We employ the idea of Hirota’s bilinear method, to obtain some new exact soliton solutions for high nonlinear form of Multiple soliton solutions of secondorder Benjamin-Ono equation. Multiple singular soliton solutions were obtained by this method. Moreover, multiple singular soliton solutions were also derived.Publisher's Versio

Najafi, Mohammad

TWMS Journal of Applied and Engineering Mathematics

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Isik University Academic Open Access

TWMS J. App. Eng. Math. V.2, N.1, 2012, pp. 94-100MULTIPLE SOLITON SOLUTIONS OF SECOND-ORDERBENJAMIN-ONO EQUATIONM. NAJAFI1 §Abstract. We employ the idea of Hirota’s bilinear method, to obtain some new ex-act soliton solutions for high nonlinear form of Multiple soliton solutions of second-order Benjamin-Ono equation. Multiple singular soliton solutions were obtained by thismethod. Moreover, multiple singular soliton solutions were also derived.Keywords: Hirota bilinear method, Benjamin-Ono equation, Multiple soliton solutions,Multiple singular soliton solutions.AMS Subject Classification: 35C07, 35C08, 35A251. IntroductionMany important phenomena and dynamic processes in physics, mechanics, chemistryand biology can be represented by nonlinear partial differential equations. The study ofexact solutions of nonlinear evolution equations plays an important role in soliton theoryand explicit formulas of nonlinear partial differential equations play an essential role inthe nonlinear science. Also, the explicit formulas may provide physical information andhelp us to understand the mechanism of related physical models.In recent years, many kinds of powerful methods have been proposed to find solutions ofnonlinear partial differential equations, numerically and/or analytically, e.g., the homo-geneous balance method [1], the tanh-coth method [2], the Exp-function method [3], thedecomposition method [4] and the improved tanh function method [5].In this paper, by means of the Hirota’s bilinear method, we will obtain some exact andnew solutions for the second-order Benjamin-Ono equation. In the following section wehave a brief review on the Hirota’s bilinear method and in Section 3 and 4, we apply theHirota’s bilinear method to obtain multiple soliton solutions and multiple singular solitonsolutions of the second-order Benjamin-Ono equation. Finally, the paper is concluded inSection 5.1 Department of Physiology, Faculty of Medicine, Kermanshah University of Medical Sciences,Kermanshah, Irane-mail: mnajafi82@gmail.com§ Manuscript received 21 January 2012.TWMS Journal of Applied and Engineering Mathematics Vol.2 No.1 c⃝ Işık University, Departmentof Mathematics 2012; all rights reserved.94M. NAJAFI: MULTIPLE SOLITON SOLUTIONS OF SECOND-ORDER BENJAMIN-ONO EQUATION 952. The Hirota bilinear methodTo formally derive N-soliton solutions for completely integrable equations, we will usethe Hirota’s direct method combined with the simplified version of [6, 7, 8]. It was provedby many that soliton solutions are just polynomials of exponentials. This will be alsoconfirmed in the coming discussions.We first substituteu(x, y, t) = ekx+my−ct, (1)into the linear terms of any equation under discussion to determine the relation betweenk,m and c. We then substitute the Cole-Hopf transformationu(x, y, t) = R (ln f(x, y, t))xx, (2)into the equation under discussion, where the auxiliary function f , for the single solitonsolution, is given byf(x, y, t) = 1 + C1 f1(x, y, t) = 1 + C1 eθ1 . (3)The steps of the Hirota’s method as summarized in [9, 10, 11, 12] are as follows:(i) For the relation between ki,mi and ci, we useu(x, y, t) = eθi , θi = ki x+mi y − ci t, (4)(ii) For single soliton, we usef = 1 + C1 eθ1 , (5)to determine R.(iii) For two-soliton solutions, we usef = 1 + C1 eθ1 + C2 eθ2 + C1C2 a12 eθ1+θ2 , (6)to determine the phase shift coefficient a12, and hence can be generalized foraij ,1 ≤ i < j ≤ 3.