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Multiple-soliton solutions and multiple-singular soliton solutions for two higher-dimensional shallow water wave equations
Applied Mathematics and Computation 211 (2009) 495–501
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Multiple-soliton solutions and multiple-singular soliton solutions for two higher-dimensional shallow water wave equations Abdul-Majid Wazwaz Department of Mathematics, Saint Xavier University, Chicago, IL 60655, United States
a r t i c l e
i n f o
Keywords: Hirota bilinear method Complete integrability Multiple-soliton solutions Multiple-singular soliton solutions
a b s t r a c t Two (3 + 1)-dimensional shallow water wave equations are studied for complete integrability. The Hirota’s bilinear method is used to determine the multiple-soliton solutions for these equations. Moreover, multiple-singular soliton solutions will also be determined for each model. The analysis highlights the capability of the direct method in handling completely integrable equations. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The (3 + 1)-dimensional shallow water wave equations [1]
uyzt þ uxxxyz 6ux uxyz 6uxz uxy ¼ 0
ð1Þ
uxzt þ uxxxyz 2ðuxx uyz þ uy uxxz Þ 4ðux uxyz þ uxz uxy Þ ¼ 0
ð2Þ
and
will be studied. Both equations reduce to the potential KdV equation for z ¼ y ¼ x. The difference between the first terms of the two models is that x replaces y in the term uyzt . The generalized shallow water wave equations studied by Ablowitz [2] and Hirota and Satsuma [3], and by Mansfield and Clarkson [1] arise as reduction of these two equations. A variety of distinct methods are used for classification of integrable equations. The Painlevé analysis, the inverse scattering method, the Bäcklund transformation method, the conservation laws method, and the Hirota bilinear method [4–13] are mostly used in the literature for investigating complete integrability. The Hirota’s bilinear method is rather heuristic and possesses significant features that make it ideal for the determination of multiple-soliton solutions for a wide class of nonlinear evolution equations. Two goals are set for this work. We first aim to use Hirota’s bilinear method to emphasize its power and reliability for handling integrable equations. We second aim to determine multiple-soliton solutions and multiple-singular soliton solutions [14–22] for these Eqs. (1) and (2). The Hereman’s simplified form [12,13] will be combined with the Hirota’s method to achieve these two goals.
2. The Hirota’s bilinear method To formally derive N-soliton solutions for completely integrable equations, we will use the Hirota’s direct method combined with the simplified version of Hereman et al. [12,13]. It was proved by many that soliton solutions are just polynomials of exponentials. This will be also confirmed in the coming discussions.
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.01.071
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A.-M. Wazwaz / Applied Mathematics and Computation 211 (2009) 495–501
We first substitute
uðx; y; z; tÞ ¼ ekxþryþszct ;
ð3Þ
into the linear terms of the equation under discussion to determine the dispersion relation between k, r, s and c. We then substitute the single-soliton solution
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð4Þ
into the equation under discussion, where the auxiliary function f ðx; y; z; tÞ is given by
f ðx; y; z; tÞ ¼ 1 þ f1 ðx; y; z; tÞ ¼ 1 þ eh1 ;
ð5Þ
hi ¼ ki x þ ri y þ si z ci t;
ð6Þ
where
i ¼ 1; 2; . . . ; N;
and solving the resulting equation to determine the numerical value for R. Notice that the N-soliton solutions can be obtained for Eqs. (1) and (2), by using the following forms for f ðx; y; z; tÞ into (4): (i) For dispersion relation, we use
uðx; y; z; tÞ ¼ ehi ;
hi ¼ ki x þ r i y þ si z ci t:
ð7Þ
(ii) For single-soliton, we use
f ¼ 1 þ eh1 :
ð8Þ
(iii) For two-soliton solutions, we use
f ¼ 1 þ eh1 þ eh2 þ a12 eh1 þh2 :
ð9Þ
(iv) For three-soliton solutions, we use
f ¼ 1 þ eh1 þ eh2 þ eh3 þ a12 eh1 þh2 þ a23 eh2 þh3 þ a13 eh1 þh3 þ b123 eh1 þh2 þh3 :
ð10Þ
