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Research Articleð¹-Expansion Method and New Exact Solutions ofthe Schrödinger-KdV Equation
Ali Filiz,1 Mehmet Ekici,2 and Abdullah Sonmezoglu2
1 Department of Mathematics, Faculty of Science and Arts, Adnan Menderes University, 09010 Aydin, Turkey2Department of Mathematics, Faculty of Science and Arts, Bozok University, 66100 Yozgat, Turkey
Correspondence should be addressed to Ali Filiz; afiliz@adu.edu.tr
Received 17 August 2013; Accepted 27 October 2013; Published 29 January 2014
Academic Editors: A. K. Sharma and C. Yiu
Copyright © 2014 Ali Filiz et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ð¹-expansionmethod is proposed to seek exact solutions of nonlinear evolution equations.With the aid of symbolic computation, wechoose the SchroÌdinger-KdV equation with a source to illustrate the validity and advantages of the proposed method. A number ofJacobi-elliptic function solutions are obtained including theWeierstrass-elliptic function solutions.When themodulusm of Jacobi-elliptic function approaches to 1 and 0, soliton-like solutions and trigonometric-function solutions are also obtained, respectively.The proposed method is a straightforward, short, promising, and powerful method for the nonlinear evolution equations inmathematical physics.
1. Introduction
Nonlinear evolution equations are widely used to describecomplex phenomena in many scientific and engineeringfields, such as fluid dynamics, plasma physics, hydrodynam-ics, solid state physics, optical fibers, and acoustics.Therefore,finding solutions of such nonlinear evolution equations isimportant. However, determining solutions of nonlinearevolution equations is a very difficult task and only in certaincases one can obtain exact solutions. Recently,many powerfulmethods to obtain exact solutions of nonlinear evolutionequations have been proposed, such as the inverse scatteringmethod [1], the BaÌcklund transformation method [2, 3], theHirota bilinear scheme [4, 5], the Painlev expansion [6],the homotopy perturbation method [7, 8], the homogenousbalance method [9], the variational method [10â12], thetanh function method [13â16], the trial function and thesine-cosine method [17], (ðº/ðº)-expansion method [18, 19],the trial equation method [20â28], the auxiliary equationmethod [29], the Jacobian-elliptic function method [30â33], the ð¹-expansion method [34â38], and the Exp-functionmethod [39â42].
In the present research, we shall apply the the ð¹-expan-sion method to obtain 52 types of exact solution: six for
the Weierstrass-elliptic function solutions and the rest forJacobian-elliptic function solutions of the Schrdinger-KdVequation:
ðð¢ð¡= ð¢
ð¥ð¥+ ð¢V, V
ð¡+ 6VV
ð¥+ V
ð¥ð¥ð¥= (|ð¢|
2)ð¥. (1)
Among themethodsmentioned above, the auxiliary equationmethod [29] is based on the assumption that the travellingwave solutions are in the form
ð¢ (ð) =
ð
â
ð=0
ððð§ð(ð) , ð = ðŒ (ð¥ â ðœð¡) , (2)
where ð§(ð) satisfies the following auxiliary ordinary differen-tial equation:
(ðð§
ðð)
2
= ðð§2(ð) + ðð§
3(ð) + ðð§
4(ð) , (3)
where ð, ð, and ð are real parameters. Although many exactsolutions were obtained in [29] via the auxiliary equation (3),all these solutions are expressed only in terms of hyperbolicand trigonometric functions. In this paper, we want togeneralize the work in [29]. We propose a new auxiliary
Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 534063, 14 pageshttp://dx.doi.org/10.1155/2014/534063
2 The Scientific World Journal
equation which has more general exact solutions in termsof Jacobian-elliptic and the Weierstrass-elliptic functions.Moreover, many exact solutions in terms of hyperbolic andtrigonometric functions can be also obtained when themodulus of Jacobian-elliptic functions tends to one and zero,respectively.
The rest of the paper is arranged as follows. In Section 2,we briefly describe the auxiliary equation method (ð¹-expansion method) for nonlinear evolution equations. Byusing the method proposed in Section 2, Jacobian-ellipticand theWeierstrass-elliptic functions solutions are presentedin Sections 3 and 4, respectively. Soliton-like solutions andtrigonometric-function solutions are listed in Sections 5 and6, respectively. Some conclusions are given in Section 7. Thepaper is ended by Appendices AâD which play an importantrole in obtaining the solutions.
2. Description of the ð¹-Expansion Method
Consider a nonlinear partial differential equation (PDE) withindependent variables ð¥, ð¡ and dependent variable ð¢:
ð(ð¢, ð¢ð¡, ð¢ð¥, ð¢ð¥ð¥, . . .) = 0. (4)
Assume that ð¢(ð¥, ð¡) = ð¢(ð), where the wave variable ð =ð¥ + ðð¡. By this, the nonlinear PDE (4) reduces to an ordinarydifferential equation (ODE):
ð(ð¢, ðð¢, ð¢, ð¢, . . .) = 0. (5)
Then we seek its solutions in the form
ð¢ (ð) =
ð
â
ð=0
ððð¹ð
(ð) , (6)
where ðð, ð = 0, 1, 2, . . . , ð, are constants to be determined,ð
is a positive integer which can be evaluated by balancing thehighest order nonlinear term(s) and the highest order partialderivative of ð¢ in (4), andð¹(ð) satisfies the following auxiliaryequation:
ð¹
(ð) = ðâðð¹4(ð) + ðð¹
2(ð) + ð , (7)
where ð = ±1 andð,ð, andð are constants.The last equationhence holds for ð¹(ð):
ð¹= 2ðð¹
3+ ðð¹,
ð¹= (6ðð¹
2+ ð)ð¹
,
ð¹(4)= 24ð
2ð¹5+ 20ððð¹
3+ (12ðð + ð
2) ð¹,
ð¹(5)= (120ð
2ð¹4+ 60ððð¹
2+ 12ðð + ð
2) ð¹
...
(8)
In Appendices A and B, we present 52 types of exact solutionfor (7) (see [34â37, 43] for details). In fact, these exactsolutions can be used to construct more exact solutions for(1).
3. New Exact Jacobian-Elliptic FunctionSolutions of the Schrödinger-KdV Equation
The coupled SchroÌdinger-KdV equation
ðð¢ð¡â ð¢
ð¥ð¥â ð¢V = 0, V
ð¡+ 6VV
ð¥+ V
ð¥ð¥ð¥â (|ð¢|
2)ð¥= 0 (9)
is known to describe various processes in dusty plasma, suchas Langmuir, dust-acoustic wave, and electromagnetic waves[44â47]. Exact solution of (9) was studied by many authors[48â51]. Here the ð¹-expansion method is applied to system(9) and gives some new solutions. Let
ð¢ = ðððð (ð) , V = ð (ð) ,
ð = ðŒð¥ + ðœð¡, ð = ð¥ + ðð¡,
(10)
where ðŒ, ðœ, and ð are constants.Substituting (10) into (9), we find that ð = 2ðŒ and ð, ð
satisfy the following coupled nonlinear ordinary differentialsystem:
ð+ (ðœ â ðŒ
2)ð + ðð = 0,
2ðŒð+ 6ðð
+ ð
â (ð
2)
= 0.
