Journal of Mathematics Research; Vol. 6, No. 4; 2014 ISSN 1916-9795 E-ISSN 1916-9809 Pulished y Canadian Center of Science and Education New Exact Solutions for a Class of High-order Dispersive Cuic-quintic Nonlinear Schrödinger Equation Ying Huang 1 & Peng Liu 1 1 School of Mathematics and Statistics, Chuxiong Normal University, Chuxiong, P.R.China Correspondence: Ying Huang, School of Mathematics and Statistics, Chuxiong Normal University, Chuxiong, P.R.China. E-mail: huang11261001@163.com Received: Septemer 27, 2014 Accepted: Octoer 11, 2014 Online Pulished: Novemer 6, 2014 doi:10.5539/jmr.v6n4p104 Astract URL: http://dx.doi.org/10.5539/jmr.v6n4p104 In the paper, the exact solutions to the cuic-quintic nonlinear Schrödinger equation with third and fourth-order dispersion terms is considered. The improved homogeneous alance method is used for constructing a series of new exact envelop wave solutions, including envelop solitary wave solutions, envelop periodic wave solutions and an envelop rational solution. Keywords: exact solution, high dispersive, nonlinear Schrödinger equation 1. Introduction Since third and fourth-order dispersion terms play a crucially important role in descriing the propagation of extremely short pulses, the generalized nonlinear Schrödinger equation iq z β 2 2 q tt + γ 1 q 2 q = i β 3 6 q ttt + β 4 24 q tttt γ 2 q 4 q (1) is taken as a model for a su-picoseconds pulse propagation in a medium which exhiits a paraolic nonlinearity law (Shundong, 2007, Karpman,1997). In particular, q(z, t) is the slowly varying envelope of the electromagnetic field, β 2 is the parameter of the group velocity dispersion, β 3 and β 4 are third-order and fourth-order dispersion, respectively, γ 1 and γ 2 are the nonlinearity coefficients (Karpman, 1998). As to this equation, its modulation instaility of optical wave was numerically investigated (Woo-Pyo,2002), the relative exact analytic solutions were studied very few, the researchers mainly studied its several special cases ecause of itself complications. For the case β 3 = 0, a series of analytical exact solutions are constructed y means of F-expansion method (Gui-Qiong, 2011). In the case β 3 = γ 1 = 0, a dark solitary wave solution with amplitude only depending on the time was presented (Azzouzi,2014). In contrast, when β 3 = β 4 = γ 2 = 0, various types exact solution were discovered (Azzouzi, 2009, Huiqing,2009, Wenxiu,2009, Anjan,2010 ). Two exact periodic solutions and two kink soliton solutions had een constructed with the homogeneous alance method (Shundong, 2007). Here, we construct new the exact solutions with the aid of the improved homogeneous alance method. In section 2, we introduce the general method for solving complex partial differential equations, and solve the new analytical solutions to high-order dispersive cuic-quintic nonlinear Schrödinger equation in section 3. 2. The Main Method In this section, we propose a general method to otain exact solutions for complex nonlinear partial differential equations, ased on the homogeneous alance method. Consider a given complex partial differential equation with independent variales z and t H(q, q z, q zz,...q t,...) = 0, (2) where H is in general a complex polynomial in q and its various partial derivatives. The main steps are: First. Taking the transformation 104