Notice that we use (7) to determine the dispersion relation, (9) to determine the factor a12 to be generalized for the other factors aij , and finally we use (10) to determine b123 , which is given by b123 ¼ a12 a23 a13 for completely integrable equations. The determination of three-soliton solutions confirms the fact that N-soliton solutions exist for any order. In the following, we will apply the aforementioned steps to the (3 + 1)-dimensional shallow water wave Eqs. (1) and (2). However, for the multiple-singular soliton solutions, we follow the following steps: (i) For dispersion relation, we use
uðx; tÞ ¼ ehi ;
hi ¼ ki x xi t:
ð11Þ
(ii) For single-soliton, we use
f ðx; tÞ ¼ 1 eh1 :
ð12Þ
(iii) For two-soliton solutions, we use
f ðx; tÞ ¼ 1 eh1 eh2 þ a12 eh1 þh2 :
ð13Þ
(iv) For three-soliton solutions, we use
f ðx; tÞ ¼ 1 eh1 eh2 eh3 þ a12 eh1 þh2 þ a23 eh2 þh3 þ a13 eh1 þh3 b123 eh1 þh2 þh3 :
ð14Þ
3. The first (3 + 1)-dimensional equation In this section, we apply the Hirota’s bilinear method to the (3 + 1)-dimensional shallow water wave equation
uyzt þ uxxxyz 6ux uxyz 6uxz uxy ¼ 0:
ð15Þ
It is interesting to point out that this equation can be reduced to the potential KdV equation by setting z ¼ y ¼ x and integrating twice with respect to x. 3.1. Multiple-soliton solutions To determine multiple-soliton solutions for Eq. (15), we follow the steps presented above. We first substitute
uðx; y; z; tÞ ¼ ehi ;
hi ¼ ki x þ r i y þ si z ci t;
ð16Þ
A.-M. Wazwaz / Applied Mathematics and Computation 211 (2009) 495–501
497
into the linear terms of (15), and solving the resulting equation we obtain the dispersion relation 3
c i ¼ ki ;
i ¼ 1; 2; . . . ; N;
ð17Þ
and hence hi becomes 3
hi ¼ ki x þ r i y þ si z ki t:
ð18Þ
Notice that the dispersion relation ci depends only on the coefficients ki of x and does not depend on the coefficients r i and si of the spatial variables y and z, respectively. To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð19Þ
3
where f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 t into Eq. (15) and solve to find that R ¼ 2. This means that the single-soliton solution is given by 3
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 t : ¼ 3 @x 1 þ ek1 xþk1 yþs1 zk1 t
ð20Þ
For the two-soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð21Þ
where
f ðx; y; z; tÞ ¼ 1 þ eh1 þ eh2 þ a12 eh1 þh2 ;
ð22Þ
into Eq. (15), where h1 and h2 are given in (18), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð23Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
1 6 i < j 6 N:
;
ð24Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variable x. Moreover, we point out that the first shallow water wave equation does not show any resonant phenomenon [9] because the phase shift term a12 in (23) cannot be 0 or 1 for jk1 j – jk2 j. This in turn gives 3
3
f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 t þ ek2 xþr2 yþs2 zk2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
3
3
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 þk2 Þt :
ð25Þ
To determine the two-soliton solutions explicitly, we substitute (25) into the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three-soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð26Þ
þ b123 expðh1 þ h2 þ h3 Þ; into (21) and substitute it into Eq. (15) to find that
b123 ¼ a12 a13 a23 :
ð27Þ
To determine the three-soliton solutions explicitly, we substitute the last result for f ðx; y; z; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the first (3 + 1)-dimensional shallow water wave equation (15) is completely integrable and gives rise to multiplesoliton solutions of any order. 3.2. Multiple-singular soliton solutions To determine multiple-singular soliton solutions we proceed as before to find that the dispersion relation is given by 3
c i ¼ ki ;
i ¼ 1; 2; . . . N;
ð28Þ
498
A.-M. Wazwaz / Applied Mathematics and Computation 211 (2009) 495–501
and hence hi becomes 3
hi ¼ ki x þ ri y þ si z ki t:
ð29Þ
To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð30Þ
3
where f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 t into Eq. (15) and solve to find that R ¼ 2. This means that the single singular soliton solution is given by 3
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 t : ¼ 3 @x 1 ek1 xþk1 yþs1 zk1 t
ð31Þ
The singular behavior arises when the denominator vanishes. For the two singular soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð32Þ
where
f ðx; y; z; tÞ ¼ 1 eh1 eh2 þ a12 eh1 þh2 ;
ð33Þ