(11)
Balancing the highest nonlinear terms and the highest orderderivative terms in (11), we find ð = 2 and ð = 2. Therefore,we suppose that the solution of (11) can be expressed by
ð (ð) = ð0+ ð
1ð¹ (ð) + ð
2ð¹2
(ð) ,
ð (ð) = ð0+ ð1ð¹ (ð) + ð
2ð¹2
(ð) ,
(12)
where ð0, ð1, ð2, ð0, ð1, and ð
2are constants to be determined
later and ð¹(ð) is a solution of ODE (7). Inserting (12) into(11) with the aid of (7), the left-hand side of (11) becomespolynomials inð¹(ð) if cancelingð¹ and setting the coefficientsof the polynomial to zero yields a set of algebraic equations,ð0, ð1, ð2, ð0, ð1, and ð
2. Solving the system of algebraic
equations with the aid of Mathematica, we obtain
ð0= 0, ð
1= ±2âð (ð â ðŒ â 3ðŒ2 + 3ðœ), ð
2= 0,
ð0= ðŒ
2â ðœ â ð, ð
1= 0, ð
2= â2ð.
(13)
Substituting these results into (12), we have the followingformal solution of (11):
ð = ±2âð (ð â ðŒ â 3ðŒ2 + 3ðœ)ð¹ (ð) ,
ð = ðŒ2â ðœ â ð â 2ðð¹
2
(ð) , where ð = ð¥ + ðð¡.(14)
With the aid of Appendix A and from the formal solution of(14) along with (10), one can deduce more general combinedJacobian-elliptic function solutions of (1). Hence, the follow-ing exact solutions are obtained.
The Scientific World Journal 3
Case 1. ð = ð2, ð = â(1 + ð2), ð = 1, ð¹(ð) = ð ðð,
ð¢1= ð
ðð{±2ðââ1 â ð2 â ðŒ â 3ðŒ2 + 3ðœð ðð} ,
V1= ðŒ
2â ðœ + 1 + ð
2â 2ð
2ð ð2ð.
(15)
Case 2. ð = ð2, ð = â(1 + ð2), ð = 1, ð¹(ð) = ððð,
ð¢2= ð
ðð{±2ðââ1 â ð2 â ðŒ â 3ðŒ2 + 3ðœððð} ,
V2= ðŒ
2â ðœ + 1 + ð
2â 2ð
2ðð2ð.
(16)
Case 3. ð = âð2, ð = 2ð2 â 1, ð = 1 â ð2, ð¹(ð) = ððð,
ð¢3= ð
ðð{±2ðââ2ð2 + 1 + ðŒ + 3ðŒ2 â 3ðœððð} ,
V3= ðŒ
2â ðœ â 2ð
2+ 1 + 2ð
2ðð2ð.
(17)
Case 4. ð = â1, ð = 2 â ð2, ð = ð2 â 1, ð¹(ð) = ððð,
ð¢4= ð
ðð{±2ââ2 + ð2 + ðŒ + 3ðŒ2 â 3ðœððð} ,
V4= ðŒ
2â ðœ â 2 + ð
2+ 2ðð
2ð.
(18)
Case 5. ð = 1, ð = â(1 + ð2), ð = ð2, ð¹(ð) = ðð ð,
ð¢5= ð
ðð{±2ââ1 â ð2 â ðŒ â 3ðŒ2 + 3ðœðð ð} ,
V5= ðŒ
2â ðœ + 1 + ð
2â 2ðð
2ð.
(19)
Case 6. ð = 1, ð = â(1 + ð2), ð = ð2, ð¹(ð) = ððð,
ð¢6= ð
ðð{±2ââ1 â ð2 â ðŒ â 3ðŒ2 + 3ðœððð} ,
V6= ðŒ
2â ðœ + 1 + ð
2â 2ðð
2ð.
(20)
Case 7. ð = 1 â ð2, ð = 2ð2 â 1, ð = âð2, ð¹(ð) = ððð,
ð¢7= ð
ðð{±2â(1 â ð2) (2ð2 â 1 â ðŒ â 3ðŒ2 + 3ðœ)ððð} ,
V7= ðŒ
2â ðœ â 2ð
2+ 1 â 2 (1 â ð
2) ðð
2ð.
(21)
Case 8. ð = ð2 â 1, ð = 2 â ð2, ð = â1, ð¹(ð) = ððð,
ð¢8= ð
ðð{±2â(ð2 â 1) (2 â ð2 â ðŒ â 3ðŒ2 + 3ðœ)ððð} ,
V8= ðŒ
2â ðœ â 2 + ð
2â 2 (ð
2â 1) ðð
2ð.
(22)
Case 9. ð = 1 â ð2, ð = 2 â ð2, ð = 1, ð¹(ð) = ð ðð,
ð¢9= ð
ðð{±2â(1 â ð2) (2 â ð2 â ðŒ â 3ðŒ2 + 3ðœ)ð ðð} ,
V9= ðŒ
2â ðœ â 2 + ð
2â 2 (1 â ð
2) ð ð
2ð.
(23)
Case 10. ð = âð2(1 â ð2), ð = 2ð2 â 1, ð = 1, ð¹(ð) = ð ðð,
ð¢10= ð
ðð{±2ðâ(ð2 â 1) (2ð2 â 1 â ðŒ â 3ðŒ2 + 3ðœ)ð ðð} ,
V10= ðŒ
2â ðœ â 2ð
2+ 1 + 2ð
2(1 â ð
2) ð ð
2ð.
(24)
Case 11. ð = 1, ð = 2 â ð2, ð = 1 â ð2, ð¹(ð) = ðð ð,
ð¢11= ð
ðð{±2â2 â ð2 â ðŒ â 3ðŒ2 + 3ðœðð ð} ,
V11= ðŒ
2â ðœ â 2 + ð
2â 2ðð
2ð.
(25)
Case 12. ð = 1, ð = 2ð2 â 1, ð = âð2(1 â ð2), ð¹(ð) = ðð ð,
ð¢12= ð
ðð{±2â2ð2 â 1 â ðŒ â 3ðŒ2 + 3ðœðð ð} ,
V12= ðŒ
2â ðœ â 2ð
2+ 1 â 2ðð
2ð.
(26)
Case 13. ð = 1/4,ð = (1â2ð2)/2, ð = 1/4, ð¹(ð) = ðð ð±ðð ð,
ð¢13= ð
ðð{
{
{
±â1 â 2ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ
2(ðð ð ± ðð ð)
}
}
}
,
V13=1
2{2ðŒ
2â 2ðœ â 1 + 2ð
2â (ðð ð ± ðð ð)
2
} .