(iv) For three-soliton solutions, we usef = 1 + C1 eθ1 + C2 eθ2 + C3 eθ3 + C1C2 a12 eθ1+θ2+C1C3 a13 eθ1+θ3 + C2C3 a23 eθ2+θ3+C1C2C3 b123 eθ1+θ2+θ3 ,(7)to determine b123. Pekcan proved in [13], b123 = a12 a23 a13, then the equation gives riseto three-soliton solutions.In the following, we will apply the aforementioned steps to second-order Benjamin-Onoequation. Multiple soliton solutions are obtained for C1 = C2 = C3 = 1. However,multiple singular soliton solutions are obtained if C1 = C2 = C3 = −1.3. Multiple soliton solutions of the second-order Benjamin-Ono equation:In this paper, we investigate explicit formula of soliton solutions of the following highnonlinear form of second-order Benjamin-Ono equation given in [14],utt + α (u2)xx + β uxxxx = 0 (8)96 TWMS J. APP. ENG. MATH. V.2, N.1, 2012where u = u(x, t) : Rx × Rt → R.To determine multiple-soliton solutions for Eq. (8), we follow the steps presented above.We first consider C1 = C2 = C3 = 1. Substitutingu (x, t) = eθi , θi = kix− wit (9)into the linear terms of Eq.(8) to find the relationwi = ±√−βki2, i = 1, 2, . . . , N (10)and consequently, θi becomesθi = kix±√−β k2i t. (11)To determine R, we substituteu(x, t) = R (ln f(x, t))xx (12)wheref(x, t) = 1 + f1(x, t) = 1 + ek1x±√−β ki2 tinto Eq.(8) and solve to find that R = −6βα.This means that the single singular soliton solution is given byu (x, t) = −6βα k21 ek1x±√βk21t(1 + ek1x±√−βk21t)2 . (13)For the two-soliton solutions, we substituteu(x, t) = −6βα(ln f(x, t))xx, (14)wheref(x, t) = 1 + eθ1 + eθ2 + a12 eθ1+θ2 , (15)into Eq.(8), where θ1 and θ2 are given in Eq.(11) to obtaina12 =(k1 − k2)2k21 + k1k2 + k22, (16)andws = ±√−β k2s , s = 1, 2,henceaij =(ki − kj)2k2i + kikj + k2j, (17)andws = ±√−β k2s , s = 1, 2, 3.This in turn givesf (x, t) = 1 + eθ1 + eθ2 +(k1 − k2)2k21 + k1k2 + k22eθ1+θ2 , (18)whereθi = kix±√−β k2i t, i = 1, 2, (19)which is a two soliton solution(Fig. 1).Similarly, to determine the three soliton solutions, we setM. NAJAFI: MULTIPLE SOLITON SOLUTIONS OF SECOND-ORDER BENJAMIN-ONO EQUATION 97Figure 1. 2-soliton solution, the parameters:α = β = −1, k1 = −1.2, k2 = 2.Figure 2. 3-soliton solution, the parameters:α = β = −1, k1 = −5, k2 =3, k3 = −2.f(x, t) = 1 + eθ1 + eθ2 + eθ3 + a12 eθ1+θ2 + a13 eθ1+θ3 + a23 eθ2+θ3+a12 a23 a13 eθ1+θ2+θ3 .(20)To determine the three soliton solutions explicitly, we substitute the last result for f(x, t)into Eq. (14), (See Fig. 2).The higher level soliton solutions, for n ≥ 4 can be obtained in a parallel manner. Theobtained results confirm that the second-order Benjamin-Ono equation is completely in-tegrable and possesses multiple soliton solutions of any order.98 TWMS J. APP. ENG. MATH. V.2, N.1, 20124. Multiple singular soliton solutions of the second-order Benjamin-Onoequation :We first consider C1 = C2 = C3 = −1. Substitutingu (x, t) = eθi , θi = kix− wit (21)into the linear terms of Eq.(8) to find the relationwi = ±√−βki2, i = 1, 2, . . . , N (22)and consequently, θi becomesθi = kix±√−β k2i t. (23)To determine R, we substituteu(x, t) = R (ln f(x, t))xx (24)wheref(x, t) = 1− f1(x, t) = 1− ek1x±√−β ki2 tinto Eq.(8) and solve to find that R = −6βα.This means that the single singular soliton solution is given byu (x, t) = −6βα k21 ek1x±√βk21t(1 + ek1x±√−βk21t)2 . (25)For the two-soliton solutions, we substituteu(x, t) = −6βα(ln f(x, t))xx, (26)wheref(x, t) = 1− eθ1 − eθ2 + a12 eθ1+θ2 , (27)into Eq.(8), where θ1 and θ2 are given in Eq.(23) to obtaina12 =(k1 − k2)2k21 + k1k2 + k22, (28)andws = ±√−β k2s , s = 1, 2,henceaij =(ki − kj)2k2i + kikj + k2j, (29)andws = ±√−β k2s , s = 1, 2, 3.This in turn givesf (x, t) = 1− eθ1 − eθ2 + (k1 − k2)2k21 + k1k2 + k22eθ1+θ2 , (30)whereθi = kix±√−β k2i t, i = 1, 2, (31)which is a two soliton solution(Fig. 3).Similarly, to determine the three soliton solutions, we setM. NAJAFI: MULTIPLE SOLITON SOLUTIONS OF SECOND-ORDER BENJAMIN-ONO EQUATION 99Figure 3. 2-soliton solution, the parameters:α = β = −0.9, k1 = 1.1, k2 = −1.2.Figure 4. 