into Eq. (15), where h1 and h2 are given in (29), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð34Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
;
1 6 i < j 6 N:
ð35Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variables x. This in turn gives 3
3
f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 t ek2 xþr2 yþs2 zk2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
3
3
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 þk2 Þt :
ð36Þ
To determine the two singular soliton solutions explicitly, we substitute (36) into the formula uðx; y; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three singular soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð37Þ
þ b123 expðh1 þ h2 þ h3 Þ; into (32) and substitute it into the Eq. (15) to find that
b123 ¼ a12 a13 a23 :
ð38Þ
To determine the three singular soliton solutions explicitly, we substitute the last result for f ðx; y; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the first (3 + 1)-dimensional shallow water wave equation (15) gives rise to multiple-soliton solutions and multiple-singular soliton solutions of any order. 4. The second (3 + 1)-dimensional equation In this section, we apply the Hirota’s bilinear method to the (3 + 1)-dimensional shallow water wave equation
uxzt þ uxxxyz 2ðuxx uyz þ uy uxxz Þ 4ðux uxyz þ uxz uxy Þ ¼ 0:
ð39Þ
It is interesting to point out that this equation can be reduced to the potential KdV equation by setting z ¼ y ¼ x and integrating twice with respect to x. 4.1. Multiple-soliton solutions To determine multiple-soliton solutions for Eq. (39), we follow the steps presented above. We first substitute
A.-M. Wazwaz / Applied Mathematics and Computation 211 (2009) 495–501
uðx; y; z; tÞ ¼ ehi ;
hi ¼ ki x þ r i y þ si z ci t;
499
ð40Þ
into the linear terms of the (39), and solving the resulting equation we obtain the dispersion relation 2
c i ¼ ki r i ;
i ¼ 1; 2; . . . ; N;
ð41Þ
and hence hi becomes 2
hi ¼ ki x þ r i y þ si z ki r i t:
ð42Þ
Notice that the dispersion relation ci depends only on the coefficients ki and r i of the spatial variables x and y, respectively, and does not depend on the coefficients si of the spatial variables z. To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð43Þ
2
where f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 r1 t into Eq. (39) and solve to find that R ¼ 2. This means that the single-soliton solution is given by 2
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 r1 t : ¼ 2 @x 1 þ ek1 xþk1 yþs1 zk1 r1 t
ð44Þ
For the two-soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð45Þ
where
f ðx; y; z; tÞ ¼ 1 þ eh1 þ eh2 þ a12 eh1 þh2 ;
ð46Þ
into Eq. (39), where h1 and h2 are given in (42), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð47Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
1 6 i < j 6 N:
;
ð48Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variable x. Moreover, we point out that the first shallow water wave equation does not show any resonant phenomenon [9] because the phase shift term a12 in (47) cannot be 0 or 1 for jk1 j – jk2 j. This in turn gives 2
2
f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 r1 t þ ek2 xþr2 yþs2 zk2 r2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ2
2
2
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 r1 þk2 r2 Þt :
ð49Þ
To determine the two-soliton solutions explicitly, we substitute (49) into the formula uðx; y; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three-soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð50Þ
þ b123 exp ðh1 þ h2 þ h3 Þ; into (45) and substitute it into Eq. (39) to find that
b123 ¼ a12 a13 a23 :
ð51Þ
To determine the three-soliton solutions explicitly, we substitute the last result for f ðx; y; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the second (3 + 1)-dimensional shallow water wave equation (39) is completely integrable and gives rise to multiple-soliton solutions of any order. An important remark can be made here. The single-soliton solution for Eq. (39) can be obtained always if the sum of the coefficients of the terms ðuxx uyz þ uy uxxz Þ and ðux uxyz þ uxz uxy Þ is always 6. In other words, if the coefficients of these two terms are given by n and n 6, for n P 0, only single-soliton solutions can be derived. However, for complete integrability, it is justified only if the coefficients are 2 and 4.
