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Short Communication

Solitons and singular solitons for a variety of Boussinesq-like equations Abdul-Majid Wazwaz n Department of Mathematics, Saint Xavier University, Chicago, IL 60655, United States

a r t i c l e i n f o

abstract

Article history: Received 16 February 2012 Accepted 9 June 2012 Editor-in-Chief: A.I. Incecik

In this work we study a variety of Boussinesq-like equations. One soliton solution and another singular soliton solution are obtained for each equation. We employ independently two distinct approaches to formally derive standard and singular solitons. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Boussinesq equations Solitons solutions Singular soliton solutions

1. Introduction The study of wave propagation on the surface of water has been a subject of considerable theoretical and practical importance during the past decades (Biswas et al., 2009; Christov et al., 1996; Dehghan and Salehi, 2012; Esfahani and Farah, 2012; Wazwaz, 2009a). Later the Boussinesq equation utt uxx ðu2 Þxx buxxxx ¼ 0,

ð1Þ

where b ¼ 71. The Boussinesq equation (1) appeared not only in the study of the dynamics of thin inviscid layers with free surface but also in the study of the nonlinear string, the shape-memory alloys, the propagation of waves in elastic rods and in the continuum limit of lattice dynamics or coupled electrical circuits (Biswas et al., 2009; Christov et al., 1996; Dehghan and Salehi, 2012; Esfahani and Farah, 2012; Wazwaz, 2009a). Boussinesq introduced this equation to describe the propagation of long waves in shallow water. This equation also arises in other physical applications such as nonlinear lattice waves, iron sound waves in a plasma, and in vibrations in a nonlinear string. Moreover, it was applied to problems in the percolation of water in porous subsurface strata. It is obvious that Eq. (1) adds a fourth-order dispersion term to certain nonlinearity. The Boussinesq equation is completely integrable and gives multiple soliton solutions and has an inﬁnite laws of conservation. Nonlinear phenomena appear in a wide variety of scientiﬁc applications such as plasma physics, solid state physics, ﬂuid dynamics and chemical kinetics (Wazwaz, 2006, 2008a). Because of the increased interest in the theory of solitary waves, a broad range of analytical and numerical methods were used in the

analysis of these scientiﬁc models. Various methods (Hirota, 2004, Wazwaz, 2006, 2007, 2008a,b, 2009a,b, 2011a,b, 2012a,b; Shokri and Dehghan, 2010; Shakeri and Dehghan, 2008) have been used to conduct analysis on the nonlinear evolution equations. Examples of the methods that have been used so far are the Hirota bilinear method, the B¨acklund transformation method, Darboux transformation, Pfafﬁan technique, the inverse scattering method, the Painleve´ analysis, the generalized symmetry method, the subsidiary ordinary differential equation method (subODE for short), the coupled amplitude-phase formulation, sine–cosine method, sech–tanh method, the mapping and the deformation approach, and many other methods. In Shokri and Dehghan (2010), a numerical simulation of the improved Boussinesq equation is obtained using collocation and approximating the solution by radial basis functions based on the third-order time discretization. In Shakeri and Dehghan (2008), the homotopy perturbation method was used to handle a generalized Boussinesq equation. The Hirota’s bilinear method (Hirota, 2004) and the Hereman– Nuseir simpliﬁed form (Hereman and Nuseir, 1997) are rather heuristic and signiﬁcant. These approaches possess powerful features that make it practical for the determination of multiple soliton solutions for a wide class of nonlinear evolution equations. In this paper, a variety of Boussinesq equations will be investigated aiming to determine one soliton solution and one singular soliton solution for each equation. To achieve our goal we introduce the following Boussinesq-like equations: utt uxx ð6u2 ux þ uxxx Þx ¼ 0,

ð2Þ

utt uxx ð6u2 ux þ uxtt Þx ¼ 0,

ð3Þ

utt uxt ð6u2 ux þuxxt Þx ¼ 0

ð4Þ

and n

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0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.06.012

utt ð6u2 ux þuxxx Þx ¼ 0:

ð5Þ

2

A.-M. Wazwaz / Ocean Engineering 53 (2012) 1–5

Eq. (2) involves the dissipative term uxx and the fourth spatial derivative uxxxx. In Eq. (3), the fourth spatial derivative uxxxx is replaced by a mixed spatio-temporal one of the same order uxxtt and the dissipative term is retained. For Eq. (4), the dissipative term uxx is replaced by uxt and the fourth spatial derivative uxxxx is replaced by a mixed spatio-temporal one of the same order in the form uxxxt. However, Eq. (5) retained the fourth spatial derivative uxxxx but the dissipative term uxx was eliminated. It is worth noting that these improved replacements keep intact the physical assumptions of the standard Boussinesq equation but then make the models less amenable to the analytical techniques since these new equations are no more fully integrable (Christov et al., 1996; Esfahani and Farah, 2012).

