Dark-Bright Soliton Solutions for Some Evolution Equations

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.16(2013) No.3,pp Dark-Bright Soliton Solutions for Some Evolution Equations Adem C. Çevikel a, Esin Aksoy a, Özkan üner b, Ahmet Bekir c a Yıldız Technical University, Education Faculty, Department of Mathematics Education, Istanbul-TURKEY b Dumlupınar University, School of Applied Sciences, Department of Management Information Systems, Kutahya-TURKEY c Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir-TURKEY (Received 19 March 2013, accepted 16 May 2013) Abstract: In this paper, the topological (dark) as well as non-topological (bright) soliton solutions of nonlinear evolution equations are obtained by the solitary wave ansatz method. Under some parameter conditions, exact solitary wave solutions are obtained. Note that, it is always useful and desirable to construct exact solutions especially soliton-type (dark, bright, kink, anti-kink, etc...) envelope for the understanding of most nonlinear physical phenomena. Keywords: Exact solutions; Dark-bright soliton; KP-BBM equation; Landau-inzburg-Higgs equation 1 Introduction There are various nonlinear evolution equations (NLEEs) that are studied in past several decades [1]. Nonlinear evolution equations are special classes of the category of partial differential equations.there are a lot of NLEEs that are integrated using various mathematical techniques. There are soliton solutions, compactons, pekons, cuspons, stumpons, cnoidal waves, singuler solitons, and more other solutions that have been found. These types of solutions are very important and arise in various areas of natural, physical sciences, applied mathematics and engineering. It is well known that seeking explicit solutions for nonlinear evolution equations, by using different numerous methods, is of important significance in mathematical physics and becomes one of the most exciting and extremely active areas of research investigation. In fact, when we want to understand the physical mechanism of phenomena in nature, described by nonlinear partial differential equations (PDEs), exact solution for the nonlinear PDEs have to be explored [2]. In particular, the traveling wave solutions play an important role in the study of the models arising from various natural phenomena and scientific and engineering fields; for instance, the wave phenomena observed in fluid dynamics, elastic media, optical fibers, nuclear physics, high-energy physics, plasma physics, gravitation and in statistical and condensed matter physics, biology, solid state physics, chemical kinematics, chemical physics and geochemistry, etc. There are various methods of integrability of NLEEs that are nowadays known. Some of these common methods of integrability are tanh-sech method [3 5], extended tanh method [6, 7], sine-cosine method [8, 9], first integral method ( [10, 11], extented mapping method [12, 13], ) -expansion method [14 16], exponential function method [17 19] and F-expansion method [20, 21] were used to develop nonlinear dispersive and dissipative problems. Solitons are ubiquitous in nature, appearing in diverse systems such as shallow water waves, DNA excitations, matter waves in Bose Einstein condensates, and ultrashort pulses (or laser beams) in nonlinear optics [22, 23]. Two different types of envelope solitons, bright and dark, can propagate in nonlinear dispersive media. Compared with the bright soliton which is a pulse on a zero-intensity background, the dark soliton appears as an intensity dip in an infinitely extended constant background [24, 25]. Remarkably, interest in dark and bright solitons has grown steadily in recent years [26 30]. In this paper, there will be one such NLEE that will be studied. For Kadomtsov Petviashvili Benjamin Bona Mahony (KP-BBM) and Landau-inzburg-Higgs (LH) equations, the one-soliton solution will be derived. Both bright and dark soliton solutions will be revealed. Corresponding author. address: abekir@ogu.edu.tr Copyright c World Academic Press, World Academic Union IJNS /759

