Applied Mathematics and Computation 215 (2009) 1548–1552

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

A (3 + 1)-dimensional nonlinear evolution equation with multiplesoliton solutions and multiple singular soliton solutions

Abdul-Majid WazwazDepartment of Mathematics, Saint Xavier University, Chicago, IL 60655, USA

a r t i c l e i n f o

Keywords:Nonlinear (3 + 1)-dimensional equationHirota bilinear methodMultiple soliton solutionsMultiple singular soliton solutionsResonance

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.07.008

E-mail address: wazwaz@sxu.edu

a b s t r a c t

In this work, a (3 + 1)-dimensional nonlinear evolution equation is investigated. The Hiro-ta’s bilinear method is applied to determine the necessary conditions for the complete inte-grability of this equation. Multiple soliton solutions are established to confirm thecompatibility structure. Multiple singular soliton solutions are also derived. The resonancephenomenon does not exist for this model.

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1. Introduction

For completely integrable evolution equations, many powerful methods were developed and used. The algebraic-geomet-ric method [1,2], the inverse scattering method, the Bäcklund transformation method, the Darboux transformation method,the Hirota bilinear method [3–13], and others were thoroughly used to derive the multiple soliton solutions of these equa-tions. The Hirota’s bilinear method is rather heuristic and possesses significant features that make it practical for the deter-mination of multiple soliton solutions [3–13], and for multiple singular soliton solutions [14–26] for a wide class of nonlinearevolution equations in a direct method. The computer symbolic systems such as Maple and Mathematica allow us to performcomplicated and tedious calculations.

In this work, a (3 + 1)-dimensional nonlinear evolution equation [1,2]

3wxz � ð2wt þwxxx � awwx � bwwyÞy þ cðwx@�1x wyÞx ¼ 0; ð1Þ

will be studied to establish N-soliton solutions in order to confirm its integrability, where a; b; and c are constants, whereaþ bþ – 0. The inverse operator @�1

x is defined by

ð@�1x f ÞðxÞ ¼

Z x

�1f ðtÞdt; ð2Þ

under the decaying condition at infinity. Note that @x@�1x ¼ @

�1x @x ¼ 1. The necessary conditions for the integrability of Eq. (1)

will be determined. For a ¼ 2; b ¼ 0; and c ¼ 2, Eq. (1) reduces to the model solved in [1,2]. In Ref. [1], the (3 + 1)-dimensionalnonlinear evolution equation is decomposed into systems of solvable ordinary differential equations with the help of the(1 + 1)-dimensional AKNS equation. In Ref. [2], the perturbation technique and the Wronskian determinant of solutions wereused to determine the N-soliton solutions for the reduced equation.

Our aim is to derive multiple regular soliton solutions and multiple singular soliton solutions for the (3 + 1)-dimensionalEq. (1). Hirota and Ito [3] examined the phenomenon of two solitons near resonant state, two solitons at the resonant state,and two solitons become after colliding with each other, where it was proved in [3] that two solitons become singular after

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A.-M. Wazwaz / Applied Mathematics and Computation 215 (2009) 1548–1552 1549

colliding with each other, in the sense that regular solitons with sech2 profiles are transmitted into singular solitons withcosech2 profiles through the interaction.

2. The Hirota’s bilinear method

The Hirota’s direct method is well known, and it gives soliton solutions by polynomials of exponentials. We only summa-rize the necessary steps, where details can be found in [3–26] among many others.

We first substitute

uðx; y; z; tÞ ¼ ekxþryþsz�xt; ð3Þ

into the linear terms of the equation under discussion to determine the dispersion relation between k; r; s and c. We thensubstitute the single soliton solution

uðx; y; z; tÞ ¼ R@ ln f ðx; y; z; tÞ

@x¼ R

fxðx; y; z; tÞf 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 1þ C1f1ðx; y; z; tÞ ¼ 1þ C1eh1 ; ð5Þ

where

hi ¼ kixþ riyþ siz�xit; i ¼ 1;2; . . . ;N; ð6Þ

and solving the resulting equation to determine the numerical value for R. Notice that the N-soliton solutions can be ob-tained for Eq. (1), 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 kixþ riyþ siz�xit: ð7Þ

(ii) For single soliton, we use

f ðx; y; z; tÞ ¼ 1þ C1eh1 : ð8Þ

(iii) For two-soliton solutions, we use

f ðx; y; z; tÞ ¼ 1þ C1eh1 þ C2eh2 þ C1C2a12eh1þh2 : ð9Þ

(iv) For three-soliton solutions, we use

f ðx; y; z; tÞ ¼ 1þ C1eh1 þ C2eh2 þ C3eh3 þ C1C2a12eh1þh2 þ C2C3a23eh2þh3 þ C1C3a13eh1þh3 þ C1C2C3b123eh1þh2þh3 : ð10Þ

Notice that we use Eq. (7) to determine the dispersion relation, Eq. (9) to determine the phase shift a12 to be generalized forthe other factors aij, and finally we use Eq. (10) to determine b123, which is given by b123 ¼ a12a23a13 for completely integrableequations. The determination of three-soliton solutions confirms the fact that N-soliton solutions exist for any order. In thefollowing, we will apply the aforementioned steps to Eq. (1). Multiple soliton solutions are obtained for C1 ¼ C2 ¼ C3 ¼ þ1.However, multiple singular soliton solutions [3,14–26] are obtained if C1 ¼ C2 ¼ C3 ¼ �1.

