X

An ;

…7†

nˆ0

where the components un …x; t† will be determined recurrently, and An are the so-called Adomian polynomials of u0 ; u1 ; . . . ; un de®ned by " 1 dn F An ˆ n! dkn

1 X iˆ0

!# i

k ui

; kˆ0

n ˆ 0; 1; 2; . . .

…8†

A.M. Wazwaz / Chaos, Solitons and Fractals 12 (2001) 1549±1556

1551

It is now well known in the literature that these polynomials can be constructed for all classes of nonlinearity according to algorithms set by Adomian [23,24] and recently developed by an alternative approach in [21]. Operating with the integral operator L 1 on both sides of (3) and using the initial conditions we ®nd
…9† u…x; t† ˆ f …x† ‡ tg…x† ‡ L 1 uxx ‡ 3…u2 †xx ‡ uxxxx : Substituting (6) and (7) into the functional equation (9) gives 1 X

un …x; t† ˆ f …x† ‡ tg…x† ‡ L

1

nˆ0

1 X

! un

nˆ0

‡3

1 X

An ‡

nˆ0

xx

1 X nˆ0

!

!

un

:

…10†

xxxx

Following the modi®ed decomposition method [18±22], we introduce the recurrence relation: u0 …x; t† ˆ f …x†;

u1 …x; t† ˆ tg…x† ‡ L 1 …3A0 ‡ u0xx ‡ u0xxxx †;

uk‡1 …x; t† ˆ L 1 …3Ak ‡ ukxx ‡ ukxxxx †;

…11†

k P 1;

where Ak are Adomian polynomials that represent the nonlinear term …u2 †xx and can be derived as A0 ˆ …u20 †xx ;

A1 ˆ 2u0 u1xx ‡ 4u0x u1x ‡ 2u0xx u1 ; 2

…12†

A2 ˆ 2u0 u2xx ‡ 4u0x u2x ‡ 2u0xx u2 ‡ 2u1 u1xx ‡ 2…u1x † ; A3 ˆ 2u0 u3xx ‡ 4u0x u3x ‡ 2u0xx u3 ‡ 2u1 u2xx ‡ 4u1x u2x ‡ 2u1xx u2 :

Other polynomials can be generated in a like manner. It is worth noting that the recurrence relation (11) introduces a slight variation from the original recurrence relation developed by Adomian [23,24]. Useful works in [18±22] had shown that, although this change in the formulation of the recurrence relation is slight, it introduces a qualitative tool that accelerates the convergence of the solution and minimizes the volume of calculations. The ®rst few components of un …x; t† follows immediately upon setting: u1 …x† ˆ tg…x† ‡ L 1 …3A0 ‡ u0xx ‡ u0xxxx †;

u0 …x† ˆ f …x†;

u2 …x† ˆ L 1 …3A1 ‡ u1xx ‡ u1xxxx †;

u3 …x† ˆ L 1 …3A2 ‡ u2xx ‡ u2xxxx †:

…13†

The scheme in (13) determines the components un …x; t†; n P 0. It is, in principle, possible to calculate more components in the decomposition P series to enhance the approximation. Consequently, one can recursively 1 determine every term of the series nˆ0 un …x; t†, and hence the solution u…x; t† is readily obtained in a series form. It is interesting to note that we obtained the series solution by using the initial conditions only. For a detailed description of Adomian decomposition method and the modi®ed decomposition algorithm, we refer the reader to [18±24]. 3. Applications In the following, we discuss three classes of solutions of Boussinesq equation. Class 1. Soliton solutions. Consider the Boussinesq equation utt ˆ uxx ‡ 3…u2 †xx ‡ uxxxx ;

60 6 x 6 60

…14†

with u ˆ u…x; t† is a suciently often di€erentiable function. The initial conditions associated with the Boussinesq equation (14) are assumed to have the form u…x; 0† ˆ 2

ak 2 ekx …1 ‡

2 aekx †

;

ut …x; 0† ˆ

2

p ak 3 1 ‡ k 2 ekx …aekx x …1 ‡

3 aekx †

1†

:

…15†

1552

A.M. Wazwaz / Chaos, Solitons and Fractals 12 (2001) 1549±1556

Applying the inverse operator L yields 1 X

un …x; t† ˆ 2

ak 2 ekx

…1 ‡ aekx †2 ! 1 X ‡ un

nˆ0

nˆ0

1

of (5) on both sides of (14) and using the decomposition series (6) and (7)

2

p ak 3 1 ‡ k 2 ekx …aekx x

1†

…1 ‡ aekx †3

!