(27)
Case 14. ð = (1 â ð2)/4, ð = (1 + ð2)/2, ð = (1 â ð2)/4,ð¹(ð) = ððð ± ð ðð,
ð¢14= ð
ðð{
{
{
±â(1 â ð
2) (1 + ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
à (ððð ± ð ðð)
}
}
}
,
V14=1
2{2ðŒ
2â 2ðœ â 1 â ð
2â (1 â ð
2) (ððð ± ð ðð)
2
} .
(28)
Case 15. ð = 1/4,ð = (ð2â2)/2,ð = ð2/4,ð¹(ð) = ðð ð±ðð ð,
ð¢15= ð
ðð{
{
{
±âð2â 2 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(ðð ð ± ðð ð)
}
}
}
,
V15=1
2{2ðŒ
2â 2ðœ â ð
2+ 2 â (ðð ð ± ðð ð)
2
} .
(29)
Case 16. ð = ð2/4, ð = (ð2 â 2)/2, ð = ð2/4, ð¹(ð) = ð ðð ±ðððð,
ð¢16= ð
ðð{
{
{
±ðâð2â 2 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(ð ðð ± ðððð)
}
}
}
,
V16=1
2{2ðŒ
2â 2ðœ â ð
2+ 2 â ð
2
(ð ðð ± ðððð)2
} .
(30)
4 The Scientific World Journal
Case 17. ð = ð2/4, ð = (ð2 â 2)/2, ð = ð2/4, ð¹(ð) =â1 â ð2ð ðð ± ððð,
ð¢17= ð
ðð{
{
{
±ðâð2â 2 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
à (â1 â ð2ð ðð ± ððð)
}
}
}
,
V17=1
2{2ðŒ
2â2ðœ â ð
2+2 â ð
2(â1 â ð2ð ðð ± ððð)
2
} .
(31)
Case 18. ð = 1/4, ð = (1 â ð2)/2, ð = 1/4, ð¹(ð) = ðððð ±ðâ1 â ð2ððð,
ð¢18= ð
ðð{
{
{
±â1 â ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ
2
à (ðððð ± ðâ1 â ð2ððð)
}
}
}
,
V18=1
2{2ðŒ
2â 2ðœ â 1 + ð
2
â (ðððð ± ðâ1 â ð2ððð)
2
} .
(32)
Case 19. ð = 1/4, ð = (1 â 2ð2)/2, ð = 1/4, ð¹(ð) = ðð ðð ±ðððð,
ð¢19= ð
ðð{
{
{
±â1 â 2ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ
2
à (ðð ðð ± ðððð)
}
}
}
,
V19=1
2{2ðŒ
2â 2ðœ â 1 + 2ð
2â (ðð ðð ± ðððð)
2
} .
(33)
Case 20. ð = 1/4, ð = (1 â ð2)/2, ð = 1/4, ð¹(ð) =â1 â ð2ð ðð ± ðððð,
ð¢20= ð
ðð{
{
{
±â1 â ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ
2
à (â1 â ð2ð ðð ± ðððð)
}
}
}
,
V20=1
2{2ðŒ
2â2ðœ â 1+ ð
2â(â1 â ð2ð ðð ± ðððð)
2
} .
(34)
Case 21. ð = (ð2 â 1)/4, ð = (ð2 + 1)/2, ð = (ð2 â 1)/4,ð¹(ð) = ðð ðð ± ððð,
ð¢21= ð
ðð{
{
{
±â(ð
2â 1) (ð
2+ 1 â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
à (ðð ðð ± ððð)
}
}
}
,
V21=1
2{2ðŒ
2â 2ðœ â ð
2â 1 â (ð
2â 1)
à (ðð ðð ± ððð)2
} .
(35)
Case 22. ð = ð2/4,ð = (ð2 â2)/2, ð = 1/4, ð¹(ð) = ð ðð/(1±ððð),
ð¢22= ð
ðð{
{
{
±ðâð2â 2 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
ð ðð
1 ± ððð
}
}
}
,
V22=1
2{2ðŒ
2â 2ðœ â ð
2+ 2 â ð
2(
ð ðð
1 ± ððð)
2
} .
(36)
Case 23. ð = â1/4, ð = (ð2 + 1)/2, ð = (1 â ð2)2/4, ð¹(ð) =ðððð ± ððð,
ð¢23=ððð{
{
{
±ââð
2â 1 + 2ðŒ + 6ðŒ
2â 6ðœ
2(ðððð ± ððð)
}
}
}
,
V23=1
2{2ðŒ
2â 2ðœ â ð
2â 1 + (ðððð ± ððð)
2
} .
(37)
Case 24. ð = (1 â ð2)2/4, ð = (ð2 + 1)/2, ð = 1/4, ð¹(ð) =ðð ð ± ðð ð,
ð¢24= ð
ðð{
{
{
±(1 â ð2)â
ð2+ 1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
à (ðð ð ± ðð ð)
}
}
}
,
V24=1
2{2ðŒ
2â 2ðœ â ð
2â 1 â (1 â ð
2)2
(ðð ð ± ðð ð)2
} .
(38)
The Scientific World Journal 5
Case 25. ð = ð4(1 âð2)/2(2 âð2),ð = 2(1 âð2)/(ð2 â 2),ð = (1 â ð
2)/2(2 â ð
2), ð¹(ð) = ððð ± â1 â ð2ððð,
ð¢25
= ððð{±
â2ð2
2 â ð2
à â(ð2 â 1) [2 (1 â ð2) + (ð2 â 2) (âðŒ â 3ðŒ2 + 3ðœ)]
à (ððð ± â1 â ð2ððð) } ,
V25
=1
ð2 â 2{ (ð
2â 2) (ðŒ
2â ðœ) + (1 â ð
2)
à [â2 + ð4(ððð ± â1 â ð2ððð)
2
]} .
(39)
Case 26. ð > 0, ð < 0, ð = ð2ð2/(1 + ð2)2ð, ð¹(ð) =ââð2ð/(1 + ð2)ðð ð(ââð/(1 + ð2)ð),
ð¢26= ð
ðð{
{
{
±2ðâð(âð + ðŒ + 3ðŒ
2â 3ðœ)
1 + ð2
à ð ð(ââð
1 + ð2ð)
}
}
}
,
V26= ðŒ
2â ðœ â ð +
2ð2ð
ð2 + 1ð ð2(â
âð
1 + ð2ð) .
(40)
Case 27. ð < 0, ð > 0, ð = (1 â ð2)ð2/(ð2 â 2)2ð, ð¹(ð) =ââð/(2 â ð2)ððð(âð/(2 â ð2)ð),
ð¢27= ð
ðð{
{
{
±2âð(âð + ðŒ + 3ðŒ
2â 3ðœ)
2 â ð2
à ðð(âð
2 â ð2ð)
}
}
}
,
V27= ðŒ
2â ðœ â ð +
2ð
2 â ð2ðð2(â
ð
2 â ð2ð) .
(41)
Case 28. ð < 0, ð > 0, ð = ð2(ð2 â 1)ð2/(2ð2 â 1)2ð,ð¹(ð) = ââð2ð/(2ð2 â 1)ððð(âð/(2ð2 â 1)ð),
ð¢28= ð
ðð{
{
{
±2ðâð(âð + ðŒ + 3ðŒ
2â 3ðœ)
2ð2 â 1ðð
à (âð
2ð2 â 1ð)
}
}
}
,
V28= ðŒ
2â ðœ â ð +
2ð2ð
2ð2 â 1ðð2(â
ð
2ð2 â 1ð) .