q(x, t) = u(ξ)e i(p 1z+p 2 t+c 0 ), ξ = k 1 z + k 2 t + ξ 0, (3) where k i, p i, i = 1, 2 are constants to e determined, c 0, ξ 0 are aritrary constants. Sustituting (3) into (2) yields a complex ordinary differential equation, in which the real and imaginary parts read, respectively, Second. The next crucial step is to take H R (u, u, u,...) = 0, (4) H I (u, u, u,...) = 0. (5) u(ξ) = n a i F i (ξ), (6) i=1 where a i are constants to e determined later, n is a positive integer that can e determined y alancing the highest-order derivative terms with the highest power nonlinear terms in the equations, and F(ξ) satisfies the first kind elliptic equation where i are aritrary constants, ut 3 is not equal to zero. (F (ξ)) 2 = 1 + 2 F 2 (ξ) + 3 F 4 (ξ). (7) Third. Sustitute Eq.(7) with (6) into (4) and (5), the left hand sides of (4) and (5) can e converted into two finite series in F i, respectively, and equating each coefficient of F i to zero yields a system of algeraic equations for k i, p i, a i. Solve it, k i, p i and a i can e expressed y i and the coefficients of Eq.(2). Fourth. Solving Eq.(7) and compounding it s solutions with (6) and (3), we can otain a series of envelop solitary wave solutions, triangle functions and rational solutions of Eq.(2). 3. Exact solutions to Eq. (1) In this section, we will construct the exact solutions for Eq.(1). Making the transformation q = u(ξ) e i(ωt+θz), ξ = µt + αz + ξ 0, (8) for Eq.(1), where ω, θ, µ and α are constants to e determined. Then separating the real and imaginary parts of the resulting complex ordinary differential equation, we otain β 4 µ 4 24 u ξξξξ + µ 2 ( β 2 2 β 4 4 ω2 β 3 2 ω)u ξξ + ( β 4 24 ω4 + β 3 6 ω3 + θ β 2 2 ω2 )u γ 1 u 3 γ 2 u 5 = 0; µ 3 ( β 3 6 + β 4 6 ω)u ξξξ + (β 2 ωµ α β 3 2 µω2 β 4 6 µω3 )u ξ = 0. (9) Especially, the method of dealing with the aove relations differs from Shundong s (Shundong,2007). Setting we finally get { µ 3 ( β 3 6 + β 4 6 ω) = 0, β 2 ωµ α β 3 2 µω2 β 4 6 µω3 = 0. (10) β 4 µ 4 u ξξξξ + 6µ 2 (2β 2 β 4 ω 2 2β 3 ω)u ξξ + (β 4 ω 4 + 4β 3 ω 3 + 24θ 12β 2 ω 2 )u 24γ 1 u 3 24γ 2 u 5 = 0. (11) Next, our task is to solve the aove ordinary differential equation, y alancing the highest order derivative terms and the highest power nonlinear terms, we choose the ansatz u = af(ξ), a 0, (12) 105
where a is a constant to e determined, and F(ξ) must satisfy the following auxiliary equation ( df dξ )2 = + cf 2 + df 4, (13) where, c and d are aritrary constants. Sustituting (12) with (13) into (11) and letting each coefficient of F i (i = 1, 2, 3) to e zero, we get algeraic equations β 4 µ 4 (c 2 + 12d) + 6µ 2 (2β 2 β 4 ω 2 2β 3 ω)c + β 4 ω 4 + 4β 3 ω 3 + 24θ 12β 2 ω 2 = 0 20cdβ 4 µ 4 + 12dµ 2 (2β 2 β 4 ω 2 2β 3 ω) 24γ 1 a 2 = 0 24d 2 β 4 µ 4 24γ 2 a 4 = 0. Solving the algeraic equations (10) and (14) yields (14) ω = β 3 ; a = ± 1 β 4 β 4 µ = ± 1 6γ 1 d β β4 4 β 4 θ = β2 3 (β2 3 + 4β 2β 4 ) 8β 3 4 6γ1 dβ 2 4 3γ β4 2 d γ 2 (2β 2 β 4 + β 2 3 ) ; 5cγ 2 γ 2 3d(2β 2 β 4 + β 2 3 ) 5cd ; α = µ β 3(3β 2 β 4 + β 2 3 ) 3β 3 4 β 4(c 2 + 12d) µ 4 c(2β 2β 4 + β 2 3 ) µ 2. (15) 24 4β 4 Solving F(ξ) from (13) one y one and sustituting them into (15) together with (8) yields the following exact solutions. Envelop period wave solutions of Eq.(1) have the form: d tan 4 dξ e i(ωt+θz), (16) d cot 4 dξ e i(ωt+θz), (17) where c = 2 d. Envelop kink soliton solutions are given y: d tanh 4 dξ e i(ωt+θz), (18) d coth 4 dξ e i(ωt+θz), (19) where c = 2 d. Envelop periodic wave solutions of the form: where = 0, c < 0, d > 0. d sec ξ e i(ωt+θz), (20) d csc ξ e i(ωt+θz), (21) 106