1

t‡L

3

1 X

! An

‡

nˆ0

1 X

! un

nˆ0

xx

:

…16†

xxxx

Following the modi®ed decomposition method [18±22] leads to the recurrence relation: u0 …x; t† ˆ 2

ak 2 ekx

; 2 …1 ‡ aekx † p ak 3 1 ‡ k 2 ekx …aekx x u1 …x; t† ˆ 2 …1 ‡ aekx †3

1†

uk‡1 …x; t† ˆ L 1 …3Ak ‡ ukxx ‡ ukxxxx †;

…17†

t ‡ L 1 …3A0 ‡ u0xx ‡ u0xxxx †;

k P 1:

Consequently, we obtain ak 2 ekx

u0 …x; t† ˆ 2

; …1 ‡ aekx †2 p ak 3 1 ‡ k 2 ekx …aekx x u1 …x; t† ˆ 2 …1 ‡ aekx †3 p ak 3 1 ‡ k 2 ekx …aekx x ˆ 2 …1 ‡ aekx †3

1† 1†

t‡

ak 4 ekx …1 ‡ k 2 †…a2 e2kx …1 ‡

4aekx ‡ 1†

aekx †4

t2 ;

1 ak 5 …1 ‡ k 2 †3=2 ekx …a3 e3kx 11a2 e2kx ‡ 11aekx 3 …1 ‡ aekx †5

u2 …x; t† ˆ L 1 …3A1 ‡ u1xx ‡ u1xxxx † ˆ ‡

t ‡ L 1 …3A0 ‡ u0xx ‡ u0xxxx †

1†

t3

1 6 w…x† t4 ; ak …1 ‡ k 2 †ekx 12 …1 ‡ aekx †8

u3 …x; t† ˆ L 1 …3A2 ‡ u2xx ‡ u2xxxx † ˆ 4a2 k 8 …1 ‡ k 2 †e2kx

10aekx ‡ 20a2 e2kx

…1

…1 ‡

10a3 e3kx ‡ a4 e3kx †

aekx †8

t4 ‡ O…t5 †; …18†

where 24a…1 ‡ 3k 2 †ekx ‡ 15a2 …1 ‡ 33k 2 †e2kx ‡ 80a3 …1

w…x† ˆ 1

11k 2 †e3kx ‡ 15…a4 ‡ 33k 2 †e4kx

24a5 …1 ‡ 3k 2 †e5kx ‡ a6 …1 ‡ k 2 †e6kx :

…19†

Other components can be easily determined. In view of (18), the solution in a series form is given by u…x; t† ˆ 2

ak 2 ekx …1 ‡

2 aekx †

2ak 3

p …aekx x 1† …a2 e2kx 4aekx ‡ 1† 2 t ‡ ak 4 …1 ‡ k 2 †ekx t 1 ‡ k 2 ekx 3 4 …1 ‡ aekx † …1 ‡ aekx †

1 5 …a3 e3kx 3=2 ak …1 ‡ k 2 † ekx 3 ‡

1 6 …1 2 ak …1 ‡ k 2 † ekx 12

11a2 e2kx ‡ 11aekx …1 ‡

5 aekx †

26aekx ‡ 66a2 e2kx

1†

t3

26a3 e3kx ‡ a4 e3kx †

…1 ‡ aekx †

6

…20† t4 ‡




A.M. Wazwaz / Chaos, Solitons and Fractals 12 (2001) 1549±1556

1553

and consequently, the exact solution p ak 2 exp…kx ‡ k 1 ‡ k 2 t† u…x; t† ˆ 2 p …1 ‡ a exp…kx ‡ k 1 ‡ k 2 t††2

…21†

is readily obtained. The single soliton solution (21) is in full agreement with the results of [15,16]. It is important to note that Kaptsov [15], among others, approached the standard form of the Boussinesq equation (1) by using general conditions. The soliton solution obtained by Kaptsov [15] is of the form u…x; t† ˆ 2

o2 ln f ; ox2

…22†

where f ˆ f …x; t† is de®ned for the single soliton by p f ˆ 1 ‡ a exp…kx  k 1 ‡ k 2 t†;