2 196 International Journal of Nonlinear Science, Vol.16(2013), No.3, pp (KP-BBM) equation Consider the following nonlinear KP-BBM equation is given by [31] (u t + u x a(u 2 ) x bu xxt ) x + ku yy = 0, (1) which was studied in [32 34]. Wazwaz [32, 33] obtained some solitons solutions and periodic solutions by using the sine cosine method and the extended tanh method. Abdou [34] used the extended mapping method with symbolic computation to obtain some periodic solutions, solitary wave solution and triangular wave solution. The non-topological (bright) soliton solution: solitary wave ansatz of the form [28] and To obtain the non-topological 1-soliton solution of (1), we assume the u(x, y, t) = λsech p τ, (2) τ = η 1 x + η 2 y vt (3) Further λ is the soliton amplitude and η i (i = 1, 2) are the inverse width of the soliton and v is the soliton velocity. The unknown exponent p will be obtained in course of derivation of the exact solution. Therefore from (2) and (3), (u x ) 2 = λ 2 p 2 η 2 1sech 2p τ λ 2 p 2 η 2 1sech 2p+2 τ (4) u xt = p 2 λη 1 vsech p τ + p(p + 1)λη 1 vsech p+2 τ (5) u xx = p 2 λη 2 1sech p τ p(p + 1)λη 2 1sech p+2 τ (6) u yy = p 2 λη 2 2sech p τ p(p + 1)λη 2 2sech p+2 τ (7) uu xx = p 2 λ 2 η 2 1sech 2p τ p(p + 1)λ 2 η 2 1sech 2p+2 τ (8) where τ = η 1 x + η 2 y vt. Substituting (4)-(9) into (1), gives u xxxt = λp 4 η 3 1vsech p τ + 2p(p + 1)(p 2 + 2p + 2)λη 3 1vsech p+2 τ p(p + 1)(p + 2)(p + 3)λη 3 1vsech p+4 τ p 2 λη 1 vsech p τ + p(p + 1)λη 1 vsech p+2 τ +p 2 λη 2 1sech p τ p(p + 1)λη 2 1sech p+2 τ 2ap 2 λ 2 η 2 1sech 2p τ + 2ap(p + 1)λ 2 η 2 1sech 2p+2 τ 2aλ 2 p 2 η 2 1sech 2p τ + 2aλ 2 p 2 η 2 1sech 2p+2 τ +bλp 4 η 3 1vsech p τ 2bp(p + 1)(p 2 + 2p + 2)λη 3 1vsech p+2 τ +bp(p + 1)(p + 2)(p + 3)λη 3 1vsech p+4 τ +kp 2 λη 2 2sech p τ kp(p + 1)λη 2 2sech p+2 τ = 0, (9) (10) From (10) equating the exponents 2p + 2 and p + 4 leads to 2p + 2 = p + 4 (11) so that p = 2 (12) From (10), setting the coefficients of sech 2p+2 τ and sech p+4 τ terms to zero we obtain 20λ 2 aη bvλη1 3 = 0, (13) which gives after some calculations, we have η 1 = λa 6bv (14) IJNS for contribution: editor@nonlinearscience.org.uk

3 Adem C. Çevikel, Esin Aksoy etal: Dark-Bright Soliton Solutions for Some Evolution Equations 197 Similary, the soliton velocity v is found from setting the cofficients of sech p τ terms to zero in Eq. (11) that we get after some calculations and eq. (14), 4λkη λη 2 1 4λη 1 v + 16λη 3 1bv = 0, (15) v = ± 2 (λa + 6kη 2 2 b)(2λab 3b) b(λa + 6kη 2 2 b) (16) We find from setting the coefficients of sech 2p τ and sech p+2 τ terms in Eq. (10) to zero, and from Eq. (14),(16) that have: 6λkη 2 2 6λη λ 2 aη λη 1 v 120λbvη 3 1 = 0, (17) Consequently, λ = ± 6 a, λ = 6kbη2 2 a we construct a family of the one-soliton-type, exact analytic solutions for Eq. (1) as follows: (18) u(x, y, t) = λsech 2 η 1 x + η 2 y vt)}, (19) where the velocity of the solitons v is given in (16), the soliton amplitude λ is given in (18) and the inverse width of the soliton η 1 is given in (14). The topological(dark) soliton solution: Let us begin the analysis by assuming an ansatz solution of the from [35]: and u(x, y, t) = λ tanh p τ, (20) τ = η 1 x + η 2 y vt (21) where λ, η i (i = 1, 2) are the free parameters of soliton and v is the velocity of soliton. The exponent p is also unknown. These will be determined. From Eqs. (21) and (22), we have: u t = λpvtanh p+1 τ tanh p 1 τ} (22) u x = λpη 1 tanh p 1 τ tanh p+1 τ } (23) (u x ) 2 = λ 2 p 2 η 2 1 tanh 2p 2 τ 2 tanh 2p τ + tanh 2p+2 τ } (24) u xt = pλvη 1 (p + 1) tanh p+2 τ 2p tanh p τ + (p 1) tanh p 2 τ } (25) u xx = λpη 2 1 (p 1) tanh p 2 τ 2p tanh p τ + (p + 1) tanh p+2 τ } (26) uu xx = λ 2 pη 2 1 (p 1) tanh 2p 2 τ 2p tanh 2p τ + (p + 1) tanh 2p+2 τ } (27) u yy = λpη 2 2 (p 1) tanh p 2 τ 2p tanh p τ + (p + 1) tanh p+2 τ } (28) u xxxt = λη 3 1vp(p 1)(p 2)(p 3) tanh p 4 τ + 4λη 3 1vp(p 1)(p 2 2p + 2) tanh p 2 τ 2λη 3 1vp 2 (3p 2 + 5) tanh p τ + 4λη 3 1vp(p + 1)(p 2 + 2p + 2) tanh p+2 τ λη 3 1vp(p + 1)(p + 2)(p + 3) tanh p+4 τ (29) where τ = η 1 x + η 2 y vt. Substituting Eqs. (22)-(29) into Eq.(1), we obtain IJNS homepage:

