CC BY
12
raÄ®
www.volsu.ru
DOI: https://doi.Org/10.15688/mpcm.jvolsu.2021.3.2
UDC 51-71,51-73 LBC 22.161.2
Submitted: 09.03.2021 Accepted: 30.07.2021
EXACT SOLUTIONS OF THE GENERALIZED NONLINEAR SCHRODINGER EQUATION 1
Gaukhar N. Shaikhova
PhD, Associate Professor, Department of General and Theoretical Physics,
L.N. Gumilyov Eurasian National University
https://orcid.org/0000-0002-0819-5338
K. Munaitpasova St, 22, 010008 Nur-Sultan, Kazakhstan
Arailym M. Syzdykova
Senior Lecturer, Department of General and Theoretical Physics, L.N. Gumilyov Eurasian National University syzdykova'[email protected]
K. Munaitpasova St, 22, 010008 Nur-Sultan, Kazakhstan
o Samgar Daulet
co Student, Department of General and Theoretical Physics,
2 L.N. Gumilyov Eurasian National University
, K. Munaitpasova St, 22, 010008 Nur-Sultan, Kazakhstan
Abstract. In this work, the generalized nonlinear Schrodinger equation is investigated. Exact solutions are derived by the sine-cosine method. This method cio is used to obtain the exact solutions for different types of nonlinear partial dif-
^ ferential equations. Graphs of obtained solutions are presented. The obtained
^ solutions are found to be important for the explanation of some practical physical
> problems.
-a
я .с
CO
©
Key words: generalized nonlinear Schrodinger equation, ODE, PDE, sine-cosine method, exact solution.
Introduction
The study exact solutions for nonlinear partial differential equations play an important role in many phenomena in physics such as hydrodynamics, fluid mechanics, condensed matter physics, plasma physics, optics and so on. Many effective and powerful methods have been established and improved, such as the Jacobi elliptic method [6], the (G0/G)-expansion method [4], the Exp-function method [5], the tanh-function method [7], the Darboux transformation method [3].
In this work we study the generalized nonlinear Schrodinger equation,
qz = i(qtt + 2lql2q + aq) + yqt, (1)
where q is a complex valued function of the spatial coordinate z and the time t, the subscripts denote the partial derivatives with respect to the variables z,t. The equation (1) is a typical soliton equation with rich physical and mathematical applications where a denotes the amplification or absorption and y relates to the group velocity [1;2].
The aim of this paper is to construct some new exact solutions for equation (1). We study the equation (1) by the sine-cosine method that have been extensively studied and widely applied for a wide variety of nonlinear problems [8-10].
The paper is organized as follows. In Section 2, we present the description of sine-cosine method. In Section 3, the sine-cosine method is applied to obtain exact solutions for generalized nonlinear Schrodinger equation.
1. Description of Sine-cosine Method
In this section we describe the sine-cosine method [8-10]. According to the sine-cosine method by using a wave variable
u(z,t) = u(z — ct), (2)
the partial differential equation (PDE)
Ei(u,ut,uz ,uzz ,uzzz ,...) = 0, (3)
can be converted to ordinary differential equation (ODE)
E2(U, —CU,U,U ,U ,...) = 0. (4)
So, one can immediately reduce the one-dimensional PDE (3) into ODE (4) by transformation (2). Then the equation (4) is integrated as long as all terms contain derivatives where integration constants are considered zeros. The solutions of ODE (4) can be expressed in the form
u(z,t) = A cosß |£|< —, (5)
or
u(z,t) = Л sinß(^), Щ< -, (6)
Ц
where 4 = £ — ct and the parameters A, h and p will be determined, and h is wave number and c is wave speed respectively. The derivatives of (5) become
(un)' = —npHAracosrap-1(H4)sin(H4), (7)
(un)'' = — n2H2p2An cosrap(n4) + nH2Anp(np — l) cosrap-2(H^). (8)
And the derivatives of (6) have next forms
(un)' = — npHArasmrap-1(H4)cos(n4), (9)
(un)" = —n2H2p2An sinrap(n4) + nH2Anp(np — l)sinrap-2(n4), (10)
and so on for the other derivatives. Applying (5)-(10) into the reduced ODE (4) we obtain a trigonometric equation of cosp(h4) or sinp(H4) terms. Then, we determine the parameters by first balancing the exponents of each pair of cosine or sine to determine p. Next, we collect all coefficients of the same power in cosfc(h4) or sinfc(h4), where these coefficients have to vanish. The system of algebraic equations among the unknown p, A, and h will be given and from that we can determine coefficients.