500
A.-M. Wazwaz / Applied Mathematics and Computation 211 (2009) 495–501
4.2. Multiple-singular soliton solutions To determine multiple-singular soliton solutions we proceed as before to find that the dispersion relation is given by 2
c i ¼ ki r i ;
i ¼ 1; 2; . . . ; N;
ð52Þ
and hence hi becomes 2
hi ¼ ki x þ ri y þ si z ki r i t:
ð53Þ
To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð54Þ
2
where f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 r1 t into Eq. (39) and solve to find that R ¼ 2. This means that the single singular soliton solution is given by 2
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 r1 t : ¼ 2 @x 1 ek1 xþk1 yþs1 zk1 r1 t
ð55Þ
The singularity comes when the denominator vanishes. For the two singular soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð56Þ
where
f ðx; y; z; tÞ ¼ 1 eh1 eh2 þ a12 eh1 þh2 ;
ð57Þ
into Eq. (39), where h1 and h2 are given in (53), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð58Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
;
1 6 i < j 6 N:
ð59Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variables x. This in turn gives 2
2
f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 r1 t ek2 xþr2 yþs2 zk2 r2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
2
2
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 r1 þk2 r2 Þt :
ð60Þ
To determine the two singular soliton solutions explicitly, we substitute (60) into the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three-soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð61Þ
þ b123 exp ðh1 þ h2 þ h3 Þ; into (56) and substitute it into Eq. (39) to find that
b123 ¼ a12 a13 a23 :
ð62Þ
To determine the three singular soliton solutions explicitly, we substitute the last result for f ðx; y; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the second (3 + 1)-dimensional shallow water wave gives rise to multiple-soliton solutions and multiple-singular soliton solutions of any order.
5. Discussion A combination of Hirota’s method and Hereman’s method were used to formally derive multiple-soliton solutions and multiple-singular soliton solutions of two completely integrable (3 + 1)-dimensional equations. The Hirota’s bilinear method
A.-M. Wazwaz / Applied Mathematics and Computation 211 (2009) 495–501
501
possesses significant features that make it ideal for the determination of multiple-soliton solutions for a wide class of nonlinear evolution equations. The study revealed the power of the bilinear method. References [1] P.A. Clarkson, E.L. Mansfield, On a shallow water wave equation, Nonlinearity 7 (1994) 975–1000. [2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, The Inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249–315. [3] R. Hirota, J. Satsuma, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn. 40 (2) (1976) 611–612. [4] R. Hirota, A new form of Bäcklund transformations and its relation to the inverse scattering problem, Prog. Theor. Phys. 52 (5) (1974) 1498–1512. [5] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004. [6] R. Hirota, Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (18) (1971) 1192–1194. [7] R. Hirota, Exact solutions of the modified Korteweg–de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (5) (1972) 1456–1458. [8] R. Hirota, Exact solutions of the Sine–Gordon equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (5) (1972) 1459–1463. [9] R. Hirota, M. Ito, Resonance of solitons in one dimension, J. Phys. Soc. Jpn. 52 (3) (1983) 744–748. [10] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28 (8) (1987) 1732– 1742. [11] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations, J. Math. Phys. 28 (9) (1987) 2094–2101. [12] W. Hereman, W. Zhaung, Symbolic software for soliton theory, Acta Applicandae Mathematicae, Phys. Lett. 76 (1980) 95–96. [13] W. Hereman, A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Sim. 43 (1997) 13–27. [14] A.M. Wazwaz, New solitary-wave special solutions with compact support for the nonlinear dispersive Kðm; nÞ equations, Chaos, Solitons and Fractals 13 (2) (2002) 321–330. [15] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method, Appl. Math. Comput. 190 (2007) 633–640. [16] A.M. Wazwaz, Multiple-front solutions for the Burgers equation and the coupled Burgers equations, Appl. Math. Comput. 190 (2007) 1198–1206. [17] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method, Appl. Math. Comput. 190 (2007) 633–640. [18] A.M. Wazwaz, Multiple-soliton solutions for the Boussinesq equation, Appl. Math. Comput. 