3. Multiple soliton solutions

In this section we apply the Hirota’s bilinear method for C1 ¼ C2 ¼ C3 ¼ þ1 to study the (3 + 1)-dimensional nonlinearequation

3wxz � ð2wt þwxxx � awwx � bwwyÞy þ cðwx@�1x wyÞx ¼ 0: ð11Þ

We first remove the integral term in (11) by introducing the potential

wðx; y; z; tÞ ¼ uxðx; y; z; tÞ; ð12Þ

to carry (11) to the equation

3uxxz � ð2uxt þ uxxxx � auxuxx � buxuxyÞy þ cðuxxuyÞx ¼ 0: ð13Þ

Substituting

uðx; y; z; tÞ ¼ ehi ; hi ¼ kixþ riyþ siz�xit; ð14Þ

into the linear terms of (13), and solving the resulting equation for xi we obtain the dispersion relation

1550 A.-M. Wazwaz / Applied Mathematics and Computation 215 (2009) 1548–1552

xi ¼12

k3i �

3kisi

2ri; i ¼ 1;2; . . . N; ð15Þ

and hence hi becomes

hi ¼ kixþ riyþ siz�12

k3i �

3kisi

2ri

� �t: ð16Þ

To determine R, we substitute

uðx; y; z; tÞ ¼ R@ ln f ðx; y; z; tÞ

@x¼ R

fxðx; y; z; tÞf ðx; y; z; tÞ ; ð17Þ

where

f ðx; y; z; tÞ ¼ 1þ ek1xþr1yþs1z� 1

2k31�

3k1 s12r1

� �t; ð18Þ

into Eq. (13) and solve to find that

R ¼ � 12aþ bþ c

; aþ bþ c – 0: ð19Þ

Moreover, the complete integrability is satisfied only if

ri ¼ ki; si ¼ k3i : ð20Þ

The last results reduce the dispersion relation (15) to

xi ¼ �k3i ; i ¼ 1;2; . . . N: ð21Þ

Consequently, the auxiliary function (18) becomes

f ðx; y; z; tÞ ¼ 1þ ek1xþk1yþk31zþk3

1t : ð22Þ

Substituting (22) into (17) gives

uðx; y; z; tÞ ¼ � 12aþ bþ c

@ ln f ðx; y; z; tÞ@x

¼ � 12k1ek1xþk1yþk31zþk3

1t

ðaþ bþ cÞ 1þ ek1xþk1yþk31zþk3

1 t� � : ð23Þ

Consequently, the single soliton solution for the (3 + 1)-dimensional nonlinear evolution Eq. (11) is given by

wðx; y; z; tÞ ¼ � 12k21ek1xþk1yþk3

1zþk31t

ðaþ bþ cÞ 1þ ek1xþk1yþk31zþk3

1 t� �2 ; ð24Þ

obtained upon using the potential defined in (12).For the two-soliton solutions, we substitute

uðx; y; z; tÞ ¼ � 12aþ bþ c

@ ln f ðx; y; z; tÞ@x

; ð25Þ

where

f ðx; y; z; tÞ ¼ 1þ eh1 þ eh2 þ a12eh1þh2 ; ð26Þ

into Eq. (13), where h1 and h2 are given in (16), to obtain the phase shift a12 by

a12 ¼ðk1 � k2Þ2

ðk1 þ k2Þ2; ð27Þ

and hence

aij ¼ðki � kjÞ2

ðki þ kjÞ2; 1 6 i < j 6 2: ð28Þ

We point out that the (3 + 1)-dimensional Eq. (11) does not show any resonant phenomenon [3] because the phase shift terma12 in (27) cannot be 0 or 1 for jk1j– jk2j.