Envelop ell soliton solutions are given y: where = 0, c > 0, d < 0, and also where = 0, c > 0, d > 0. q = ± d sech cξ e i(ωt+θz), (22) c d csch cξ e i(ωt+θz), (23) Envelop rational solution is ±1 dξ e i(ωt+θz), (24) where = c = 0, d > 0, and this is singular at ξ = 0. In (16)-(24), ξ = µt + αz + ξ 0, a, θ, ω, α and µ are given y (15). 3. Conclusions In summary, Shundong, Z. early considered the exact envelop wave solutions to the high dispersive cuic-quintic nonlinear Schrödinger equation which descries the propagation of extremely short pulses. Based on the improved homogeneous alance method, we otain some exact envelop wave solutions y taking the first elliptic equation as auxiliary ordinary differential equation, some physical phenomena of extremely short pulses can e etter understood with the help of these new solutions. Acknowledgements This work was supported y Chinese Natural Science Foundation Grant (11261001) and Yunnan Provincial Department of Education Research Foundation Grant (2012Y130). References Anjan, B., Allison, M., Daniela, M., Fayequa, M., & Keka, C. B. (2010). An exact solution for the modified nonlinear Schrödingers equation for Davydov solitons in -helix proteins. Mathematical Biosciences, 227(1), 68-71. http://dx.doi.org/10.1016/j.ms.2010.05.008 Azzouzi, F., Triki, H., & Grelu, P. (2014). Dipole soliton solution for the homogeneous high-order nonlinear Schrödinger equation with cuiccquinticcseptic non-kerr terms. Applied Mathematical Modelling, (in press) http://dx.doi.org/10.1016/j.apm.2014.08.011 Azzouzi, F., Triki, H., Mezghich, K., & EI Akrmi, A. (2009). Solitary wave solutions for higher dispersive cuicquintic nonlinear Schrödinger equation. Chaos Solitons Fract., 39(3), 1304-1307. http://dx.doi.org/10.1016/j.chaos.2007.06.024 Gui-Qiong, X. (2011). New types of exact solutions for the fourth-order dispersive cuiccquintic nonlinear Schrödinger equation. Applied Mathematics and Computation, 217(12), 5967-5971. http://dx.doi.org/10.1016/j.amc.2010.12.008 Huiqun, Z. (2009). New exact complex travelling wave solutions to nonlinear Schrödinger(NLS) equation. Commun Nonlinear Sci Numer Simulat., 14(3), 668-673. http://dx.doi.org/10.1016/j.cnsns.2007.11.014 Karpman, V. I., & Shagalov, A. G. (1997). Solitons and their staility in high dispersive system.i. Fourthorder nonlinear Schrödinger-type equations with power-law nonlinearities. Physics letter A, 228(1-2), 59-65. http://dx.doi.org/10.1016/s0375-9601(97)00063-7 Karpman, V. I. (1998). Evolution of solitons descried y high-order nonlinear Schrödinger equations. letter A, 244(5), 397-400. http://dx.doi.org/10.1016/s0375-9601(98)00251-5 Physics Shundong, Z. (2007). Exact solutions for the high-order dispersive cuic-quintic nonlinear Schrödinger equation y the extended hyperolic auxiliary equation method. Chaos Solitons Fract, 34(5), 1608-1612. http://dx.doi.org/10.1016/j.chaos.2006.05.001 107
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