2. Two distinct approaches for soliton solutions In this section we present the necessary steps for two distinct schemes to determine one soliton solution and one singular soliton solution for each Boussinesq-like equations (2)–(4). The two approaches will be applied independently. For the two approaches, the dispersion relation is determined in a like manner. We ﬁrst substitute uðx,tÞ ¼ ekxct

ð6Þ

into the linear terms of the evolution equation to determine the dispersion relation between k and c.

3. The ﬁrst Boussinesq-like equation We ﬁrst study the ﬁrst Boussinesq-like equation given by utt uxx ð6u2 ux þ uxxx Þx ¼ 0:

ð11Þ

3.1. One soliton solution Substituting uðx,tÞ ¼ ey ,

y ¼ kxot

ð12Þ

into the linear terms of (11) gives the dispersion relation by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð13Þ o ¼ k 1 þk2 , and as a result we obtain qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y ¼ kxk 1 þ k2 t:

ð14Þ

Following the analysis presented earlier we assume that the one soliton solution reads uðx,tÞ ¼ lðarctanðf ðx,tÞÞÞx ,

ð15Þ

where the auxiliary function f ðx,tÞ for the one soliton solution is given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 f ðx,tÞ ¼ ekxk 1 þ k t : ð16Þ Substituting (15) and (16) into (11) and solving for l we ﬁnd

l ¼ 7 2:

2.1. One soliton solution To determine the single soliton solution, we substitute uðx,tÞ ¼ lðarctan f ðx,tÞÞx ,

ð7Þ

where the auxiliary function f ðx,tÞ reads f ðx,tÞ ¼ ekxct

Combining (15)–(17) gives the one soliton and the one antisoliton solution by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 kx 1 þ k t 2ke pﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 : uðx,tÞ ¼ 7 ð18Þ 1 þe2kx2k 1 þ k t

ð8Þ

into the equation under discussion to determine the numerical value of l. Having determined the dispersion relation c and the coefﬁcient l, the single soliton solution follows immediately upon substituting these results in (7).

3.2. One singular soliton solutions In this section, we will determine one singular soliton solution for the Boussinesq-like equation utt uxx ð6u2 ux þ uxxx Þx ¼ 0,

2.2. One singular soliton solution We ﬁrst point out that the singular soliton solution for the Boussinesq-like equations (2)–(4) exists only if the nonlinear term 6u2 ux in these four models is replaced by 6u2 ux . This is identical to case of the modiﬁed KdV equation as presented in Wazwaz (2009a). To determine the single singular soliton solution, we substitute Fðx,tÞ uðx,tÞ ¼ m ln , ð9Þ Gðx,tÞ x where the auxiliary functions Fðx,tÞ and Gðx,tÞ read Fðx,tÞ ¼ 1þ ekxct , Gðx,tÞ ¼ 1ekxct

ð17Þ

ð10Þ

into the equation under discussion to determine the numerical value of m. Having determined the dispersion relation c and the coefﬁcient m, the singular soliton solution follows immediately upon substituting these results in (9).

ð19Þ

where singular soliton exists only if the nonlinear term 6u2 ux in (11) is replaced by 6u2 ux . Because there is no change in the linear terms, the dispersion relation remains the same, i.e. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð20Þ o ¼ k 1 þk2 : Following the assumption made earlier, we assume that the one singular soliton is given by Fðx,tÞ uðx,tÞ ¼ m ln , ð21Þ Gðx,tÞ x where the auxiliary functions Fðx,tÞ and Gðx,tÞ for the singular soliton solution are given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 Fðx,tÞ ¼ 1 þ ekxk 1 þ k t , pﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 Gðx,tÞ ¼ 1ekxk 1 þ k t :

ð22Þ

Substituting (21) and (22) into (19) and solving for m we ﬁnd

m ¼ 71:

ð23Þ

A.-M. Wazwaz / Ocean Engineering 53 (2012) 1–5

Combining (21)–(23) gives the singular soliton and the antisingular soliton solution by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ kx 1 þ k2 t 2ke pﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 : uðx,tÞ ¼ 7 ð24Þ 1e2kx2k 1 þ k t pﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 It is clear that the solution blows up when e2kx2k 1 þ k t ¼ 1.