192 (2007) 479–486. [19] A.M. Wazwaz, N-soliton solutions for the combined KdV–CDG equation and the KdV–Lax equation, Appl. Math. Comput. 203 (2008) 402–407. [20] A.M. Wazwaz, Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equation, Appl. Math. Comput. 203 (2008) 592–597. [21] A.M. Wazwaz, Single and multiple-soliton solutions for the (2 + 1)-dimensional KdV equation, Appl. Math. Comput. 204 (1) (2008) 20–26. [22] A.M. Wazwaz, Solitons and singular solitons for the Gardner–KP equation, Appl. Math. Comput. 204 (1) (2008) 162–169.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Multiple-soliton solutions and multiple-singular soliton solutions for two higher-dimensional shallow water wave equations Abdul-Majid Wazwaz Department of Mathematics, Saint Xavier University, Chicago, IL 60655, United States
a r t i c l e
i n f o
Keywords: Hirota bilinear method Complete integrability Multiple-soliton solutions Multiple-singular soliton solutions
a b s t r a c t Two (3 + 1)-dimensional shallow water wave equations are studied for complete integrability. The Hirota’s bilinear method is used to determine the multiple-soliton solutions for these equations. Moreover, multiple-singular soliton solutions will also be determined for each model. The analysis highlights the capability of the direct method in handling completely integrable equations. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The (3 + 1)-dimensional shallow water wave equations [1]
uyzt þ uxxxyz 6ux uxyz 6uxz uxy ¼ 0
ð1Þ
uxzt þ uxxxyz 2ðuxx uyz þ uy uxxz Þ 4ðux uxyz þ uxz uxy Þ ¼ 0
ð2Þ
and
will be studied. Both equations reduce to the potential KdV equation for z ¼ y ¼ x. The difference between the first terms of the two models is that x replaces y in the term uyzt . The generalized shallow water wave equations studied by Ablowitz [2] and Hirota and Satsuma [3], and by Mansfield and Clarkson [1] arise as reduction of these two equations. A variety of distinct methods are used for classification of integrable equations. The Painlevé analysis, the inverse scattering method, the Bäcklund transformation method, the conservation laws method, and the Hirota bilinear method [4–13] are mostly used in the literature for investigating complete integrability. The Hirota’s bilinear method is rather heuristic and possesses significant features that make it ideal for the determination of multiple-soliton solutions for a wide class of nonlinear evolution equations. Two goals are set for this work. We first aim to use Hirota’s bilinear method to emphasize its power and reliability for handling integrable equations. We second aim to determine multiple-soliton solutions and multiple-singular soliton solutions [14–22] for these Eqs. (1) and (2). The Hereman’s simplified form [12,13] will be combined with the Hirota’s method to achieve these two goals.
2. The Hirota’s bilinear method To formally derive N-soliton solutions for completely integrable equations, we will use the Hirota’s direct method combined with the simplified version of Hereman et al. [12,13]. It was proved by many that soliton solutions are just polynomials of exponentials. This will be also confirmed in the coming discussions.
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We first substitute
uðx; y; z; tÞ ¼ ekxþryþszct ;
ð3Þ
into the linear terms of the equation under discussion to determine the dispersion relation between k, r, s and c. We then substitute the single-soliton solution
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð4Þ
into the equation under discussion, where the auxiliary function f ðx; y; z; tÞ is given by
f ðx; y; z; tÞ ¼ 1 þ f1 ðx; y; z; tÞ ¼ 1 þ eh1 ;
ð5Þ
hi ¼ ki x þ ri y þ si z ci t;
ð6Þ
where
i ¼ 1; 2; . . . ; N;
and solving the resulting equation to determine the numerical value for R. Notice that the N-soliton solutions can be obtained for Eqs. (1) and (2), by using the following forms for f ðx; y; z; tÞ into (4): (i) For dispersion relation, we use
uðx; y; z; tÞ ¼ ehi ;
hi ¼ ki x þ r i y þ si z ci t:
ð7Þ
(ii) For single-soliton, we use
f ¼ 1 þ eh1 :
ð8Þ
(iii) For two-soliton solutions, we use
f ¼ 1 þ eh1 þ eh2 þ a12 eh1 þh2 :
ð9Þ
(iv) For three-soliton solutions, we use
f ¼ 1 þ eh1 þ eh2 þ eh3 þ a12 eh1 þh2 þ a23 eh2 þh3 þ a13 eh1 þh3 þ b123 eh1 þh2 þh3 :
ð10Þ
Notice that we use (7) to determine the dispersion relation, (9) to determine the factor a12 to be generalized for the other factors aij , and finally we use (10) to determine b123 , which is given by b123 ¼ a12 a23 a13 for completely integrable equations. The determination of three-soliton solutions confirms the fact that N-soliton solutions exist for any order. In the following, we will apply the aforementioned steps to the (3 + 1)-dimensional shallow water wave Eqs. (1) and (2). However, for the multiple-singular soliton solutions, we follow the following steps: (i) For dispersion relation, we use