…23†

where a and k are arbitrary constants. Substituting (23) into (22) gives the general soliton solution (21). Fig. 1 below shows the single soliton solution (21) for a ˆ 1; k ˆ 1, 0 6 t 6 60 and 80 6 x 6 80. However, Fig. 2 shows the single soliton solution (21) for a ˆ 1; k ˆ 1, t ˆ 0 and 10 6 x 6 10. Class 2. Periodic solutions. In [15], the one- and two-soliton solutions are produced by the solutions of fourth-order ordinary di€erential equation dx …dx

k1 †…dx

k2 †…dx

k1

k2 †f ˆ 0;

…24†

where dx is a derivative with respect to x, k1 and k2 are arbitrary constants, and f satis®es the bilinear equation [15] ftt f

ft2

ffxxxx ‡ 4fx fxxx

3fxx2

ffxx ‡ fx2 ˆ 0:

In the purely imaginary constants k1 ˆ ik and k2 ˆ form f ˆ sin…kx† ‡ et

p k4 k2

‡

4k 2 4…k 2

1 e 1†

…25† ik, Kaptsov [15] derived the solution to (24) in the

p t k4 k2

…26†

Fig. 1. The surface shows the theoretical solution (21) of the Boussinesq equation for 0 6 t 6 60 and Fig. 2. The graph shows the theoretical solution of the Boussinesq equation for t ˆ 0 and

80 6 x 6 80.

10 6 x 6 10.

1554

A.M. Wazwaz / Chaos, Solitons and Fractals 12 (2001) 1549±1556

and hence, the periodic solution in x , u…x; t† ˆ

2

2…k sin…kx††

sin…kx† ‡ e

p t k4 k2

, 2

2

2…k cos …kx††

sin…kx† ‡ e

‡

p t k4 k2

…4k 2

‡

p k4 k2

1†e t 4k 2 4

…4k 2

!

p k4 k2

1†e t 4k 2 4

!2 …27†

is readily obtained upon substituting (28) into (22). To (24), there corresponds another solution, where by following Kaptsov's approach, we obtained another solution in the form: f ˆ cos…kx† ‡ et

p k4 k2

‡

4k 2 4…k 2

1 e 1†

p t k4 k2

:

…28†

To (28), there corresponds a periodic solution in x to the Boussinesq equation given by , p ! 4 2 p …4k 2 1†e t k k 2 t k4 k2 u…x; t† ˆ 2…k cos…kx†† cos…kx† ‡ e ‡ 4k 2 4 , p !2 4 2 p …4k 2 1†e t k k 2 2 t k4 k2 2…k sin …kx†† cos…kx† ‡ e ‡ : 4k 2 4

…29†

It is important to note that the solution (29) is periodic in x and is called transversal impulse. Fig. 3 shows the periodic solution (29) for k ˆ 1:5, 4 6 t 6 4 and 6 6 x 6 6. Class 3. Polynomial solutions. In this example, we consider the initial conditions of the Boussinesq equation to be of the form u…x; 0† ˆ a ‡ bx;

ut …x; 0† ˆ A ‡ Bx;

…30†

where a; b; A and B are arbitrary constants. Following the discussions presented above we ®nd: u0 …x; t† ˆ a ‡ bx;

u1 …x; t† ˆ At ‡ Bxt ‡ 3b2 t2 ;

u2 …x; t† ˆ 2Bbt3 ;

1 u3 …x; t† ˆ B2 t4 ; 2

…31†

where other components vanish as a result. Accordingly, the exact solution is given by the polynomial 1 u…x; t† ˆ a ‡ bx ‡ …A ‡ Bx†t ‡ 3b2 t2 ‡ 2Bbt3 ‡ B2 t4 : 2

Fig. 3. The graph shows the periodic solution (29) for

…32†

4 6 t 6 4 and

6 6 x 6 6.