4. The second Boussinesq-like equation

utt uxx ð6u2 ux þuxtt Þx ,

ð25Þ

where the fourth spatial derivative uxxxx in the preceding model is replaced by a mixed spatio-temporal one of the same order uxxtt. 4.1. One soliton solutions Substituting uðx,tÞ ¼ ey1 ,

y1 ¼ kxo1 t

ð26Þ

into the linear terms of (25) gives the dispersion relation by k

o1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ , k a0, k2 o1,

Proceeding as before, we assume that the one singular soliton is given by Fðx,tÞ uðx,tÞ ¼ m1 ln , ð35Þ Gðx,tÞ x where the auxiliary functions Fðx,tÞ and Gðx,tÞ for the singular soliton solution are given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 Fðx,tÞ ¼ 1 þ ekxðk= 1k Þt , pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 Gðx,tÞ ¼ 1ekxðk= 1k Þt :

We next study the second Boussinesq-like equation given by

3

ð36Þ

Substituting (35) and (36) into (33) and solving for m1 we ﬁnd 1

m1 ¼ 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ :

ð37Þ

1k

This again shows that m1 depends on the coefﬁcient k of the spatial variable x. Combining (35)–(37) gives a singular soliton and the antisingular soliton solutions by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 kxðk= 1k Þt 2ke pﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð38Þ uðx,tÞ ¼ 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 1k ð1e2kxð2k= 1k Þt Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 It is clear that the solution blows up when e2kxð2k= 1k Þt ¼ 1.

ð27Þ

1k

5. The third Boussinesq-like equation

and as a result we obtain We next study the third Boussinesq-like equation given by

k

y ¼ kx pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t: 2

ð28Þ

1k

Proceeding as before, we assume that the one soliton reads uðx,tÞ ¼ l1 ðarctanðf ðx,tÞÞÞx ,

ð29Þ

where the auxiliary function f ðx,tÞ for the one soliton solution is given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 f ðx,tÞ ¼ ekxðk= 1k Þt : ð30Þ Substituting (29) and (30) into (25) and solving for l1 we ﬁnd 2

l1 ¼ 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : 2

ð31Þ

1k

utt uxf ð6u2 ux þ uxxt Þx ,

where the dissipative term uxx and the fourth spatial derivative uxxxx of the ﬁrst Boussinesq-like equation (11) are replaced by uxt and a mixed spatio-temporal one of the same order in the form uxxxt respectively. 5.1. One soliton solutions Substituting uðx,tÞ ¼ ey2 ,

y2 ¼ kxo2 t

o2 ¼ ðk þ k3 Þ,

ð41Þ

and as a result we obtain

y ¼ kx þðk þk3 Þt: uðx,tÞ ¼ l2 ðarctanðf ðx,tÞÞÞx ,

In this section, we will determine one singular soliton solution for the Boussinesq-like equation ð33Þ

where singular soliton exists only if the nonlinear term 2ðu3 Þx in (25) is replaced by 2ðu3 Þx . The dispersion relation reads k

1k

ð34Þ

ð43Þ

where the auxiliary function f ðx,tÞ for the one soliton solution is given by 3

4.2. One singular soliton solutions

ð42Þ

The one soliton reads

f ðx,tÞ ¼ ekx þ ðk þ k Þt :

o1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ , k2 o 1:

ð40Þ

into the linear terms of (39) gives the dispersion relation by

This means that the coefﬁcient l1 is not ﬁxed as in the ﬁrst equation, but it depends on the coefﬁcient k of the spatial variable x. Combining (29)–(31) gives a family of one soliton and one anti-soliton solution by pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 kxðk= 1k Þt 2ke pﬃﬃﬃﬃﬃﬃﬃﬃﬃ : uðx,tÞ ¼ 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð32Þ 2 2 1k ð1 þe2kxð2k= 1k Þt Þ

utt uxx ð2ðu3 Þx þ uxtt Þx ¼ 0,

ð39Þ

ð44Þ

Substituting (43) and (44) into (39) and solving for l2 we ﬁnd qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð45Þ l2 ¼ 72 1 þ k2 : This means that the coefﬁcient l2 is not ﬁxed as in the ﬁrst equation, but it depends on the coefﬁcient k of the spatial variable x. Combining (43)–(45) gives a one soliton and one anti-soliton solution by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2 2k 1 þ k ekx þ þ ðk þ k Þt uðx,tÞ ¼ 7 : ð46Þ 3 1 þe2kx þ 2ðk þ k Þt