(42)
Case 29. ð = 1, ð = 2 â 4ð2, ð = 1, ð¹(ð) = ð ððððð/ððð,
ð¢29= ð
ðð{±2â2 â 4ð2 â ðŒ â 3ðŒ2 + 3ðœ
ð ððððð
ððð} ,
V29= ðŒ
2â ðœ â 2 + 4ð
2â2ð ð
2ððð
2ð
ðð2ð.
(43)
Case 30. ð = ð2, ð = 2, ð = 1, ð¹(ð) = ð ððððð/ððð,
ð¢30= ð
ðð{±2ðâ2 â ðŒ â 3ðŒ2 + 3ðœ
ð ððððð
ððð} ,
V30= ðŒ
2â ðœ â 2 â
2ð2ð ð2ððð
2ð
ðð2ð.
(44)
Case 31. ð = 1, ð = ð2 + 2, ð = 1 â 2ð2 + ð4, ð¹(ð) =ðððððð/ð ðð,
ð¢31= ð
ðð{±2âð2 + 2 â ðŒ â 3ðŒ2 + 3ðœ
ðððððð
ð ðð} ,
V31= ðŒ
2â ðœ â ð
2â 2 â
2ðð2ððð
2ð
ð ð2ð.
(45)
Case 32. ð = ðŽ2(ð â 1)2/4, ð = (ð2 + 1)/2 + 3ð, ð = (ð â1)2/4ðŽ
2, ð¹(ð) = ðððððð/ðŽ(1 + ð ðð)(1 + ðð ðð),
ð¢32= ð
ðð{
{
{
± (ð â 1)âð2+ 6ð + 1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
Ãðððððð
(1 + ð ðð) (1 + ðð ðð)
}
}
}
,
V32=1
2{2ðŒ
2â 2ðœ â ð
2â 6ð â 1
â(ð â 1)
2ðð2ððð
2ð
(1 + ð ðð)2
(1 + ðð ðð)2} .
(46)
Case 33. ð = ðŽ2(ð + 1)2/4, ð = (ð2 + 1)/2 â 3ð, ð = (ð +1)2/4ðŽ
2, ð¹(ð) = ðððððð/ðŽ(1 + ð ðð)(1 â ðð ðð),
ð¢33= ð
ðð{
{
{
± (ð + 1)âð2â 6ð + 1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
Ãðððððð
(1 + ð ðð) (1 â ðð ðð)
}
}
}
,
V33=1
2{2ðŒ
2â 2ðœ â ð
2+ 6ð â 1
â(ð + 1)
2ðð2ððð
2ð
(1 + ð ðð)2
(1 â ðð ðð)2} .
(47)
6 The Scientific World Journal
Case 34. ð = â4/ð, ð = 6ð âð2 â 1, ð = â2ð3 + ð4 + ð2,ð¹(ð) = ððððððð/(ðð ð
2ð + 1),
ð¢34= ð
ðð{±4âð2 â 6ð + 1 + ðŒ + 3ðŒ2 â 3ðœ
âððððððð
ðð ð2ð + 1} ,
V34= ðŒ
2â ðœ + ð
2â 6ð + 1 +
8ððð2ððð
2ð
(ðð ð2ð + 1)2.
(48)
Case 35. ð = 4/ð, ð = â6ð â ð2 â 1, ð = 2ð3 + ð4 + ð2,ð¹(ð) = ððððððð/(ðð ð
2ð â 1),
ð¢35= ð
ðð{ ± 4ââð2 â 6ð â 1 â ðŒ â 3ðŒ2 + 3ðœ
Ãâððððððð
ðð ð2ð â 1} ,
V35= ðŒ
2â ðœ + ð
2+ 6ð + 1 â
8ððð2ððð
2ð
(ðð ð2ð â 1)2.
(49)
Case 36. ð = 1/4,ð = (1 â 2ð2)/2, ð = 1/4, ð¹(ð) = ð ðð/(1 ±ððð),
ð¢36= ð
ðð{
{
{
±â1 â 2ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ
2
ð ðð
1 ± ððð
}
}
}
,
V36=1
2{2ðŒ
2â 2ðœ â 1 + 2ð
2â
ð ð2ð
(1 ± ððð)2} .
(50)
Case 37. ð = (1 â ð2)/4, ð = (1 + ð2)/2, ð = (1 â ð2)/4,ð¹(ð) = ððð/(1 ± ð ðð),
ð¢37= ð
ðð{
{
{
±â(1 â ð
2) (1 + ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
Ãððð
1 ± ð ðð} ,
V37=1
2{2ðŒ
2â 2ðœ â 1 â ð
2â
(1 â ð2) ðð
2ð
(1 ± ð ðð)2} .
(51)
Case 38. ð = 4ð1, ð = 2 + 6ð
1â ð
2, ð = 2 + 2ð1â ð
2,ð¹(ð) = ð
2ð ððððð/(ð
1â ðð
2ð),
ð¢38= ð
ðð{ ± 4âð
1(2 + 6ð
1â ð2 â ðŒ â 3ðŒ2 + 3ðœ)
Ãð2ð ððððð
ð1â ðð2ð
} ,
V38= ðŒ
2â ðœ â 2 â 6ð
1+ ð
2â8ð
1ð4ð ð2ððð
2ð
(ð1â ðð2ð)
2.
(52)
Case 39. ð = â4ð1, ð = 2 â 6ð
1â ð
2, ð = 2 â 2ð1â ð
2,ð¹(ð) = âð
2ð ððððð/(ð
1+ ðð
2ð),
ð¢39= ð
ðð{ ± 4âð
1(â2 + 6ð
1+ ð2 + ðŒ + 3ðŒ2 â 3ðœ)
Ãð2ð ððððð
ð1+ ðð2ð
} ,
V39= ðŒ
2â ðœ â 2 + 6ð
1+ ð
2+8ð
1ð4ð ð2ððð
2ð
(ð1+ ðð2ð)
2.
(53)
Case 40. ð = (2âð2 â2ð1)/4,ð = ð2/2â1â3ð
1, ð = (2â
ð2â2ð
1)/4, ð¹(ð) = ð2ð ððððð/(ð ð2ð+ (1+ð
1)ðððâ1âð
1),
ð¢40
= ððð{
{
{
±â(2 â ð
2â 2ð
1) (ð
2â 2 â 6ð
1â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
Ãð2ð ððððð
ð ð2ð + (1 + ð1) ððð â 1 â ð
1
}
}
}
,
V40=1
2{2ðŒ
2â 2ðœ â ð
2+ 2 + 6ð
1
âð4(2 â ð
2â 2ð
1) ð ð
2ððð
2ð
[ð ð2ð + (1 + ð1) ððð â 1 â ð
1]2} .