4 198 International Journal of Nonlinear Science, Vol.16(2013), No.3, pp pλvη 1 (p + 1) tanh p+2 τ 2p tanh p τ + (p 1) tanh p 2 τ } +λpη1 2 (p 1) tanh p 2 τ 2p tanh p τ + (p + 1) tanh p+2 τ } 2aλ 2 p 2 η1 2 tanh 2p 2 τ 2 tanh 2p τ + tanh 2p+2 τ } 2aλ 2 pη1 2 (p 1) tanh 2p 2 τ 2p tanh 2p τ + (p + 1) tanh 2p+2 τ } +bλpη1v(p 3 1)(p 2)(p 3) tanh p 4 τ 4(p 1)(p 2 2p + 2) tanh p 2 τ +2p(3p 2 + 5) tanh p τ 4(p + 1)(p 2 + 2p + 2) tanh p+2 τ +(p + 1)(p + 2)(p + 3) tanh p+4 τ} +kλpη2 2 (p 1) tanh p 2 τ 2p tanh p τ + (p + 1) tanh p+2 τ } = 0, (30) By equating the highest exponents of tanh p+4 τ and tanh 2p+2 τ terms in Eq (30), one gets: that yields: 2p + 2 = p + 4 (31) p = 2 (32) From (30), setting the coefficient of tanh 2p+2 τ and tanh p+4 τ terms to zero and after some calculations, the expression η 1 = λa (33) 6vb Again, from (30) setting the coefficients of tanh p 2 τ terms to zero and (33), one obtains 2bλa( 6kη 2 v = ± 2 b + λa)(4λa 3) 6b( 6kη2 2b + λa) (34) Consequently, we can determine the dark soliton solution for the constant coefficient KP-BBM equation when the above expressions of p, v and η 1 are given by Eqs (32), (34) and (33) are substituted in (1) as: which exist provided that p > 0. u(x, y, t) = λ tanh 2 η 1 x + η 2 y vt} (35) 3 LH equation In this section, we present the multi-symplectic form of the Landau-inzburg-Higgs equation in detail, [36] which is a typical nonlinear evolution equation, where m and k are real constants. Now, the bright and dark soliton solution of this equation will be obtained. The bright soliton solution : ansatz: u tt u xx m 2 u + k 2 u 3 = 0, (36) To find exact bright soliton solution for Eq. (36), we introduce the following solitary wave u(x, t) = λsech p τ, (37) where τ = η(x vt) which λ, η are the amplitude, the inverse width of the soliton, respectively, and v is the velocity of the soliton. The unknown index p will be determined during the course of derivation of the solution of this Eq. (37). From the ansatz (37), we obtions: u tt = p 2 λη 2 v 2 sech p τ p(p + 1)λη 2 v 2 sech p+2 τ (38) u xx = p 2 λη 2 sech p τ p(p + 1)λη 2 sech p+2 τ, (39) u 3 = λ 3 sech 3p τ (40) IJNS for contribution: editor@nonlinearscience.org.uk