uðx; tÞ ¼ ehi ;
hi ¼ ki x xi t:
ð11Þ
(ii) For single-soliton, we use
f ðx; tÞ ¼ 1 eh1 :
ð12Þ
(iii) For two-soliton solutions, we use
f ðx; tÞ ¼ 1 eh1 eh2 þ a12 eh1 þh2 :
ð13Þ
(iv) For three-soliton solutions, we use
f ðx; tÞ ¼ 1 eh1 eh2 eh3 þ a12 eh1 þh2 þ a23 eh2 þh3 þ a13 eh1 þh3 b123 eh1 þh2 þh3 :
ð14Þ
3. The first (3 + 1)-dimensional equation In this section, we apply the Hirota’s bilinear method to the (3 + 1)-dimensional shallow water wave equation
uyzt þ uxxxyz 6ux uxyz 6uxz uxy ¼ 0:
ð15Þ
It is interesting to point out that this equation can be reduced to the potential KdV equation by setting z ¼ y ¼ x and integrating twice with respect to x. 3.1. Multiple-soliton solutions To determine multiple-soliton solutions for Eq. (15), we follow the steps presented above. We first substitute
uðx; y; z; tÞ ¼ ehi ;
hi ¼ ki x þ r i y þ si z ci t;
ð16Þ
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497
into the linear terms of (15), and solving the resulting equation we obtain the dispersion relation 3
c i ¼ ki ;
i ¼ 1; 2; . . . ; N;
ð17Þ
and hence hi becomes 3
hi ¼ ki x þ r i y þ si z ki t:
ð18Þ
Notice that the dispersion relation ci depends only on the coefficients ki of x and does not depend on the coefficients r i and si of the spatial variables y and z, respectively. To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð19Þ
3
where f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 t into Eq. (15) and solve to find that R ¼ 2. This means that the single-soliton solution is given by 3
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 t : ¼ 3 @x 1 þ ek1 xþk1 yþs1 zk1 t
ð20Þ
For the two-soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð21Þ
where
f ðx; y; z; tÞ ¼ 1 þ eh1 þ eh2 þ a12 eh1 þh2 ;
ð22Þ
into Eq. (15), where h1 and h2 are given in (18), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð23Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
1 6 i < j 6 N:
;
ð24Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variable x. Moreover, we point out that the first shallow water wave equation does not show any resonant phenomenon [9] because the phase shift term a12 in (23) cannot be 0 or 1 for jk1 j – jk2 j. This in turn gives 3
3
f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 t þ ek2 xþr2 yþs2 zk2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
3
3
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 þk2 Þt :
ð25Þ
To determine the two-soliton solutions explicitly, we substitute (25) into the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three-soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð26Þ
þ b123 expðh1 þ h2 þ h3 Þ; into (21) and substitute it into Eq. (15) to find that
b123 ¼ a12 a13 a23 :
ð27Þ
To determine the three-soliton solutions explicitly, we substitute the last result for f ðx; y; z; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the first (3 + 1)-dimensional shallow water wave equation (15) is completely integrable and gives rise to multiplesoliton solutions of any order. 3.2. Multiple-singular soliton solutions To determine multiple-singular soliton solutions we proceed as before to find that the dispersion relation is given by 3
c i ¼ ki ;
i ¼ 1; 2; . . . N;
ð28Þ
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and hence hi becomes 3
hi ¼ ki x þ ri y þ si z ki t:
ð29Þ
To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð30Þ
3
where f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 t into Eq. (15) and solve to find that R ¼ 2. This means that the single singular soliton solution is given by 3
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 t : ¼ 3 @x 1 ek1 xþk1 yþs1 zk1 t
ð31Þ
The singular behavior arises when the denominator vanishes. For the two singular soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð32Þ
where
f ðx; y; z; tÞ ¼ 1 eh1 eh2 þ a12 eh1 þh2 ;
ð33Þ
into Eq. (15), where h1 and h2 are given in (29), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð34Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
;
1 6 i < j 6 N:
ð35Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variables x. This in turn gives 3
3
f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 t ek2 xþr2 yþs2 zk2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
3
3
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 þk2 Þt :
ð36Þ
