EXACT SOLUTIONS OF THE HIROTA EQUATION USING THE SINE-COSINE METHOD Текст научной статьи по специальности «Математика»

DOI: 10.14529/mmph210306

EXACT SOLUTIONS OF THE HIROTA EQUATION USING THE SINE-COSINE METHOD

G.N. Shaikhova, Y.S. Kalykbay

L.N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan E-mail: g.shaikhova@gmail.com

Nonlinear partial differential equations of mathematical physics are considered to be major subjects in physics. The study of exact solutions for nonlinear partial differential equations plays an important role in many phenomena in physics. Many effective and viable methods for finding accurate solutions have been established.

In this work, the Hirota equation is examined. This equation is a nonlinear partial differential equation and is a combination of the nonlinear Schrodinger equation and the complex modified Korteweg-de Vries equation. The nonlinear Schrodinger equation is the physical model and occurs in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. The complex modified Korteweg-de Vries equation has been applied as a model for the nonlinear evolution of plasma waves and represents the physical model that incorporates the propagation of transverse waves in a molecular chain model and in a generalized elastic solid.

To find exact solutions of the Hirota equation, the sine-cosine method is applied. This method is an effective tool for searching exact solutions of nonlinear partial differential equations in mathematical physics. The obtained solutions can be applied when explaining some of the practical problems of physics.

Keywords: Hirota equation; sine-cosine method; solution; ordinary differential equation; partial differential equation; nonlinearity.

Introduction

Nonlinear partial differential equations (PDEs) are widely used as models to describe physical phenomena in various fields of sciences such as biology, solid state physics, fluid mechanics, plasma physics, plasma wave, condensed matter physics, chemical physics, optical fibers, and chemical physics [1]. Various powerful methods such as, Darboux transformation method [2], Hirota's method [3] and sine-cosine method [1, 4-6], modification of the truncated expansion method [7], have been developed to obtain exact solutions of these equations.

In this work, we study the Hirota equation

q + qxx+2I q\2 q+iaqxxx+6\q\2 qx = ° (1)

where q (x,t) is a complex valued function of the spatial coordinate x and the time t, a is a real constant, i is imaginary unit. The equation was introduced in [8] and studied in [9-11]. It is a combination of the nonlinear Schrodinger equation and the complex modified Korteveg-de Vrise equation. When a = 0 the Hirota equation (1) can be reduced to the nonlinear Schrodinger equation.

1. Review of the sine-cosine method

In this section, we describe the sine-cosine method [1, 4-6]. According to the sine-cosine method by using a wave transformation

u ( x, t) = u = (x - ct), (2)

the partial differential equation (PDE)

E1 (ut ,Ux, Uxx, Uxxx >•••) = ° (3)

can be converted to ordinary differential equation (ODE)

E2(u,U',U",U'",...) = 0 (4)

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 [1, 4-6]

u ( x, t) =

V 1 2 ц (5)

0, otherwise,

or

u ( x, t) =

"-'WM't- (6)

0, otherwise,

where g = x - ct, the parameters p and ft will be determined, and p is wave number and c is wave speed respectively [1]. The derivatives of (5) become

(un (fig)) = -n PX cosnP-1 (fg) sin (fg), (7)

(un (p£))" =-n2y2p2Xn cosnP (fg) + ny2rp(np -1) cosnP-2 (fg), (8)

and the derivatives of (6) have next forms

(un (fg)) = n pXn sinnP-1 (fg) cos (fg), (9)

(un (p£ ))" =-n2f2p2Xn sinnP (fg) + ny2rp(np -1) sinnP-2 (fg), (10)

and so on for the other derivatives. Applying (5)—(10) into the reduced ordinary differential equation (4) we obtain a trigonometric equation of cosr (g) or sinr (g) terms. Then, we determine the parameters by first balancing the exponents of each pair of cosine or sine to determine ft. Next, we collect all coefficients of the same power in cosr (fg) or sinr (fg), where these coefficients have to vanish. The

system of algebraic equations among the unknown ft and p will be given and from that, we can determine coefficients.

2. Implementation of the sine-cosine method

We consider the Hirota equation (1). By transformation

q = el(ax+d%(x,t), (11)

where a,d are real constants, the equation (1) can be converted to

2 3 2 3 3 2

-du + iut + 2iauv + m - a u + 2u - 3a iauv - 3aaw + a au + iau^ - 6aau + 6iau uv = 0. (12)

t x xx x xx xxx x v '

By separating real and imaginary parts in the equation (12) we obtain the system of equations

-du + uxx - a2u + 2u3 - 3aauxx + a3au - 6aaU = 0, (13)

ut + 2aux - 3a2aux + auxxx + 6au2ux = 0. (14)

Substituting the wave transformation

u (x,t) = u (£) = u (x - ct), (15)

into system of equations (13)—(14) we get the following two ordinary differential equations:

(a3a - a2 - d)u + (1 - 3aa)u"+ 2 (1 - 3aa)u3 = 0, (16)

(-c + 2a -3a2a)u}+au"'+ 2a(u3) = 0. (17)

Integrate equation (17) once, with respect to £, yields

(-c + 2a - 3a2a) u + au"+ 2au3 = L, (18)

where L is constant of integration.