5 Adem C. Çevikel, Esin Aksoy etal: Dark-Bright Soliton Solutions for Some Evolution Equations 199 Substituting Eqs. (38)-(40) into Eq.(36), we get p 2 λη 2 v 2 sech p τ p(p + 1)λη 2 v 2 sech p+2 τ p 2 λη 2 sech p τ + p(p + 1)λη 2 sech p+2 τ m 2 λsech p τ + k 2 λ 3 sech 3p τ = 0. (41) so that Equating the exponents of sech p+2 τ and sech 3p τ term in equation (41), one obtains Setting the coefficients of sech p τ terms in Eq.(41) to zero and from Eq. (43), we have which gives p + 2 = 3p (42) p = 1 (43) λv 2 λ m 2 λ = 0, (44) v = ± 1 + m 2, (45) where v is the velocity of the soliton. By setting the correponding coefficients of sech p+2 τ and sech 3p τ terms to zero one gets, after some calculations and (45) leads to: 2λv 2 + 2λ + k 2 λ 3 = 0, (46) λ = ± m 2 (47) k where λ is an integration constant related to the initial pulse inverse width. Finally, we get the bright soliton solution for the constant coefficient LH equation, when the above expressions of p, λ, η and v given by Eqs (43), (45) are (47) substituted in (37) as : u(x, t) = λsechη(x vt)}, (48) The dark soliton solution: In this section, we are interested in finding the dark solitary wave solution, as defined in, [35] for the considered LH equation (37). In order to construct dark soliton solutions for Eq. (37), we use an ansatz solution of the form and u(x, t) = λ tanh p τ, (49) τ = η(x vt) (50) where λ, η are unknown constant parameters and v is the velocity of the soliton, that will be determined. The exponent p is also unknown. From Eqs. (49) and (50), we get: u t = λpηvtanh p+1 τ tanh p 1 τ} (51) u x = λpη tanh p 1 τ tanh p+1 τ } (52) u tt = λη 2 v 2 p(p + 1) tanh p+2 τ 2p 2 tanh p τ + p(p 1) tanh p 2 τ } (53) u xx = λη 2 p(p 1) tanh p 2 τ 2p 2 tanh p τ + p(p + 1) tanh p+2 τ } (54) IJNS homepage:

6 200 International Journal of Nonlinear Science, Vol.16(2013), No.3, pp u 3 = λ 3 tanh 3p τ (55) where τ = η(x vt). Substituting Eqs. (51)-(55) into Eq.(36), we obtain λη 2 v 2 p(p + 1) tanh p+2 τ 2p 2 tanh p τ + p(p 1) tanh p 2 τ } λη 2 p(p 1) tanh p 2 τ 2p 2 tanh p τ + p(p + 1) tanh p+2 τ } m 2 λ tanh p τ + k 2 λ 3 tanh 3p τ = 0. (56) By equating the highest exponents of tanh 3p τ and tanh p+2 τ terms in Eq (57), one gets: 3p = p + 2 (57) which yields the following analytical condition: p = 1 (58) Setting the coefficients of tanh p τ terms in Eq.(56) to zero and from Eq. (58), we have 2λη 2 v 2 + 2λη 2 m 2 λ = 0 (59) which gives 4η2 2m 2 v = ± 2η, (60) and equating the exponents of tanh p+2 τ and tanh 3p τ term in equation (56), one obtains 2λη 2 v 2 2λη 2 + k 2 λ 3 = 0, (61) and from Eq. (60) resulted as; λ = ± m k (62) where λ is an free constant. Lastly, we can determine the topological solution for the u(x, t) = λ tanh η(x vt)} (63) where the velocity of the solitons v is given in (60), free parameter λ is given by (62). 4 Conclusions In this paper, the KM-BBM and LH equations are solved analytically and 1-soliton solutions are obtained. This paper will study a generalized form of the NLSE that will be useful in various areas of physical sciences and engineering. This method can also be applied to other kinds of nonlinear partial differential equations. References [1] Ablowitz, M.J., Segur, H., Solitons and inverse scattering transform, SIAM, Philadelphia, (1981). [2] ibbon J.D., A survey of the origins and physical importance of soliton equations,philos. Trans. Roy. Soc. London Ser. A, 315(1985): [3] Malfliet, W., Hereman, W., The tanh method. I: Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54(1996): [4] Bekir, A., Cevikel, A.C., Solitary wave solutions of two nonlinear physical models by tanh coth method, Communications in Nonlinear Science and Numerical Simulation, 14(5)(2009): [5] Wazwaz, A.M., Mehanna, M., Solitons and Periodic Wave Solutions for Coupled Nonlinear Equations, International Journal of Nonlinear Science, 14(2)(2012): [6] El-Wakil, S.A., Abdou, M.A., New exact travelling wave solutions using modified extended tanh-function method, Chaos, Solitons & Fractals, 31(4)(2007): IJNS for contribution: editor@nonlinearscience.org.uk

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8 202 International Journal of Nonlinear Science, Vol.16(2013), No.3, pp Sci,6(2)(2008): [35] Triki, H., Wazwaz, A.M, Dark solitons for a combined potential KdV and Schwarzian KdV equations with t- dependent coefficients and forcing term, Applied Mathematics and Computation, 217(2011): [36] Hu, W., Deng, Z., Han, S., Fan W., Multi-symplectic Runge-Kutta methods for Landau-inzburg-Higgs equation, Applied Mathematics and Mechanics, 30(8)(2009): IJNS for contribution: editor@nonlinearscience.org.uk