To determine the two singular soliton solutions explicitly, we substitute (36) into the formula uðx; y; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three singular soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð37Þ
þ b123 expðh1 þ h2 þ h3 Þ; into (32) and substitute it into the Eq. (15) to find that
b123 ¼ a12 a13 a23 :
ð38Þ
To determine the three singular soliton solutions explicitly, we substitute the last result for f ðx; y; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the first (3 + 1)-dimensional shallow water wave equation (15) gives rise to multiple-soliton solutions and multiple-singular soliton solutions of any order. 4. The second (3 + 1)-dimensional equation In this section, we apply the Hirota’s bilinear method to the (3 + 1)-dimensional shallow water wave equation
uxzt þ uxxxyz 2ðuxx uyz þ uy uxxz Þ 4ðux uxyz þ uxz uxy Þ ¼ 0:
ð39Þ
It is interesting to point out that this equation can be reduced to the potential KdV equation by setting z ¼ y ¼ x and integrating twice with respect to x. 4.1. Multiple-soliton solutions To determine multiple-soliton solutions for Eq. (39), we follow the steps presented above. We first substitute
A.-M. Wazwaz / Applied Mathematics and Computation 211 (2009) 495–501
uðx; y; z; tÞ ¼ ehi ;
hi ¼ ki x þ r i y þ si z ci t;
499
ð40Þ
into the linear terms of the (39), and solving the resulting equation we obtain the dispersion relation 2
c i ¼ ki r i ;
i ¼ 1; 2; . . . ; N;
ð41Þ
and hence hi becomes 2
hi ¼ ki x þ r i y þ si z ki r i t:
ð42Þ
Notice that the dispersion relation ci depends only on the coefficients ki and r i of the spatial variables x and y, respectively, and does not depend on the coefficients si of the spatial variables z. To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð43Þ
2
where f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 r1 t into Eq. (39) and solve to find that R ¼ 2. This means that the single-soliton solution is given by 2
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 r1 t : ¼ 2 @x 1 þ ek1 xþk1 yþs1 zk1 r1 t
ð44Þ
For the two-soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð45Þ
where
f ðx; y; z; tÞ ¼ 1 þ eh1 þ eh2 þ a12 eh1 þh2 ;
ð46Þ
into Eq. (39), where h1 and h2 are given in (42), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð47Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
1 6 i < j 6 N:
;
ð48Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variable x. Moreover, we point out that the first shallow water wave equation does not show any resonant phenomenon [9] because the phase shift term a12 in (47) cannot be 0 or 1 for jk1 j – jk2 j. This in turn gives 2
2
f ðx; y; z; tÞ ¼ 1 þ ek1 xþr1 yþs1 zk1 r1 t þ ek2 xþr2 yþs2 zk2 r2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ2
2
2
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 r1 þk2 r2 Þt :
ð49Þ
To determine the two-soliton solutions explicitly, we substitute (49) into the formula uðx; y; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three-soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð50Þ
þ b123 exp ðh1 þ h2 þ h3 Þ; into (45) and substitute it into Eq. (39) to find that
b123 ¼ a12 a13 a23 :
ð51Þ
To determine the three-soliton solutions explicitly, we substitute the last result for f ðx; y; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the second (3 + 1)-dimensional shallow water wave equation (39) is completely integrable and gives rise to multiple-soliton solutions of any order. An important remark can be made here. The single-soliton solution for Eq. (39) can be obtained always if the sum of the coefficients of the terms ðuxx uyz þ uy uxxz Þ and ðux uxyz þ uxz uxy Þ is always 6. In other words, if the coefficients of these two terms are given by n and n 6, for n P 0, only single-soliton solutions can be derived. However, for complete integrability, it is justified only if the coefficients are 2 and 4.
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4.2. Multiple-singular soliton solutions To determine multiple-singular soliton solutions we proceed as before to find that the dispersion relation is given by 2
c i ¼ ki r i ;
i ¼ 1; 2; . . . ; N;
ð52Þ
and hence hi becomes 2
hi ¼ ki x þ ri y þ si z ki r i t:
ð53Þ
To determine R, we substitute
uðx; y; z; tÞ ¼ R
@ ln f ðx; y; z; tÞ fx ðx; y; z; tÞ ¼R ; @x f ðx; y; z; tÞ
ð54Þ
2
where f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 r1 t into Eq. (39) and solve to find that R ¼ 2. This means that the single singular soliton solution is given by 2
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ 2k1 ek1 xþk1 yþs1 zk1 r1 t : ¼ 2 @x 1 ek1 xþk1 yþs1 zk1 r1 t
ð55Þ