As the same function u (£) satisfies both equations (16) and (18), we obtain the following constraint condition:

Shaikhova G.N., Exact Solutions of the Hirota Equation

Kalykbay Y.S. using the Sine-Cosine Method

a3a- a - d -(1 - 3aa), L = 0. (19)

—c + 2a — 3a2 a a

By using condition (19), we have

( a3 a2 - da- a2 a) (1 - 3aa)

We can rewrite second order ordinary differential equation (16) as

c = 2a — 3a2a — ------- . (20)

u "+ 2u +

^ a3 a — a2 — d^

1 — 3aa

v

u = 0. (21)

In next subsection, we solve the equation (21) by the sine-cosine method.

2.1. The sine solution

According to method the solution of the (21) can be found by transformation

utf?) = AsinP (tf;). (22)

To find sine solution we use (22) and its derivative

u "(tf) = -tf2p2AsinP (tf) + tf 2Ap(p-1)sinP-2 (tf). (23)

Substitute (22) and (23) into (21) we get

-tf 2p2AsinP (tf;) + /u2Ap(p-1)sinP-2 (tf;) + 2A3sin3p (tf;) + (a3a-a2 -d)Asin^ (tf;) = 0. (24)

Using the balance method, by equating the exponents of sin1, (24) we determine 3 :

p-1 *

P-2 = 3P ^ p = -1. (25)

Substitute (25) in (24) we obtain next equation

y-V^^y _ ft2 _ //

-tf2Asin-1 tf + 2tf 2Asin-3 + 2A3 sin-3 (tf4) + ( ^ ^-)Asin-1 = 0. (26)

Equating the coefficients of each pair of the sine functions, we find the following system of algebraic equations:

sm"1 (tf;): -tfA + (a3a-a2 -d)A = 0, (27)

1 - 3aa

sin

3 : 2A/J2 + 2A3 = 0 . (28)

From (27)-(28) we have

,a3a — a2 — d. a2 + d — A2

tf = ±J(--—-) , a = . , (29)

V 1 - 3aa a3 - 3aA2

where a,d, A are real numbers.

Substituting (25), (29) into (22) and then obtained expression into (11) we have the sine solution of the Hirota equation

q1(x, t ) = ±e'( ax+dt )Asin 1

с I з„ 2—: ^

a a — a — с

1 — 3aa

-( x — ct )

(30)

2 ( aV — da — a2a) a2 + d — A2

where c = 2a — 3a a—-------, a=—--—

(1 — 3aa) a3 — 3a A2

2.2. The cosine solution

To find cosine solution we use

u(ji£) = Acosp(^), (31)

and its second order derivative

u \g) = + cosP-2 (g (32)

Substitute (31) and (32) into (21) we get

Vp2X cosP (^) + ^2Xp (P -1)cosP-2 (^) + 2X3 cos3p (fig) + (^^ -d)XcosP ) = 0. (33)

Using the balance method, by equating the exponents of cosJ, (33) we find ¡3 :

p-1 * 0,

P-2 = 3P ^ P = -1. (34)

Substituting (34) in (33) we obtain next equation

-fXcos-1 + 2 fXcos-3 (ug) + 2X3 cos-3 + (a - d)Xcos-1 (^f) = 0. (35)

From (35) equating the coefficients of each pair of the cosine functions, we find the following system of algebraic equations:

cos-1 (g: VX + (a—-a2 - d)X = 0, (36)

1 - 3aa

cos

~-3 (jg): IX/J2 + 2X3 = 0 . (37)

Solving system (36)-(37) leads to the results,

La3a - a2 - d. a2 + d-X2

,u = ±J(a—~-), a = 3 o2 , (38)

V 1 - 3aa a3 - 3aX2

where a,d, A are real numbers.

Substituting (34), (38) into (31) and then obtained expression into (11) we have the cosine solution

q2 (x, t) = ±el(ac+dt )X cos 1

' ' 3 2---^

a a- a - с

1 - 3aa

■ ( x - ct)

(39)

(aV - da- aa) a2 + d -X2

where c = 2a - 3a2a - -------, a = -

(1 - 3aa) a3 - 3aX2

Conclusion

The sine-cosine method was effectively used for the analytic treatment of the Hirota equation. Exact solutions were derived. The obtained solutions can have an application to some practical physical problems. As the Hirota equation is a combination of the nonlinear Schrodinger equation and the complex modified Korteveg-de Vries equation in case a = 0 we can get exact solutions for the nonlinear Schrodinger equation. The applied method can be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution partial differential equations.