This in turn gives

f ðx; y; z; tÞ ¼ 1þ ek1xþk1yþk31zþk3

1t þ ek2xþk2yþk32zþk3

2t þ ðk1 � k2Þ2

ðk1 þ k2Þ2eðk1þk2Þxþðk1þk2Þyþðk3

1þk32Þzþðk

31þk3

2Þt : ð29Þ

A.-M. Wazwaz / Applied Mathematics and Computation 215 (2009) 1548–1552 1551

To determine the two-soliton solutions explicitly, we substitute (29) into the formula uðx; y; z; tÞ ¼ � 12aþbþc ðln f ðx; y; z; tÞÞx, and

then we use the potential wðx; y; z; tÞ ¼ uxðx; y; z; tÞ as defined in (12).To determine the three soliton solutions, we substitute the auxiliary function

f ðx; y; z; tÞ ¼ 1þ expðh1Þ þ expðh2Þ þ expðh3Þ þ a12 expðh1 þ h2Þ þ a23 expðh2 þ h3Þ þ a13 expðh1 þ h3Þþ b123 expðh1 þ h2 þ h3Þ; ð30Þ

and proceed as before to find that

b123 ¼ a12a13a23: ð31Þ

To determine the three-soliton solutions explicitly, we substitute the last result for f ðx; y; z; tÞ in the formulawðx; y; z; tÞ ¼ � 12

aþbþc ðln f ðx; y; z; tÞÞxx. The higher level soliton solutions, for N P 4 can be obtained in a parallel manner. Thisconfirms that the (3 + 1)-dimensional nonlinear evolution Eq. (13) is completely integrable and gives rise to multiple solitonsolutions of any order.

4. Multiple singular soliton solutions

In this section we follow the scheme presented in [3,14–26] where C1 ¼ C2 ¼ C3 ¼ �1 to derive multiple singular solitonsolutions for the (3 + 1)-dimensional nonlinear equation

Proceeding as before, we substitute

uðx; y; z; tÞ ¼ R@ ln f ðx; y; z; tÞ

@x¼ R

fxðx; y; z; tÞf ðx; y; z; tÞ ; ð32Þ

where

f ðx; y; z; tÞ ¼ 1� ek1xþk1yþk31zþk3

1t; ð33Þ

where the dispersion relation is given by

xi ¼ �k3i ; i ¼ 1;2; . . . N: ð34Þ

Substituting (33) into (32) gives

uðx; y; z; tÞ ¼ � 12aþ bþ c

@ ln f ðx; y; z; tÞ@x

¼ 12k1ek1xþk1yþk31zþk3

1t

ðaþ bþ cÞ 1� ek1xþk1yþk31zþk3

1 t� � : ð35Þ

Consequently, the single singular soliton solution for the (3 + 1)-dimensional nonlinear evolution (11) is given by

wðx; y; z; tÞ ¼ 12k21ek1xþk1yþk3

1zþk31t

ðaþ bþ cÞ 1� ek1xþk1yþk31zþk3

1t� �2 ; ð36Þ

obtained upon using the potential defined in (12).For the two singular soliton solutions, we substitute

uðx; y; z; tÞ ¼ � 12aþ bþ c

@ ln f ðx; y; z; tÞ@x

; ð37Þ

where

f ðx; y; z; tÞ ¼ 1� eh1 � eh2 þ a12eh1þh2 ; ð38Þ

into Eq. (13), where h1 and h2 are defined above, to obtain

a12 ¼ðk1 � k2Þ2

ðk1 þ k2Þ2; ð39Þ

and hence

aij ¼ðki � kjÞ2

ðki þ kjÞ2; 1 6 i < j 6 2: ð40Þ

This in turn gives

f ðx; y; z; tÞ ¼ 1� ek1xþk1yþk31zþk3

1t � ek2xþk2yþk32zþk3

2t þ ðk1 � k2Þ2

ðk1 þ k2Þ2eðk1þk2Þxþðk1þk2Þyþðk3

1þk32Þzþðk

31þk3

2Þt : ð41Þ

1552 A.-M. Wazwaz / Applied Mathematics and Computation 215 (2009) 1548–1552

To determine the two-soliton solutions explicitly, we substitute (41) into (37), and then we use the potentialwðx; y; z; tÞ ¼ uxðx; y; z; tÞ as defined in (??).

To determine the three singular soliton solutions, we use the auxiliary function

f ðx; y; z; tÞ ¼ 1� expðh1Þ � expðh2Þ � expðh3Þ þ a12 expðh1 þ h2Þ þ a23 expðh2 þ h3Þ þ a13 expðh1 þ h3Þ� b123 expðh1 þ h2 þ h3Þ; ð42Þ

and proceed as before to find that

b123 ¼ a12a13a23: ð43Þ

To determine the three singular soliton solutions explicitly, we substitute the last result for f ðx; y; z; tÞ in the formulawðx; y; z; tÞ ¼ � 12

aþbþc ðln f ðx; y; z; tÞÞxx. The higher level singular soliton solutions, for N P 4 can be obtained in a parallel man-ner. This confirms that the (3 + 1)-dimensional nonlinear evolution Eq. (13) is gives rise to multiple singular soliton solutionsof any order.

5. Discussion

In this work we emphasized the complete integrability of the (3 + 1)-dimensional nonlinear evolution equation by deter-mining the necessary conditions between the coefficients of the spatial variables. Multiple soliton solutions were formallyderived. Moreover, multiple singular soliton solutions of any order was derived as well. The results of other works are specialcases of our results.

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