2. Using the Sine-cosine Method
We consider the generalized nonlinear Schrodinger equation (1). By transformation
q = ez(az+dt)u(z,t), (11)
the equation (1) can be converted to
iau + uz + id2u + 2dut — iutt — 2iu3 — iau — idyu — yut = 0, (12)
Be separating real and imaginary parts in the equation (12) we obtain next system
uz + 2dut — yut = 0, (13)
u(a + d2 — a — yd) — utt — 2u3 = 0. (14)
Substituting the wave transformation
u(z,t) = u(4) = u(z — ct), (15)
into system of equations (13)-(14) we obtain that
u (l — 2dc + yc) = 0, (16)
u(a + d2 — a — yd) — c2u" — 2u3 = 0. (17)
From equation (16) we can get
c = o^T—. (18)
2d — y
So, finally we study ODE (17)
u(a + d2 — a — Yd) — c2u" — 2u3 = 0, (19)
where c is expressed by equation (18).
2.1. The Sine Solution
According to method the sine solution of the (19) can be found by transformation
u = A sinp(^). (20)
We use (20) and its second order derivative
u = —H2 (2A sin((^) + h2A((( — 1)sin(-2(^), (21)
and substitute (20) and (21) into (19) we obtain
A sinp(n^)(a + d2 — a — y^) + cV(2A sinp(^) —
—cVA((( — 1)sinp-2(^) — 2A3 sin3p(n^) = 0. (22)
From (22) we find (3:
(3 — 2 = 3( ^ (3 = —1. (23)
Substitute (23) in (22) we obtain equation
A sin-1(^)(a + d2 — a — y^) + c2h2A sin-1 (h4) —
—2cv2a sin-3^) — 2A3 sin-3^) = 0. (24)
From the equation (24) we have the next system
sin-1 (^4) : A(a + d2 — a — y^) + cv2A = 0, (25)
sin-3 (h4) : — 2c2h2A — 2A3 = 0. (26)
The equation (25) gives
ц = ± + d2 - a - yd), (27)
с
and from (26) we obtain
A = ±y/ (a + d2 — a — Yd). (28)
Substituting (27)-(28) into (20) and then obtained expression into (11) we have the sine solution
qi(z,t) = ±el(az+dtWa - d2 - a - ydsin-1(^ (a + ^ a yd) {z - ct)), (29)
с
Where C = 2d—Y •
2.2. The Cosine Solution
Cosine solution of the (19) can be found by transformation
u = A cosp(H4), (30)
and its second order derivative is
u = —h2P2A cosp(h4) + H2AP(P — 1)cosp-2(H4). (31)
Substitute (30) and (31) into (19) we obtain
A cosp(H^)(a + d2 — a — Yd) + cVp2A cosp(h4) —
—c2H2Ap(p — 1) cosp-2(H4) — 2A3 cos3p(H4) = 0. (32)
From (32) we find p:
p — 2 = 3p ^ p = —1. (33)
Substitute (33) in (32) we obtain
A cos-1(h4)(« + d2 — a — Yd) + c2h2A cos-1(h4) —
— 2c2h2a cos-3(h4) — 2A3 cos-3(h4) = 0. (34)
From the equation (34) we have the next system
cos-1(h4) : A(a + d2 — a — y^) + c2h2A = 0, (35)
cos-3(h4) : —2c2 h2A — 2A3 = 0. (36)
From (35) we have
ц = ± ^-(a + - a - yd\ (з7)
с
and from (36) we obtain
A = ±y/ (a + d2 — a — Yd). (38)
Substituting (37)-(38) into (30) and then obtained expression into (11) we obtain the cosine solution
q2(z,t) = ±eг(■az+dt)^У^-d2—a-Y^ ^^^ (z — ct)), (39)
c
where C = .
