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

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 Schrödinger equation and the complex modified Korteweg–de Vries equation. The nonlinear Schrödinger 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.


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 sinecosine method [1,[4][5][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 where   q x,t is a complex valued function of the spatial coordinate x and the time t , α is a real constant, i is imaginary unit. The equation was introduced in [8] and studied in [9][10][11]. It is a combination of the nonlinear Schrödinger equation and the complex modified Korteveg-de Vrise equation. When 0   the Hirota equation (1) can be reduced to the nonlinear Schrödinger equation.

Review of the sine-cosine method
In this section, we describe the sine-cosine method [1,[4][5][6]. According to the sine-cosine method by using a wave transformation where , x ct   the parameters µ and β will be determined, and µ is wave number and c is wave speed respectively [1]. The derivatives of (5) and so on for the other derivatives. Applying (5)-(10) into the reduced ordinary differential equation (4)   , where these coefficients have to vanish. The system of algebraic equations among the unknown β and µ will be given and from that, we can determine coefficients.

Implementation of the sine-cosine method
We consider the Hirota equation (1). By transformation where a,d are real constants, the equation (1) can be converted to (12) By separating real and imaginary parts in the equation (12) we obtain the system of equations into system of equations (13)-(14) we get the following two ordinary differential equations: Integrate equation (17) once, with respect to ξ , yields where L is constant of integration. As the same function   u ξ satisfies both equations (16) and (18), we obtain the following constraint condition: In next subsection, we solve the equation (21) by the sine-cosine method.

The sine solution
According to method the solution of the (21)

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 Schrödinger equation and the complex modified Korteveg-de Vries equation in case α = 0 we can get exact solutions for the nonlinear Schrödinger 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.