3-soliton solution, the parameters:α = β = −1, k1 = −0.5, k2 =0.3, k3 = 1.2.f(x, t) = 1− eθ1 − eθ2 − eθ3 + a12 eθ1+θ2 + a13 eθ1+θ3 + a23 eθ2+θ3−a12 a23 a13 eθ1+θ2+θ3 .(32)To determine the three soliton solutions explicitly, we substitute the last result for f(x, t)into Eq. (26), (See Fig. 4).The higher level soliton solutions, for n ≥ 4 can be obtained in a parallel manner. Theobtained results confirm that the second-order Benjamin-Ono equation is completely in-tegrable and possesses multiple soliton solutions of any order.5. ConclusionThe study of exact soliton solutions of nonlinear evolution equations plays an impor-tant role in soliton theory and explicit formulas of nonlinear partial differential equations100 TWMS J. APP. ENG. MATH. V.2, N.1, 2012play an essential role in the nonlinear science. In this paper, by using the Hirota bilin-ear method, we obtained some explicit formulas of soliton solutions for the second-orderBenjamin-Ono equation. Multiple soliton solutions were formally derived. Moreover, mul-tiple singular soliton solutions of any order was derived as well.References[1] Fan, E. G., (2000), Two new applications of the homogeneous balance method, Phys. Lett. A., 265,353-357.[2] Mohamad Jawad A. J., Petkovic M. D. and Biswas, A., (2011), Soliton solutions for nonlinearCalaogero-Degasperis and potential Kadomtsev-Petviashvili equations, Comput. Math. Appl., 62,2621-2628.[3] Borhanifar, A. and Kabir, M. M., (2009), New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations, Journal of Computational and Applied Mathe-matics, 229, 158-167.[4] Kaya, D. and El-Sayed, S. M., (2003), Numerical soliton-like solutions of the potential Kadomt-sevPetviashvili equation by the decomposition method, Phys. Lett. A, 320, 192-199.[5] Inan, I. E. and Kaya, D., (2006), Some exact solutions to the potential Kadomtsev-Petviashvili equa-tion and to a system of shallow water wave equations, Phys. Lett. A, 355, 314-318.[6] Hereman, W. and Zhaung, W., (1980), Symbolic software for soliton theory, Acta Applicandae Math-ematicae, Phys. Lett. A, 76, 95-96.[7] Hereman, W. and Zhuang, W., (1991), A MACSYMA program for the Hirota method, 13th WorldCongress Comput. Appl. Math., 2, 842-863.[8] Hereman, W. and Nuseir, A., (1997), Symbolic methods to construct exact solutions of nonlinearpartial differential equations, Math. Comput. Sim., 43, 13-27.[9] Wazwaz A. M., (2008), Regular soliton solutions and singular soliton solutions for the modifiedKadomtsev-Petviashvili equations, Appl. Math. Comput., 204, 817-823.[10] Wazwaz A. M., (2009), A (3 + 1)-dimensional nonlinear evolution equation with multiple solitonsolutions and multiple singular soliton solutions, Appl. Math. Comput., 215, 1548-1552.[11] Wazwaz A. M., (2007), New solitons and kink solutions for the Gardner equation, Commun NonlinearSci Numer Simul., 12, 1395-1404.[12] Wazwaz A. M., (2010), Multiple soliton solutions for the (2 + 1)-dimensional asymmetric Nizhanik-Novikov-Veselov equation, Nonlinear Anal Ser A: Theory Meth. Appl., 72, 1314-1318.[13] Pekcan A. The Hirota Direct Method (a thesis master of science). Bilkent university 2005.[14] Hereman, W., Banerjee, P. P., Korpel, A., Assanto, G., Van Immerzeele, A. and Meerpole, A., (1986),Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraicmethod, J. Phys. A. Math. Gen., 19(5), 607-628.Mohammad Najafi, research scholar of Mathematics, has been workingRazi University, Iran, since 23 September 2008. He has M.Sc in Facultyof Science, Razi University, Kermanshah Iran. His area of research are ap-plied mathematics and computational sciences, wave phenomena, numericalsolution of ODE, PDE, nonlinear waves in Physics and Soliton theory.Reproduced with permission of the copyright owner. Further reproduction prohibited withoutpermission.

Multiple soliton solutions of second-order Benjamin-Ono equation

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Multiple soliton solutions of second-order Benjamin-Ono equation

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