4

A.-M. Wazwaz / Ocean Engineering 53 (2012) 1–5

5.2. One singular soliton solutions

6.2. One singular soliton solutions

In this section, we will determine one singular soliton solution for the Boussinesq-like equation

In this section, we will determine one singular soliton solution for the Boussinesq-like equation

utt uxt ð2ðu3 Þx þuxxt Þx ¼ 0,

utt ð6u2 ux þuxxx Þx ¼ 0:

ð47Þ 3

3

where the nonlinear term 2ðu Þx in (39) is replaced by 2ðu Þx . The dispersion relation reads

o2 ¼ ðk þ k3 Þ:

ð48Þ

Proceeding as before, we assume that the one singular soliton is given by Fðx,tÞ , ð49Þ uðx,tÞ ¼ m2 ln Gðx,tÞ x where the auxiliary functions Fðx,tÞ and Gðx,tÞ for the singular soliton solution are given by

ð60Þ

The dispersion relation remains the same as before, and hence we assume that the one singular soliton is given by Fðx,tÞ , ð61Þ uðx,tÞ ¼ m3 ln Gðx,tÞ x where the auxiliary functions Fðx,tÞ and Gðx,tÞ for the singular soliton solution are given by 2

Fðx,tÞ ¼ 1 þ ekx 7 k t , 2

Gðx,tÞ ¼ 1ekx 7 k t :

ð62Þ

Proceeding as before and solving for m3 we ﬁnd

3

Fðx,tÞ ¼ 1þ ekx þ ðk þ k Þt ,

m3 ¼ 7 1: 3

Gðx,tÞ ¼ 1ekx þ ðk þ k Þt :

ð50Þ

Substituting (49) and (50) into (47) and solving for m2 we ﬁnd qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð51Þ m2 ¼ 7 1 þ k2 : Combining (49)–(51) gives the singular soliton and the antisingular soliton solutions by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kx þ ðk þ k3 Þt 2 2k 1 þ k ekx þ e : ð52Þ uðx,tÞ ¼ 7 3 1e2kx þ 2ðk þ k Þt

6. The fourth Boussinesq-like equation We ﬁrst study the ﬁrst Boussinesq-like equation given by utt ð6u2 ux þ uxxx Þx ¼ 0,

ð53Þ

ð63Þ

Combining (61)–(63) gives the singular soliton and the antisingular soliton solution by uðx,tÞ ¼ 7

kx 7 k2 t

2ke

1e2kx 7 k

2

t

:

ð64Þ

7. Discussion An analytic study was conducted on four Boussinesq-like equations. We formally derived one soliton solution for each Boussinesq-like equation. However, using another distinct approach, we derived one singular soliton solution for each equation. The structures of the obtained solutions are distinct and the dispersion relations are distinct as well. Moreover, the singular soliton solutions exist only when we change the sign of the nonlinear term 6u2 ux .

where the dissipation term uxx does not exist. References

6.1. One soliton solutions Proceeding as before, the dispersion relation reads

o3 ¼ 7 k2 ,

ð54Þ

and as a result we obtain

y3 ¼ kx 7 k2 t:

ð55Þ

The one soliton can be set as uðx,tÞ ¼ l3 ðarctanðf ðx,tÞÞÞx ,

ð56Þ

where the auxiliary function f ðx,tÞ for the one soliton solution is given by 2

f ðx,tÞ ¼ ekx 7 k t :

ð57Þ

We can easily show that

l3 ¼ 72:

ð58Þ

This in turn gives the one soliton and the one anti-soliton solution by kx 7 k2 t

uðx,tÞ ¼ 7

2ke

2

1 þ e2kx 7 2k

t

:

ð59Þ

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Wazwaz, A.M., 2011b. Extended KP equations and extended system of KP equations: multiple-soliton solutions. Can. J. Phys. 89 (7), 739–743. Wazwaz, A.M., 2012a. A study on two extensions of the Bogoyavlenskii–Schieff equation. Commun. Nonlin. Sci. Numer. Simul. 17 (4), 1500–1505. Wazwaz, A.M., 2012b. Soliton solutions for two (3 þ1)-dimensional non-integrable KdV-type equations. Math. Comput. Model. 55, 1845–1848.

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