(54)
Case 41. ð = (2âð2 +2ð1)/4,ð = ð2/2 â 1+ 3ð
1, ð = (2â
ð2+2ð
1)/4,ð¹(ð) = ð2ð ððððð/(ð ð2ð+(â1+ð
1)ðððâ1âð
1),
ð¢41
= ððð{
{
{
±â(2 â ð
2+ 2ð
1) (ð
2â 2 + 6ð
1â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
Ãð2ð ððððð
ð ð2ð + (â1 + ð1) ððð â 1 â ð
1
}
}
}
,
V41=1
2{2ðŒ
2â 2ðœ â ð
2+ 2 â 6ð
1
âð4(2 â ð
2+ 2ð
1) ð ð
2ððð
2ð
[ð ð2ð + (â1 + ð1) ððð â 1 â ð
1]2} .
(55)
Case 42. ð = (ð¶2ð4â(ðµ2+ð¶2)ð2+ðµ2)/4,ð = (ð2+1)/2,ð =(ð2â 1)/4(ð¶
2ð2â ðµ
2), ð¹(ð) = (â(ðµ2 â ð¶2)/(ðµ2 â ð¶2ð2) +
ð ðð)/(ðµððð + ð¶ððð),
The Scientific World Journal 7
ð¢42
= ððð
{{
{{
{
±â[ð¶2ð4â (ðµ
2+ ð¶
2)ð
2+ ðµ
2] (ð
2+ 1 â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
Ã
â(ðµ2â ð¶
2) / (ðµ
2â ð¶
2ð2) + ð ðð
ðµððð + ð¶ððð
}}
}}
}
,
V42=1
2
{{
{{
{
2ðŒ2â 2ðœ â ð
2â 1 â [ð¶
2ð4â (ðµ
2+ ð¶
2)ð
2+ ðµ
2]
à (
â(ðµ2â ð¶
2) / (ðµ
2â ð¶
2ð2) + ð ðð
ðµððð + ð¶ððð
)
2
}}
}}
}
.
(56)
Case 43. ð = (ðµ2 + ð¶2ð2)/4, ð = 1/2 â ð2, ð =1/4(ðµ
2+ ð¶
2ð2), ð¹(ð) = (â(ð¶2ð2 + ðµ2 â ð¶2)/(ðµ2 + ð¶2ð2) +
ððð)/(ðµð ðð + ð¶ððð),
ð¢43= ð
ðð
{{
{{
{
±â(ðµ2+ ð¶
2ð2) (1 â 2ð
2â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
Ã
â(ð¶2ð2 + ðµ2 â ð¶2) / (ðµ2 + ð¶2ð2) + ððð
ðµð ðð + ð¶ððð
}}
}}
}
,
V43=1
2{2ðŒ
2â 2ðœ â 1 + 2ð
2â (ðµ
2+ ð¶
2ð2)
Ã(â(ð¶2ð2 + ðµ2 â ð¶2)/(ðµ2 + ð¶2ð2) + ððð
ðµð ðð + ð¶ððð)
2
} .
(57)
Case 44. ð = (ðµ2 +ð¶2)/4,ð = ð2/2 â 1, ð = ð4/4(ðµ2 +ð¶2),ð¹(ð) = (â(ðµ2 + ð¶2 â ð¶2ð2)/(ðµ2 + ð¶2)+ððð)/(ðµð ðð+ð¶ððð),
ð¢44= ð
ðð{
{
{
±â(ðµ2+ ð¶
2) (ð
2â 2 â 2ðŒ â 6ðŒ
2+ 6ðœ)
2
Ã
â(ðµ2 + ð¶2 â ð¶2ð2) / (ðµ2 + ð¶2) + ððð
ðµð ðð + ð¶ððð
}}
}}
}
,
V44=1
2{2ðŒ
2â 2ðœ â ð
2+ 2 â (ðµ
2+ ð¶
2)
à (â(ðµ2 + ð¶2 â ð¶2ð2)/(ðµ2 + ð¶2) + ððð
ðµð ðð + ð¶ððð)
2
} .
(58)
Case 45. ð = â(ð2 + 2ð + 1)ðµ2, ð = 2ð2 + 2, ð = (2ð âð2â 1)/ðµ
2, ð¹(ð) = (ðð ð2ð â 1)/ðµ(ðð ð2ð + 1),
ð¢45= ð
ðð{ ± 2 (ð + 1)ââ2ð
2 â 2 + ðŒ + 3ðŒ2 â 3ðœ
Ãðð ð
2ð â 1
ðð ð2ð + 1} ,
V45= ðŒ
2â ðœ â 2ð
2â 2 + 2(ð + 1)
2
à (ðð ð
2ð â 1
ðð ð2ð + 1)
2
.
(59)
Case 46. ð = â(ð2 â 2ð + 1)ðµ2, ð = 2ð2 + 2, ð = â(2ð +ð2+ 1)/ðµ
2, ð¹(ð) = (ðð ð2ð + 1)/ðµ(ðð ð2ð â 1),
ð¢46= ð
ðð{ ± 2 (ð â 1)ââ2ð
2 â 2 + ðŒ + 3ðŒ2 â 3ðœ
Ãðð ð
2ð + 1
ðð ð2ð â 1} ,
V46= ðŒ
2â ðœ â 2ð
2â 2 + 2(ð â 1)
2
à (ðð ð
2ð + 1
ðð ð2ð â 1)
2
.
(60)
We note that there is much duplication in the list of 46solutions in terms of Jacobian-elliptic functions. Here aresome examples; using the well-known identities relatingJacobian-elliptic functions (see 121.00, 129.01, 129.02, and129.03 in [52], e.g.) reveals that ð¢
1, ð¢3, and ð¢
4are identical;
V1, V3, and V
4are identical; ð¢
2, ð¢8, and ð¢
10are identical; V
2,
V8, and V
10are identical; ð¢
5, ð¢11, and ð¢
12are identical; V
5, V11,
and V12
are identical; ð¢6, ð¢7, and ð¢
9are identical; V
6, V7, and
V9are identical. Use of 162.01 in [52] reveals that ð¢
27and ð¢
28
are equivalent and V27and V
28are equivalent.
4. The New Weierstrass-Elliptic FunctionSolutions of the Schrödinger-KdV Equation
On using the solutions given in [43], mentioned inAppendix B, and from the formal solution (14) along with(10), we get then the following exact solutions.
Case 47. ð2= (4/3)(ð
2â 3ðð ), ð
3= (4ð/27)(â2ð
2+ 9ðð ),
ð¹(ð) = â(1/ð)[â(ð; ð2, ð3) â (1/3)ð],
ð¢47= ð
ðð{ ± 2âð (ð â ðŒ â 3ðŒ2 + 3ðœ)
Ãâ1
ð[â (ð; ð
2, ð3) â
1
3ð]} ,
V47= ðŒ
2â ðœ â ð â 2 [â (ð; ð
2, ð3) â
1
3ð] .