The research work was prepared with the financial support of the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan, IRN project AP08956932.

References

1. Wazwaz A. Partial Differential Equations and Solitary Waves Theory. Springer-Verlag Berlin Heidelberg, 2009, 700 p. DOI: 10.1007/978-3-642-00251-9

2. Bekova G., Yesmakhanova K., Myrzakulov R., Shaikhova G. Darboux Transformation and Soliton Solution for the (2+1)-Dimensional Complex Modified Korteweg-De Vries Equations. Journal of Physics: Conference Series, 2017, Vol. 936, p. 012045 (1-6). DOI: 10.1088/1742-6596/936/1/012045

3. Kutum B.B., Shaikhova G.N. ^-Soliton Solution for Two-Dimensional q-Toda Lattice. Bulletin of the Karaganda University. Physics series, 2019, no. 3(95), pp. 22-26. DOI: 10.31489/2019Ph3/22-26.

4. Wazwaz A.M. The Sine-Cosine Method for Obtaining Solutions with Compact and Noncompact Structures. Appl. Math. Comput., 2004, Vol. 159, Iss. 2, pp. 559-576. DOI: 10.1016/j.amc.2003.08.136

Shaikhova G.N., Kalykbay Y.S.

Exact Solutions of the Hirota Equation using the Sine-Cosine Method

5. Yusufoglu E., Bekir A. Solitons and Periodic Solutions of Coupled Nonlinear Evolution Equations by using Sine-Cosine Method. Internat. J. Comput. Math., 2006, Vol. 83, Iss. 12, pp. 915-924. DOI: 10.1080/00207160601138756

6. Wazwaz A.M. A Sine-Cosine Method for Handling Nonlinear Wave Equations. Mathematical and Computer Modeling, 2004, Vol. 40, Iss. 5, pp. 499-508. DOI: 10.1016/j.mcm.2003.12.010

7. Shaikhova G.N., Kutum B.B., Altaybaeva A.B., Rakhimzhanov B.K. Exact Solutions For the (3+1)-Dimensional Kudryashov-Sinelshchikov Equation. Journal of Physics: Conference Series, 2019, Vol. 1416, pp. 012030(1-6). DOI: 10.1088/1742-6596/1416/1/012030

8. Hirota R. Exact Envelope Solutions of a Nonlinear Wave Equation. Journal of Mathematical Physics, 1973, Vol. 14, Iss. 7, pp. 805-809. DOI: 10.1063/1.1666399

9. Sasa N., Satsuma J. New-Type of Soliton Solutions for a Higher-Order Nonlinear Schrodinger Equation. J. Phys. Soc. Jpn., 1991, Vol. 60, no. 2, pp. 409-417. DOI: 10.1143/JPSJ.60.409

10. Karpman V.I., Rasmussen J.J., Shagalov A.G., Dynamics of Solitons and Quasisolitons of the Cubic Third-Order Nonlinear Schrodinger Equation. Phys. Rev. E., 2001, Vol. 64, Iss. 2, pp. 026614(1-13). DOI: 10.1103/PhysRevE.64.026614

11. Tao Y., He J. Multisolitons, Breathers, and Rogue Waves for the Hirota Equation Generated by the Darboux Transformation. Phys. Rev. E., 2012, Vol. 85, Iss. 2, pp. 02660(1-7). DOI: 10.1103/PhysRevE.85.026601

Received March 24, 2021

Information about the authors

Shaikhova Gaukhar Nurlybekovna, PhD, Associate Professor, Department of General and Theoretical Physics, L.N. Gumilyov Eurasian national university, Nur-Sultan, Kazakhstan, ORCID iD: https://orcid.org/0000-0002-0819-5338, e-mail: g.shaikhova@gmail.com

Kalykbay Yrysbay, L.N. Gumilyov Eurasian national university, Nur-Sultan, Kazakhstan, ORCID iD: https://orcid.org/0000-0002-1465-8298.

Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2021, vol. 13, no. 3, pp. 47-52

УДК 517.1+517.3 DOI: 10.14529/mmph210306

ТОЧНЫЕ РЕШЕНИЯ УРАВНЕНИЯ ХИРОТА С ПОМОЩЬЮ МЕТОДА СИНУС-КОСИНУС

Г.Н. Шайхова, Ы.С. Калыкбай

Евразийский национальный университет имени Л.Н. Гумилева, г. Нур-Султан, Казахстан E-mail: g.shaikhova@gmail.com

Нелинейные дифференциальные уравнения в частных производных математической физики являются важным объектом в физике. Так, изучение точных решений нелинейных уравнений в частных производных играет важную роль во многих явлениях в физике. Существует множество эффективных и действенных методов нахождения точных решений.