The singularity comes when the denominator vanishes. For the two singular soliton solutions, we substitute
uðx; y; z; tÞ ¼ 2
@ ln f ðx; y; z; tÞ ; @x
ð56Þ
where
f ðx; y; z; tÞ ¼ 1 eh1 eh2 þ a12 eh1 þh2 ;
ð57Þ
into Eq. (39), where h1 and h2 are given in (53), to obtain
a12 ¼
ðk1 k2 Þ2 ðk1 þ k2 Þ2
ð58Þ
;
and hence
aij ¼
ðki kj Þ2 ðki þ kj Þ2
;
1 6 i < j 6 N:
ð59Þ
It is clear that the phase shifts aij ; 1 6 i < j 6 N depend only on the coefficients km of the spatial variables x. This in turn gives 2
2
f ðx; y; z; tÞ ¼ 1 ek1 xþr1 yþs1 zk1 r1 t ek2 xþr2 yþs2 zk2 r2 t þ
ðk1 k2 Þ2 ðk1 þ k2 Þ
2
2
2
eðk1 þk2 Þxþðr1 þr2 Þyþðs1 þs2 Þzðk1 r1 þk2 r2 Þt :
ð60Þ
To determine the two singular soliton solutions explicitly, we substitute (60) into the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . Similarly, to determine the three-soliton solutions, we set
f ðx; y; z; tÞ ¼ 1 þ expðh1 Þ þ expðh2 Þ þ expðh3 Þ þ a12 expðh1 þ h2 Þ þ a23 expðh2 þ h3 Þ þ a13 expðh1 þ h3 Þ
ð61Þ
þ b123 exp ðh1 þ h2 þ h3 Þ; into (56) and substitute it into Eq. (39) to find that
b123 ¼ a12 a13 a23 :
ð62Þ
To determine the three singular soliton solutions explicitly, we substitute the last result for f ðx; y; tÞ in the formula uðx; y; z; tÞ ¼ 2ðln f ðx; y; z; tÞÞx . The higher level soliton solutions, for n P 4 can be obtained in a parallel manner. This confirms that the second (3 + 1)-dimensional shallow water wave gives rise to multiple-soliton solutions and multiple-singular soliton solutions of any order.
5. Discussion A combination of Hirota’s method and Hereman’s method were used to formally derive multiple-soliton solutions and multiple-singular soliton solutions of two completely integrable (3 + 1)-dimensional equations. The Hirota’s bilinear method
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possesses significant features that make it ideal for the determination of multiple-soliton solutions for a wide class of nonlinear evolution equations. The study revealed the power of the bilinear method. References [1] P.A. Clarkson, E.L. Mansfield, On a shallow water wave equation, Nonlinearity 7 (1994) 975–1000. [2] M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, The Inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249–315. [3] R. Hirota, J. Satsuma, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn. 40 (2) (1976) 611–612. [4] R. Hirota, A new form of Bäcklund transformations and its relation to the inverse scattering problem, Prog. Theor. Phys. 52 (5) (1974) 1498–1512. [5] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004. [6] R. Hirota, Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (18) (1971) 1192–1194. [7] R. Hirota, Exact solutions of the modified Korteweg–de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (5) (1972) 1456–1458. [8] R. Hirota, Exact solutions of the Sine–Gordon equation for multiple collisions of solitons, J. Phys. Soc. Jpn. 33 (5) (1972) 1459–1463. [9] R. Hirota, M. Ito, Resonance of solitons in one dimension, J. Phys. Soc. Jpn. 52 (3) (1983) 744–748. [10] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28 (8) (1987) 1732– 1742. [11] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations, J. Math. Phys. 28 (9) (1987) 2094–2101. [12] W. Hereman, W. Zhaung, Symbolic software for soliton theory, Acta Applicandae Mathematicae, Phys. Lett. 76 (1980) 95–96. [13] W. Hereman, A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Sim. 43 (1997) 13–27. [14] A.M. Wazwaz, New solitary-wave special solutions with compact support for the nonlinear dispersive Kðm; nÞ equations, Chaos, Solitons and Fractals 13 (2) (2002) 321–330. [15] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method, Appl. Math. Comput. 190 (2007) 633–640. [16] A.M. Wazwaz, Multiple-front solutions for the Burgers equation and the coupled Burgers equations, Appl. Math. Comput. 190 (2007) 1198–1206. [17] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method, Appl. Math. Comput. 190 (2007) 633–640. [18] A.M. Wazwaz, Multiple-soliton solutions for the Boussinesq equation, Appl. Math. Comput. 192 (2007) 479–486. [19] A.M. Wazwaz, N-soliton solutions for the combined KdV–CDG equation and the KdV–Lax equation, Appl. Math. Comput. 203 (2008) 402–407. [20] A.M. Wazwaz, Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equation, Appl. Math. Comput. 203 (2008) 592–597. [21] A.M. Wazwaz, Single and multiple-soliton solutions for the (2 + 1)-dimensional KdV equation, Appl. Math. Comput. 204 (1) (2008) 20–26. [22] A.M. Wazwaz, Solitons and singular solitons for the Gardner–KP equation, Appl. Math. Comput. 204 (1) (2008) 162–169.