(61)
8 The Scientific World Journal
Case 48. ð2= (4/3)(ð
2â 3ðð ), ð
3= (4ð/27)(â2ð
2+ 9ðð ),
ð¹(ð) = â3ð /(3â(ð; ð2, ð3) â ð),
ð¢48= ð
ðð{ ± 2âð (ð â ðŒ â 3ðŒ2 + 3ðœ)
à â3ð
3â (ð; ð2, ð3) â ð
} ,
V48= ðŒ
2â ðœ â ð â
6ðð
3â (ð; ð2, ð3) â ð
.
(62)
Case 49. ð2= â(5ðð· + 4ð
2+ 33ððð )/12, ð
3= (21ð
2
ð· â 63ðð ð· + 20ð3â 27ððð )/216, ð¹(ð) =
â12ð â(ð; ð2, ð3) + 2ð (2ð + ð·)/(12â(ð; ð
2, ð3) + ð·),
ð¢49= ð
ðð
{{
{{
{
± 2âð (ð â ðŒ â 3ðŒ2 + 3ðœ)
Ã
â12ð â (ð; ð2, ð3) + 2ð (2ð + ð·)
12â (ð; ð2, ð3) + ð·
}}
}}
}
,
V49= ðŒ
2â ðœ â ð
â4ðð [6â (ð; ð
2, ð3) + 2ð + ð·]
[12â (ð; ð2, ð3) + ð·]
2.
(63)
Case 50. ð2= (1/12)ð
2+ ðð , ð
3= (1/216)ð(36ðð â ð
2),
ð¹(ð) = âð [6â(ð; ð2, ð3) + ð]/3â
(ð; ð
2, ð3),
ð¢50= ð
ðð{ ± 2âðð (ð â ðŒ â 3ðŒ2 + 3ðœ)
Ã6â (ð; ð
2, ð3) + ð
3â (ð; ð2, ð3)} ,
V50= ðŒ
2â ðœ â ð â
2ðð [6â (ð; ð2, ð3) + ð]
2
9[â (ð; ð2, ð3)]2
.
(64)
Case 51. ð2= (1/12)ð
2+ ðð , ð
3= (1/216)ð(36ðð â ð
2),
ð¹(ð) = 3â(ð; ð
2, ð3)/âð[6â(ð; ð
2, ð3) + ð],
ð¢51= ð
ðð{±2âð â ðŒ â 3ðŒ2 + 3ðœ
3â(ð; ð
2, ð3)
6â (ð; ð2, ð3) + ð
} ,
V51= ðŒ
2â ðœ â ð â
18[â(ð; ð
2, ð3)]2
[6â (ð; ð2, ð3) + ð]
2.
(65)
Case 52. ð = 5ð2/36ð, ð2= 2ð
2/9, ð
3= ð
3/54, ð¹(ð) =
ðââ15ð/2ðâ(ð; ð2, ð3)/(3â(ð; ð
2, ð3) + ð),
ð¢52= ð
ðð{ ± 2âð (ð â ðŒ â 3ðŒ2 + 3ðœ)
Ãðââ15ð/2ðâ (ð; ð
2, ð3)
3â (ð; ð2, ð3) + ð
} ,
V52= ðŒ
2â ðœ â ð +
15ð3â2(ð; ð
2, ð3)
[3â (ð; ð2, ð3) + ð]
2.
(66)
It should be noted that any solution that can be expressed interms of aWeierstrass-elliptic function can be also convertedinto a solution in terms of a Jacobian-elliptic function (formore details, see [53]). Consequently, Cases 47â52 are alreadycovered in Cases 1â46. For example, using 1031.01 in [52]reveals that, with the ð, ð, and ð values for Case 1, ð¢
1and
ð¢48are identical and V
1and V
48are identical.
5. New Soliton-Like Solutions of theSchrödinger-KdV Equation
Some soliton-like solutions of (1) can be obtained in thelimited case when the modulus ð â 1 (see Appendix C),as follows:
ð¢1= ð
ðð{±2ââ2 â ðŒ â 3ðŒ2 + 3ðœtanhð} ,
V1= ðŒ
2â ðœ + 2sech2ð,
ð¢3= ð
ðð{±2ââ1 + ðŒ + 3ðŒ2 â 3ðœsechð} ,
V3= ðŒ
2â ðœ â 1 + 2sech2ð,
ð¢5= ð
ðð{±2ââ2 â ðŒ â 3ðŒ2 + 3ðœcothð} ,
V5= ðŒ
2â ðœ â 2csch2ð,
ð¢11= ð
ðð{±2â1 â ðŒ â 3ðŒ2 + 3ðœcschð} ,
V11= ðŒ
2â ðœ â 1 â 2csch2ð,
ð¢13= ð
ðð{
{
{
±ââ1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(coth ð ± cschð)
}
}
}
,
V13=1
2{2ðŒ
2â 2ðœ + 1 â (coth ð ± cschð)2} ,
ð¢16= ð
ðð{
{
{
±ââ1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(tanhð ± ðsechð)
}
}
}
,
V16=1
2{2ðŒ
2â 2ðœ + 1 â (tanhð ± ðð echð)2} ,
The Scientific World Journal 9
ð¢22= ð
ðð{
{
{
±ââ1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
tanhð1 ± sechð
}
}
}
,
V22=1
2{2ðŒ
2â 2ðœ + 1 â (
tanhð1 ± sechð
)
2
} ,
ð¢23= ð
ðð{±2ââ1 + ðŒ + 3ðŒ2 â 3ðœsechð} ,
V23= ðŒ
2â ðœ â 1 + 2sech2ð,
ð¢26= ð
ðð{
{
{
±2âð(âð + ðŒ + 3ðŒ
2â 3ðœ)
2tanh(ââð
2ð)
}
}
}
,
V26= ðŒ
2â ðœ â ðsech2 (ââð
2ð) ,
ð¢27= ð
ðð{±2âð (âð + ðŒ + 3ðŒ2 â 3ðœ)sech (âðð)} ,
V27= ðŒ
2â ðœ â ð [1 â 2sech2 (âðð)] ,
ð¢30= ð
ðð{±2â2 â ðŒ â 3ðŒ2 + 3ðœtanhð} ,
V30= ðŒ
2â ðœ â 2 (2 â sech2ð) ,
ð¢31= ð
ðð{±2â3 â ðŒ â 3ðŒ2 + 3ðœsechðcschð} ,
V31= ðŒ
2â ðœ â 3 â 2(sechðcschð)2,
ð¢34= ð
ðð{±4ââ4 + ðŒ + 3ðŒ2 â 3ðœ
sech2ð1 + tanh2ð
} ,
V34= ðŒ
2â ðœ â 4 +
8sech4ð
(1 + tanh2ð)2,
ð¢40= ð
ðð{
{
{
±â(â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2)1 + sechð1 â sechð
}
}
}
,
V40=1
2{2ðŒ
2â 2ðœ + 1 â
1 + sechð1 â sechð
} ,
ð¢41= ð
ðð{
{
{
±â(â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2)1 â sechð1 + sechð
}
}
}
,
V41=1
2{2ðŒ
2â 2ðœ + 1 â
1 â sechð1 + sechð
} ,
ð¢43= ð
ðð{
{
{
±ââ1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
ðµ + âðµ2 + ð¶2sechððµtanhð + ð¶sechð
}
}
}
,
V43=1
2{2ðŒ
2â 2ðœ + 1 â (
ðµ + âðµ2 + ð¶2sechððµtanhð + ð¶sechð
)
2
} .
