EXACT SOLUTIONS OF THE LAKSHMANAN – PORSEZIAN – DANIEL EQUATION

. In this paper, the Lakshmanan – Porsezian – Daniel (LPD) equation is considered. This equation is integrable and admits Lax pair. The LPD equation is the generalization of the nonlinear Schrodinger (NLS) equation and described by Ablowitz-Kaup-Newell-Segur (AKNS) system. Using the sine-cosine method and the hyperbolic tangent method a variety of new exact solutions are obtained. These methods are effective tools for searching exact solutions of nonlinear partial differential equations in mathematical physics. The obtained solutions are found to be important for the explanation of some practical physical problems.

Studying the nonlinear excitations of the spin chains with competing bilinear and biquadratic interactions attracts is the main activity in mathematics and physics. For this reason, Lakshmanan, Porsezian, and Daniel had been studied the integrable properties of a classical one-dimensional isotropic biquadratic Heisenberg spin chain (HSC) in its continuum limit by using a geometric method in Refs. [8,15]. Researchers suggested the integrable Lakshmanan -Porsezian -Daniel (LPD) equation which has the higher-order terms (dispersions and nonlinear effects).
The LPD equation is given by [8,15], where ( , ) is a complex valued function of the spatial coordinate and the time , is real constant, the subscripts denote the partial derivatives with respect to the variables , . The LPD equation is NLS type equation with higher-order nonlinear terms, such as fourth-order dispersion, second-order dispersion, cubic and quintic nonlinearities. It also describes the effect of higher-order molecular excitations that introduce quadruple-quadruple coefficients and is a candidate of integrable. Moreover, the LPD equation demonstrates many integrability properties like Painleve analysis, Lax pair representation, soliton solutions, and so on. More clearly the LPD equation describes the nonlinear effect in Refs. [8,15,5].
In the case = 0, (1) reduces into nonlinear Schrodinger equation Linear eigenvalue problem for (1), which is obtained through the Ablowitz -Kaup -Newell-Segur (AKNS) system [2,13,14], is written as with eigenfunctions as Ψ = (Ψ 1 , Ψ 2 ) , and is a parameter, so that is equivalent to (1). Matrices , are Lax pair of (1). The compatibility condition (7) can be understood also as the zero curvature condition. Optical solitons for the local (classical) LPD equation are found by modified simple equation method in [3], by the trial equation method [4], and by Riccati equation approach [16]. Dynamical behavior of solution in integrable nonlocal LPD equation is studied via Darboux transformation in Ref. [9]. Very recently inverse scattering transform has been applied in Ref. [20] where generalized nonlocal Lakshmanan -Porsezian -Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established.
In this paper, we construct some new exact solutions for (1) by analytical methods. We study the LPD equation (1) by the sine-cosine method and the hyperbolic tangent method. Such methods have been widely applied for a wide variety of nonlinear partial differential equations to obtain different kind of solutions.

2.1
The Sine Solution. According to method the sine solution of the (9) can be found by transformation where parameters , and will be determined, and is wave number. We use (10) and its derivatives After substitution of Eqs. (10)- (13) into (9) Using the balance method, by equating the exponents of from (14) we find : Substitute (15) in (14) From (16) we have the next system .
According to method, we apply the following series expansion, where = tanh( ) and is a positive integer, in most cases, that will be determined. To determine the parameter , we usually balance the linear terms of highest-order derivative in the resulting equation with the highest-order nonlinear terms. For our (34), balancing the nonlinear term 5 , which has the exponent 5 , with the highest order derivative ′′′′ , which has the exponent + 4, yields 5 = + 4 that gives = 1. Then, the hyperbolic tangent method allows us to use the substitution and derivatives by method are

Conclusion.
In this work, the Lakshmanan -Porsezian -Daniel equation was studied by the sine-cosine method and the hyperbolic tangent method. We obtained the periodic solutions and the solitary wave solutions. These methods can also be performed to other nonlinear partial differential equations in mathematical physics.