A.M. Wazwaz / Chaos, Solitons and Fractals 12 (2001) 1549±1556

1555

It is clear that by setting the arbitrary constants by numerical values, several speci®c polynomial solutions can be obtained such as u…x; t† ˆ bx ‡ At ‡ 3b2 t2 ;

a ˆ 0; B ˆ 0;

1 u…x; t† ˆ a ‡ Bxt ‡ B2 t4 ; 2

b ˆ 0; A ˆ 0;

1 u…x; t† ˆ bx ‡ Bxt ‡ 3b2 t2 ‡ 2bBt3 ‡ B2 t4 ; 2 u…x; t† ˆ a ‡ bx ‡ At ‡ 3b2 t2 ; B ˆ 0:

…33† a ˆ 0; A ˆ 0;

It is worth noting that other polynomial solutions 3

u…x; t† ˆ …x  t† ‡ …x  t†  6t;

u…x; t† ˆ x2

t2

3

…34†

were obtained by Ablowitz and Segur [10] and Kaptsov [15] by using the inverse scattering method and the bilinear formalism, respectively.

4. Discussions The computations associated with the three classes of solutions discussed above were performed by using Maple V. The goal to obtain exact solutions of the Boussinesq equation by using the modi®ed decomposition method has been achieved. The method has been applied directly without using bilinear forms or Wronskians. Moreover, new exact periodic solutions were also obtained in a parallel manner to Kaptsov's approach [15]. Furthermore, new polynomial solutions were determined as well. The obtained results demonstrate the reliability of the algorithm and gives it a wider applicability to nonlinear dispersive equations. References [1] Hirota R. Direct methods in soliton theory. In: Bullogh RK, Caudrey PJ, editors. Solitons. Berlin: Springer; 1980. [2] Hirota R. Exact envelope-soliton solutions of a nonlinear wave. J Math Phys 1973;14(7):805±9. [3] Hirota R. Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices. J Math Phys 1973;14(7):810±4. [4] Debnath L. Nonlinear partial di€erential equations for scientists and engineers. Berlin: Birkhauser; 1998. [5] Debnath L. Nonlinear water waves. Boston: Academic Press; 1994. [6] Bhatnagar PL. Nonlinear waves in one-dimensional dispersive systems. Oxford: Clarendon Press; 1976. [7] Drazin PG, Johnson RS. Solitons: an introduction. Cambridge, New York, 1993. [8] Lamb GL. Elements of soliton theory. New York: Wiley; 1980. [9] Whitham GB. Linear and nonlinear waves. New York: Wiley; 1974. [10] Ablowitz M, Segur H. Solitons and inverse scattering transform. Philadelphia, PA: SIAM; 1981. [11] Freeman NC. Soliton interactions in two dimensions. Adv Appl Mech 1980;20:1±37. [12] Freeman NC. Soliton solutions of non-linear evolution equations. IMA J Appl Math 1984;32:125±45. [13] Nimmo JJC, Freeman NC. A method of obtaining the N-soliton solutions of the Boussinesq equation in terms of a Wronskian. Phys Lett 1983;95A:4±6. [14] Nimmo JJC, Freeman NC. The use of Backlund transformations in obtaining the N-soliton solutions in Wronskian form. J Phys A: Math General 1984;17:1415±24. [15] Kaptsov OV. Construction of exact solutions of the Bousseniseq equation. J Appl Mech and Tech Phys 1998;39(3):389±92. [16] Andreev VK, Kaptsov OV, Pukhnachov VV, Rodionov AA. Applications of group-theoretical methods in hydrodynamics. Boston: Kluwer Academic Publishers; 1998. [17] Bratsos AG. The solution of the Boussinesq equation using the method of lines. Comput Methods Appl Mech Eng 1998;157: 33±44. [18] Wazwaz AM. A ®rst course in integral equations. Singapore: World Scienti®c; 1997. [19] Wazwaz AM. Analytical approximations and Pade' approximants for Volterra's population model. Appl Math and Comput 1999;100:13±25. [20] Wazwaz AM. A reliable modi®cation of Adomian decomposition method. Appl Math and Comput 1999;102:77±86.

1556

A.M. Wazwaz / Chaos, Solitons and Fractals 12 (2001) 1549±1556

[21] Wazwaz AM. A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl Math and Comput 2000;111:53±69. [22] Wazwaz AM. The modi®ed decomposition method and Pade approximants for solving the Thomas-Fermi equation. Appl Math and Comput 1999;105:11±9. [23] Adomian G. Solving frontier problems of physics: the decomposition method. Boston: Kluwer Academic Publishers; 1994. [24] Adomian G. A review of the decomposition method in applied mathematics. J Math Anal Appl 1988;135:501±44.