(67)
Here, it should be noted that each exact solution given in (67)can be split into two solutions if one chooses the (+ve) and(âve) signs, respectively, but they have not been calculated.Also, all the exact solutions given by (67) can be verifiedby substitution. The main feature for some of these exactsolutions is the inclusion of the free parameters ð, ðµ, and ð¶.
6. New Trigonometric-Function Solutions ofthe Schrödinger-KdV Equation
Some trigonometric-function solutions of (1) can be obtainedin the limited case when the modulusð â 0. For example,
ð¢5= ð
ðð{±2ââ1 â ðŒ â 3ðŒ2 + 3ðœcscð} ,
V5= ðŒ
2â ðœ + 1 â 2csc2ð,
ð¢6= ð
ðð{±2ââ1 â ðŒ â 3ðŒ2 + 3ðœsecð} ,
V6= ðŒ
2â ðœ + 1 â 2sec2ð,
ð¢9= ð
ðð{±2â2 â ðŒ â 3ðŒ2 + 3ðœtanð} ,
V9= ðŒ
2â ðœ â 2sec2ð,
ð¢11= ð
ðð{±2â2 â ðŒ â 3ðŒ2 + 3ðœcotð} ,
V11= ðŒ
2â ðœ â 2csc2ð,
ð¢13= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(cscð ± cotð)
}
}
}
,
V13=1
2{2ðŒ
2â 2ðœ â 1 â (cscð ± cotð)2} ,
ð¢14= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(secð ± tanð)
}
}
}
,
V14=1
2{2ðŒ
2â 2ðœ â 1 â (secð ± tan ð)2} ,
ð¢24= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(cscð ± cotð)
}
}
}
,
V24=1
2{2ðŒ
2â 2ðœ â 1 + (cscð ± cotð)2} ,
ð¢32= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(secð â tanð)
}
}
}
,
V32= ðŒ
2â ðœ â secð (secð â tanð) ,
ð¢36= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(cscð ± cotð)
}
}
}
,
10 The Scientific World Journal
V36=1
2{2ðŒ
2â 2ðœ â 1 â (cscð ± cotð)2} ,
ð¢37= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2(secð ± tanð)
}
}
}
,
V37=1
2{2ðŒ
2â 2ðœ â 1 â (secð ± tanð)2} ,
ð¢42= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
âðµ2 â ð¶2 + ðµsinððµcosð + ð¶
}
}
}
,
V42=1
2{2ðŒ
2â 2ðœ â 1 â (
âðµ2 â ð¶2 + ðµsinððµcosð + ð¶
)
2
} ,
ð¢43= ð
ðð{
{
{
±â1 â 2ðŒ â 6ðŒ
2+ 6ðœ
2
âðµ2 â ð¶2 + ðµcosððµsinð + ð¶
}
}
}
,
V43=1
2{2ðŒ
2â 2ðœ â 1 â (
âðµ2 â ð¶2 + ðµcosððµsinð + ð¶
)
2
} ,
ð¢44= ð
ðð{±ââ1 â ðŒ â 3ðŒ2 + 3ðœ
2âðµ2 + ð¶2
ðµsinð + ð¶cosð} ,
V44= ðŒ
2â ðœ + 1 â
2 (ðµ2+ ð¶
2)
(ðµsinð + ð¶cosð)2.
(68)
Here, we note also that each trigonometric-function solutionobtained in this section can split into two solutions if wechoose the (+ve) and (âve) signs, respectively. Besides, allthese solutions can be verified by direct substitution. Also, themain feature for some of these exact solutions is the inclusionof the free parameters ð, ðµ, and ð¶.
7. Conclusion
In this paper, the ð¹-expansion method has been applied toconstruct 52 types of exact solution of the the SchroÌdinger-KdV equation. The main advantage of this method overother methods is that it possesses all types of exact solu-tion, including those of Jacobian-elliptic and Weierstrass-elliptic functions. Moreover, the soliton-like solutions andtrigonometric-function solutions have been also obtained asthe modulus ð of Jacobi-elliptic function approaches to 1and 0. It can be said that the results in this paper providegood supplements to the existing literature and are useful fordescribing certain nonlinear phenomena.Thismethod can beapplied to many other nonlinear evolution equations. Finally,it is worthwhile to mention that the proposed method is alsoa straightforward, short, promising, and powerful methodfor other nonlinear evolution equations in mathematicalphysics.
Appendices
A. Relations between Values of (ð, ð, ð ) andCorresponding ð¹(ð) in (7)
Relations between values of (ð,ð,ð ) and correspondingð¹(ð)in (7), where ðŽ, ðµ, and ð¶ are arbitrary constants and ð
1=
â1 â ð2. As shown in Table 1.
B. The Weierstrass-Elliptic Function Solutionsfor (7)
The Weierstrass-elliptic function solutions for (7), whereð· = (1/2)(â5ð ± â9ð2 â 36ðð ) and â(ð; ð
2, ð3) =
ðâ(ð; ð2, ð3)/ðð. As shown in Table 2.
C. Relations between Jacobian-EllipticFunctions and Hyperbolic Functions
The Jacobian-elliptic functions degenerate into hyperbolicfunctions whenð â 1 as follows:ð ðð â tanhð, ððð â sechð, ððð â sechð,
ð ðð â sinhð, ð ðð â sinhð, ððð â 1,
ðð ð â coth ð, ððð â coshð, ððð â coshð,
ðð ð â cschð, ðð ð â cschð, ððð â 1.(C.1)
The Jacobian-elliptic functions degenerate into trigono-metric functions whenð â 0 as follows:ð ðð â sinð, ððð â cosð, ððð â 1,
ð ðð â tanð, ð ðð â sinð, ððð â cos ð,
ðð ð â cscð, ððð â secð, ððð â 1,
ðð ð â cotð, ðð ð â cscð, ððð â secð.
(C.2)
D. Some Trigonometric and HyperbolicIdentities
Consider the following:
coth ð â cschð = tanh ð2, cscð â cotð = tan ð
2,
coth ð + cschð = coth ð2, cscð + cotð = cotð
2,
tanh ð + ðsechð = tanh [12(ð +
ðð
2)] ,
secð + tan ð = tan [12(ð +
ð
2)] ,
tanh ð â ðsechð = coth [12(ð +
ðð
2)] ,
secð â tan ð = cot [12(ð +
ð
2)] .