In Figures 1, 2 and 3, we show the profile of solutions (29) and (39) with three cases: a = y; a < y; ea > Y-
Conclusion
In this work the sine-cosine method was used to present an analytic study of the generalized nonlinear Schrodinger equation. Several exact solutions were obtained. The plots of obtained solutions are presented for cases a = y; a < y; ea > y. The performance of the scheme shows that the method is powerful and reliable. The present method is readily applicable to a large variety of such nonlinear equations.
NOTE
1 This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09057947).
REFERENCES
1. Agrawal G.P. Nonlinear Fiber Optics. San Diego, Academic Press, 2001. 467 p.
2. Allen L., Eberly J.H. Optical Resonance and Two-Level Atoms. New York, Wiley Press, 1975. xiii + 233 p.
3. Bekova G., Yesmakhanova K., Myrzakulov R., Shaikhova G. Darboux Transformation and Soliton Solution for the (2+1)-Dimensional Complex Modifed Korteweg-de Vries Equations. J.Phys.: Conference Series, 2017, vol. 54, no. 936, article ID: 012045.
4. Feng J., Li W., Wan Q. Using (G0/G)-Expansion Method to Seek Traveling Wave Solution of Kolmogorov — Petrovskii — Piskunov Equation. Appl. Math. Comput, 2011, no. 217, pp. 5860-5865.
5. He J., Wu X. Exp-Function Method for Nonlinear Wave Equations. Chaos, Solitons & Fractals, 2006, vol. 30, pp. 700-708.
6. Inc M., Ergut M. Periodic Wave Solutions for the Generalized Shallow Water Wave Equation by the Improved Jacobi Elliptic Function Method. Appl. Math. E-Notes, 2005, no. 5, pp. 89-96.
7. Malfliet W. The Tanh method. II. Perturbation Technique for Conservative Systems. Phys. Scr., 1996, vol. 54, pp. 569-575.
8. Wazwaz A.M. A Sine-Cosine Method for Handling Nonlinear Wave Equations. Math. Comput. Modelling, 2004, vol. 40 (5-6), pp. 499-508.
9. Wazwaz A.M. The Sine-Cosine Method for Obtaining Solutions with Compact and Noncompact Structures. Appl. Math. Comput, 2004, vol. 159 (2), pp. 559-576.
10. Wazwaz A.M. Solitons and Periodic Solutions of Coupled Nonlinear Evolution Equations by Using Sine-Cosine Method. Internat. J. Comput. Math, 2006, vol. 83 (12), pp. 915924.
ТОЧНЫЕ РЕШЕНИЯ ОБОБЩЕННОГО НЕЛИНЕЙНОГО УРАВНЕНИЯ ШРЕДИНГЕРА
Гаухар Нурлыбековна Шайхова
Доктор (PhD), доцент кафедры общей и теоретической физики,
Евразийский национальный университет им. Л.Н. Гумилева
https://orcid.org/0000-0002-0819-5338
ул. K. Мунайтпасова, 22, 010008 г. Нур-султан, Казахстан
Арайлым Мерекеновна Сыздыкова
Старший преподаватель кафедры общей и теоретической физики, Евразийский национальный университет им. Л.Н. Гумилева [email protected]
ул. К. Мунайтпасова, 22, 010008 г. Нур-султан, Казахстан
Самгар Даулет
Студент кафедры общей и теоретической физики, Евразийский национальный университет им. Л.Н. Гумилева [email protected]
ул. К. Мунайтпасова, 22, 010008 г. Нур-султан, Казахстан
Аннотация. В работе исследуется обобщенное нелинейное уравнение Шредингера. Точные решения получены методом синус-косинусов. Этот метод используется для получения точных решений для различных типов нелинейных уравнений в частных производных. Представлены графики полученных решений. Полученные решения важны для объяснения некоторых задач физики.
Ключевые слова: обобщенное нелинейное уравнение Шредингера, ОДУ, ДУЧП, метод синус-косинусов, точное решение.