В данной работе исследовано уравнение Хироты. Это уравнение является нелинейным уравнением в частных производных и представляет собой комбинацию нелинейного уравнения Шредингера и комплексного модифицированного уравнения Кортевега-де Фриза. Нелинейное уравнение Шредингера является физической моделью и встречается в различных областях физики, включая нелинейную оптику, физику плазмы, сверхпроводимость и квантовую механику. Комплексное модифицированное уравнение Кортевега-де Фриза применяется в качестве модели нелинейной эволюции плазменных волн и представляет собой физическую модель, которая включает распространение поперечных волн в модели молекулярной цепочки и в обобщенном упругом твердом теле.

Для нахождения точных решений уравнения Хироты применен метод синус-косинус. Этот метод является эффективным инструментом для поиска точных решений нелинейных уравнений в частных производных математической физики. Полученные решения могут иметь приложение для объяснения некоторых практических задач физики.

Ключевые слова: уравнение Хироты, метод синус-косинуc, решение, обыкновенное дифференциальное уравнение, дифференциальное уравнение в частных производных, нелинейность.

Литература

1. Wazwaz, A. Partial Differential Equations and Solitary Waves Theory / A. Wazwaz. - SpringerVerlag Berlin Heidelberg, 2009. - 700 p.

2. Darboux Transformation and Soliton Solution for the (2+1)-Dimensional Complex Modified Korteweg-de Vries Equations / G. Bekova, K. Yesmakhanova, R. Myrzakulov, G. Shaikhova // Journal of Physics: Conference Series. - 2017. - Vol. 936. - P. 012045 (1-6).

3. Kutum, B.B. g-Soliton Solution for Two-Dimensional g-Toda Lattice. Bulletin of the Karaganda university / B.B. Kutum, G.N. Shaikhova // Physics series. - 2019. - no. 3(95). - P. 22-26.

4. Wazwaz, A.M. The Sine-Cosine Method for Obtaining Solutions with Compact and Noncompact Structures / A.M. Wazwaz // Applied Mathematics and Computation. - 2004. - Vol. 159, Iss. 2. -P. 559-576.

5. Yusufoglu, E. Solitons and Periodic Solutions of Coupled Nonlinear Evolution Equations by using Sine-Cosine Method / E. Yusufoglu, A. Bekir // Internat. J. Comput. Math. - 2006. - Vol. 83, Iss. 12. - P. 915-924.

6. Wazwaz, A.M. A sine-cosine method for handling nonlinear wave equations / A.M. Wazwaz // Mathematical and Computer Modeling. - 2004. - Vol. 40, Iss. 5. - P. 499-508.

7. Exact solutions for the (3+1)-dimensional Kudryashov-Sinelshchikov equation / G.N. Shaikhova, B.B. Kutum, A.B. Altaybaeva, B.K. Rakhimzhanov // Journal of Physics: Conference Series. - 2019. -Vol. 1416. - P. 012030(1-6).

8. Hirota, R. Exact Envelope Solutions of a Nonlinear Wave Equation / R. Hirota // Journal of Mathematical Physics. - 1973. - Vol. 14, Iss. 7. - P. 805-809.

9. Sasa, N. New-Type of Soliton Solutions for a Higher-Order Nonlinear Schrodinger Equation / N. Sasa, J. Satsuma // J. Phys. Soc. Jpn. - 1991. - Vol. 60, no. 2. - P. 409-417

10. Karpman, V.I. Dynamics of Solitons and Quasisolitons of the Cubic Third-Order Nonlinear Schrodinger Equation / V.I. Karpman, J.J. Rasmussen, A.G. Shagalov // Phys. Rev. E. - 2001. - Vol. 64, Iss. 2. - P. 026614.

11. Tao, Y. Multisolitons, Breathers, and Rogue Waves for the Hirota Equation Generated by the Darboux Transformation / Y. Tao, J. He // Phys. Rev. E. - 2012. - Vol. 85, Iss. 2. - P. 02660(1-7).

Поступила в редакцию 24 марта 2021 г.

Сведения об авторах

Шайхова Гаухар Нурлибековна - PhD, доцент, Евразийский национальный университет имени Л.Н. Гумилева, г. Нур-Султан, Казахстан, ORCID iD: https://orcid.org/0000-0002-0819-5338, e-mail: g.shaikhova@gmail.com

Калыкбай Ырысбай - Евразийский национальный университет имени Л.Н. Гумилева, г. НурСултан, Казахстан, ORCID iD: https://orcid.org/0000-0002-1465-8298.