(D.1)
The Scientific World Journal 11
Table 1
Case ð ð ð ð¹(ð)
1 ð2
â(1 + ð2) 1 ð ðð
2 ð2
â(1 + ð2) 1 ððð =
ððð
ððð
3 âð2
2ð2â 1 1 â ð
2ððð
4 â1 2 â ð2
ð2â 1 ððð
5 1 â(1 + ð2) ð
2ðð ð = (ð ðð)
â1
6 1 â(1 + ð2) ð
2ððð =
ððð
ððð
7 1 â ð2
2ð2â 1 âð
2ððð = (ððð)
â1
8 ð2â 1 2 â ð
2â1 ððð = (ððð)
â1
9 1 â ð2
2 â ð2
1 ð ðð =ð ðð
ððð
10 âð2(1 â ð
2) 2ð
2â 1 1 ð ðð =
ð ðð
ððð
11 1 2 â ð2
1 â ð2
ðð ð =ððð
ð ðð
12 1 2ð2â 1 âð
2(1 â ð
2) ðð ð =
ððð
ð ðð
131
4
1 â 2ð2
2
1
4ðð ð ± ðð ð
141 â ð
2
4
1 + ð2
2
1 â ð2
4ððð ± ð ðð
151
4
ð2â 2
2
ð2
4ðð ð ± ðð ð
16ð2
4
ð2â 2
2
ð2
4ð ðð ± ðððð
17ð2
4
ð2â 2
2
ð2
4
â1 â ð2ð ðð ± ððð
181
4
1 â ð2
2
1
4ðððð ± ðâ1 â ð2ððð
191
4
1 â 2ð2
2
1
4ðð ðð ± ðððð
201
4
1 â ð2
2
1
4
â1 â ð2ð ðð ± ððð
21ð2â 1
4
ð2+ 1
2
ð2â 1
4ðð ðð ± ððð
22ð2
4
ð2â 2
2
1
4
ð ðð
1 ± ððð
23 â1
4
ð2+ 1
2
(1 â ð2)2
4ðððð ± ððð
24(1 â ð
2)2
4
ð2+ 1
2
1
4ðð ð ± ðð ð
25ð4(1 â ð
2)
2(2 â ð2)
2(1 â ð2)
ð2 â 2
1 â ð2
2(2 â ð2)ððð ± â1 â ð2ððð
26 ð > 0 ð < 0ð2ð2
(1 + ð2)2
ð
ââð
2ð
(1 + ð2)ðð ð(â
âð
1 + ð2ð)
12 The Scientific World Journal
Table 1: Continued.
Case ð ð ð ð¹(ð)
27 ð < 0 ð > 0(1 â ð
2)ð2
(ð2 â 2)2
ð
ââð
(2 â ð2)ððð(â
ð
2 â ð2ð)
28 ð < 0 ð > 0ð2(ð2â 1)ð
2
(2ð2 â 1)2
ð
ââð2ð
(2ð2 â 1)ððð(â
ð
2ð2 â 1ð)
29 1 2 â 4ð2
1ð ððððð
ððð
30 ð4
2 1ð ððððð
ððð
31 1 ð2+ 2 1 â 2ð
2+ ð
4ðððððð
ð ðð
32ðŽ2(ð â 1)
2
4
ð2+ 1
2+ 3ð
(ð â 1)2
4ðŽ2
ðððððð
ðŽ(1 + ð ðð)(1 + ð ð ðð)
33ðŽ2(ð + 1)
2
4
ð2+ 1
2â 3ð
(ð + 1)2
4ðŽ2
ðððððð
ðŽ(1 + ð ðð)(1 â ð ð ðð)
34 â4
ð6ð â ð
2â 1 â2ð
3+ ð
4+ ð
2ððððððð
ðð ð2ð + 1
354
ðâ6ð â ð
2â 1 2ð
3+ ð
4+ ð
2ððððððð
ðð ð2ð â 1
361
4
1 â 2ð2
2
1
4
ð ðð
1 ± ððð
371 â ð
2
4
1 + ð2
2
1 â ð2
4
ððð
1 ± ð ðð
38 4ð1
2 + 6ð1â ð
22 + 2ð
1â ð
2ð2ð ððððð
ð1â ðð2ð
39 â4ð1
2 â 6ð1â ð
22 â 2ð
1â ð
2âð2ð ððððð
ð1+ ðð2ð
402 â ð
2â 2ð
1
4
ð2
2â 1 â 3ð
1
2 â ð2â 2ð
1
4
ð2ð ððððð
ð ð2ð + (1 + ð1)ððð â 1 â ð
1
412 â ð
2+ 2ð
1
4
ð2
2â 1 + 3ð
1
2 â ð2+ 2ð
1
4
ð2ð ððððð
ð ð2ð + (â1 + ð1)ððð â 1 + ð
1
42ð¶2ð4â (ðµ
2+ ð¶
2)ð2+ ðµ
2
4
ð2+ 1
2
ð2â 1
4(ð¶2ð2 â ðµ2)
â((ðµ2 â ð¶2)/(ðµ2 â ð¶2 ð2)) + ð ðð
ðµððð + ð¶ððð
43ðµ2+ ð¶
2ð2
4
1
2â ð
21
4(ð¶2ð2 + ðµ2)
â((ð¶2ð2 + ðµ2 â ð¶2)/(ðµ2 + ð¶2ð2)) + ððð
ðµð ðð + ð¶ððð
44ðµ2+ ð¶
2
4
ð2
2â 1
ð4
4(ð¶2 + ðµ2)
â((ðµ2 + ð¶2 â ð¶2ð2)/(ðµ2 + ð¶2)) + ððð
ðµð ðð + ð¶ððð
45 â(ð2+ 2ð + 1)ðµ
22ð
2+ 2
2ð â ð2â 1
ðµ2
ðð ð2ð â 1
ðµ(ðð ð2ð + 1)
46 â(ð2â 2ð + 1)ðµ
22ð
2+ 2 â
2ð + ð2+ 1
ðµ2
ðð ð2ð + 1
ðµ(ðð ð2ð â 1)
The Scientific World Journal 13
Table 2
Case ð2
ð3
ð¹(ð)
474
3(ð2â 3ðð )
4ð
27(â2ð
2+ 9ðð ) â
1
ð(â(ð; ð
2, ð3) â
1
3ð)
484
3(ð2â 3ðð )
4ð
27(â2ð
2+ 9ðð ) â
3ð
3â(ð; ð2, ð3) â ð
49 â5ðð· + 4ð
2+ 33ððð
12
21ð2ð· â 63ðð ð· + 20ð
3â 27ððð
216
â12ð â(ð; ð2, ð3) + 2ð (2ð + ð·)
12â(ð; ð2, ð3) + ð·
501
12ð2+ ðð
1
216ð(36ðð â ð
2)
âð [6â(ð; ð2, ð3) + ð]
3â(ð; ð2, ð3)
511
12ð2+ ðð
1
216ð(36ðð â ð
2)
3â(ð; ð
2, ð3)
âð[6â(ð; ð2, ð3) + ð]
522ð
2
9
ð3
54
ðââ15ð/2ðâ(ð; ð2, ð3)
3â(ð; ð2, ð3) + ð
, ð = 5ð2
36ð
Conflict of Interests
The authors declare that there is no conflict of interests re-garding the publication of this paper.
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