Homotopic Approximate Solutions for the General Perturbed Nonlinear Schrödinger Equation

In this work, a class of perturbed nonlinear Schrödinger equation is studied by using the homotopy perturbation method. Firstly, we obtain some Jacobi-like elliptic function solutions of the corresponding typical general undisturbed nonlinear Schrödinger equation through the mapping deformation method, and secondly, a homotopic mapping transform is constructed, then the approximate solution with arbitrary degree of accuracy for the perturbed equation is researched, it is pointed out that the series of approximate solution is convergent. Finally, the efficiency and accuracy of the approximate solution is also discussed by using the fixed point theorem.


INTRODUCTION
With the development of soliton theory in nonlinear science, searching for analytical exact solutions or approximate solutions of the nonlinear partial differential equations (NLPDEs) plays an important and significant role in the study of the dynamics of those nonlinear phenomena [1].Many powerful methods have been used to handle these problems recently.For example, inverse scattering transformation [2], Hirota bilinear method [3], homogeneous balance method [4], Bäcklund transformation [5], Darboux transformation [6], projective Riccati equations method [7], the generalized Jacobi elliptic function expansion method [8] and so on [9].But because of the complexity of NLPDEs, people can't find the exact solutions for many of them especially with disturbed term.Researchers had to develop some approximate methods for nonlinear theory, such as multiple-scale method [10], variational iteration method [11], indirect matching method [12] etc.The main essence of these methods is the study of nonlinear problems dealt with linear problems by using the approximate expansion.
The homotopy analysis method (HAM) was firstly proposed in 1992 by Liao [13], which yields a fast convergence for most of the selected problems.It also showed a high accuracy and a rapid convergence to solutions of the nonlinear partial evolution equations.After this, many types of nonlinear problems were solved with HAM by others, such as discrete KdV equation [14], a smoking habit model [15], and so on.As a special case of HAM, He proposed the homotopy perturbation method [HPM] [16].Recently, based on the idea of HPM, Mo proposed the homotopic mapping method to handle some nonlinear problems with small perturbed term [17].A great quantity of works about this subject have been researched by many authors, such as perturbed Kdv-Burgers equation [18], mid-latitude stationary wind field [19] etc.
In this paper, we extend the applications of HPM to solve a class of disturbed nonlinear Schrödinger equation in the nonlinear optics.And many useful results are researched.

MODEL AND HOMOTOPIC MAPPING
Consider the following generalized nonlinear Schrödinger equation with perturbed term If we let , t x z t  ，Eq.( 1) turns to the following form Where f is a perturbed term, which is a sufficiently smooth function in a corresponding domain. ()t  and () t  are the slowly increasing dispersion coefficient and nonlinear coefficient respectively, () t  represents the heat-insulating amplification or loss.The transmission of soliton in the real communication system of optical soliton is described by Eq.( 2) with 0 f  [20][21][22][23][24][25].
In order to obtain the approximate solution of Eq.( 2),we make the transformation With the following consistency conditions 0 () Where , , , , k k a a c are arbitrary nonzero constants.Substituting (4) into (2),we have If we let , Eq.( 6) becomes When 0 f  , we have '' 3 24 20 By using the general mapping deformation method [9], we can obtain the following solutions of the corresponding undisturbed Eq. ( 3) Where j F is an arbitrary solution of the equation obtain the twenty-two classes of solutions j F from Ref. [26], for example, if we let ) . We find that 0 u turns to the solution 31 u in Ref. [24],and which was degenerated to the famous bright-soliton solutions 1 u in Ref. [25] when 1 m  .
Eq.( 8) has the solution In order to obtain the solution of Eq.( 2), We introduce the following homotopic mapping   , Where  is an initial approximate solution to Eq.( 8), and the linear operator L is expressed as Obviously, from mapping (9),   ,1 0 H   is the same as Eq.( 7).Thus the solution of Eq.( 7) is the same as the solution of

APPROXIMATE SOLUTION
In order to obtain the solution of Eq.( 7), set If we let 00   ，noticed the analytic properties of 0 , f  and mapping (9), we can deduce that the series of ( 11) are uniform convergence when   0,1 p  [16].
Substituting expression (11) into   ,0 H u p  , expanding nonlinear terms into the power series in powers of p , we compare the coefficients of the same power of p on both sides of the equation, we have 0 00 :   From (12) we have If we select Where From (4),( 5),( 11),( 15),( 16), (17) and mapping (9) we have the first and second order approximate Jacobi-like elliptic function solutions x t of the generalized disturbed nonlinear Schrödinger equation (2) as follows: , With the same process, we can also obtain the N order approximate solution Where 2 12 .

Remark 2:
The N-order approximate solution x t is degenerated to the solitary wave approximate solution and trigonometric function approximate solution when the modulus 1 m  or 0 m  .If selecting different 0  , we can obtain the other fifty-one types of approximate solutions of Eq.(2).

COMPARISION OF ACCURACY
In order to explain the accuracy of the expressions of the approximate solution represented by Eq.( 18), we consider the small perturbation term From the discussion of Section 3, we obtain the second order approximate Jacobi-like elliptic function solution of Eq.( 19) as follows ,, to be an exact solution of Eq.( 6), noticed that 2hom ()

CONCLUSION
We researched a class of disturbed nonlinear Schrödinger equation with variable coefficients by using the homotopic mapping method, which is much more simple and efficient than some other asymptotic method such as perturbation method etc, the Jacobi-like function approximate solution with arbitrary degree of accuracy for the disturbed equation is researched, which shown that this method can be used to the soliton equation with complex variables, but it is still worth to research that whether or not the method can be used to the system with high dimension and high order.

Figure 1 .
Figure 1.Comparison between the curves of solutions 1hom () u

Figure 2 .
Figure 2. Comparison between the curves of solutions 1hom () u  (solid line) and

Table 1 ,
(19)e2, Fig.1and Fig.2.From Figs.1-2, it is easy to see that as This behaviour is coincident with that for the approximate solution of the weakly disturbed evolution equation(19).
u  are very close to each other.