On Perturbative Cubic Nonlinear Schrodinger Equations under Complex Nonhomogeneities and Complex Initial Conditions

A perturbing nonlinear Schrodinger equation is studied under general complex nonhomogeneities and complex initial conditions for zero boundary conditions. The perturbation method together with the eigenfunction expansion and variational parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a power series solution is proved to exist. Using Mathematica, the symbolic solution algorithm is tested through computing the possible approximations under truncation procedures. The method of solution is illustrated through case studies and figures.


Introduction
The nonlinear Schrodinger NLS equation is the principal equation to be analyzed and solved in many fields, see 1-5 , for examples.In the last two decades, there are a lot of NLS problems depending on additive or multiplicative noise in the random case 6, 7 or a lot of solution methodologies in the deterministic case.Wang with initial condition u x, 0 g x and boundary conditions u x L 0 , t u x L 1 , t 0, which gives rise to solitary solutions using variational iteration method.Zhu 11 used the extended hyperbolic auxiliary equation method in getting the exact explicit solutions to the higherorder NLS: q tt γ 1 q 2 q i β 2 6 q ttt β 3 24 q tttt − γ 2 q 4 q, 1.4 without any conditions.Sun et al. 12 solved the NLS: with the initial condition ψ x, 0 ψ 0 x using Lie group method.By using coupled amplitude phase formulation, Porsezian and Kalithasan 13 constructed the quartic anharmonic oscillator equation from the coupled higher-order NLS.Two-dimensional grey solitons to the NLS were numerically analyzed by Sakaguchi and Higashiuchi 14 .The generalized derivative NLS was studied by Huang et al. 15 introducing a new auxiliary equation expansion method.Abou Salem and Sulem 16 studied the effective dynamics of solitons for the generalized Schrodinger equation in a random potential.El-Tawil 17 considered a nonlinear Schrodinger equation with random complex input and complex initial conditions.Colin et al. 18 considered three components of nonlinear Schrodinger equations related to the Raman amplification in a plasma.In 19 , Jia-Min and Yu-Lu constructed an appropriate transformations and an extended elliptic subequation approach to find some exact solutions for variable coefficient cubic-quintic nonlinear Schrodinger equation with an external potential.
In this paper, a straight forward solution algorithm is introduced using the transformation from a complex solution to a coupled equations in two real solutions, eliminating one of the solutions to get separate independent and higher-order equations, and finally introducing a perturbative approximate solution to the system.

The Linear Case
Consider the nonhomogeneous linear Schrodinger equation: Eliminating one of the variables in 2.3 , one can get the following independent equations: where

2.5
Using the eigenfunction expansion technique 20 , the following solutions for 2.4 are obtained: where T n z and τ n z can be got through the applications of initial conditions and then solving the resultant second-order differential equations using the method of the variational 4 D i fferential Equations and Nonlinear Mechanics parameter 21 .The final expressions can be got as follows where

2.9
The following conditions should also be satisfied:

2.10
Finally, the following solution is obtained:

The Nonlinear Case
Consider the homogeneous nonlinear Schrodinger equation: where u t, z is a complex-valued function which is subjected to the initial and boundary conditions 2.2 .Lemma 3.1.The solution of 3.1 with the constraints 2.2 is a power series in ε if the solution exists.
Proof.At ε 0, the following linear equation is got:  which has the solution, see the previous section, u 0 t, z e −γz w 0 t, z iv 0 t, z .Following Pickard approximation, 3.13 can be rewritten as

3.3
At n 1, the iterative equation takes the following form: which can be solved as a linear case with zero initial and boundary conditions.The following general solution can be obtained:

3.6
At n 2, the following equation is obtained: which can be solved as a linear case with zero initial and boundary conditions.The following general solution can be obtained:

3.8
Continuing like this, one can get 3.9 As n → ∞, the solution if exists can be reached as u t, z lim n → ∞ u n t, z .Accordingly, the solution is a power series in ε.
According to the previous lemma, one can assume the solution of 3.1 as the following: 3.10 Let u t, z ψ t, z iφ t, z , ψ, φ: real valued functions.The following coupled equations are got:
Substituting 3.12 into 3.11 and then equating the equal powers of ε, one can get the following set of coupled equations: 14 15 3.17 and so on.The prototype equations to be solved are

3.19
where ψ i t, 0 δ i,0 f 1 t , φ i t, 0 δ i,0 f 2 t and all other corresponding conditions are zeros.The nonhomogeneity functions G 1 i and G 2 i are functions computed from previous steps.Following the solution algorithm described in the previous section for the linear case, the general symbolic algorithm in Figure 39 can be simulated through the use of a symbolic package, mathematica-5 is used in this paper.

The Zero-Order Approximation
In this case, u 0 t, z ψ 0 iφ 0 , 3.20  where   The absolute value of the zero-order approximation is got from  where The absolute value of the first-order approximation can be got using

The First-Order Approximation
3.27 where   where the constants and variables A 21 z , C 22 , B 21 z , A 22 z , C 22 , and B 22 z can be evaluated similarly as the previous approximation.The absolute value of the second-order approximation can be got using

Case Studies
To examine the proposed solution algorithm, see Figure 39, some case studies are illustrated.

Case Study 1
Taking f 1 t 0, f 2 t 0, F 1 t, z 1, F 2 t, z 0, and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 1, 2, and 3.

Case Study 2
Taking f 1 t 0, f 2 t 0, F 1 t, z 0, F 2 t, z 1 and following the solution algorithm, it has been noticed that the same results for the case study 1 are got.

Case Study 3
Taking f 1 t 1, f 2 t 0, F 1 t, z 0, F 2 t, z 0 and following the solution algorithm, the selective results for the first-zero approximation are got in Figures 4, 5, and 6.
One can notice the decrease of the solution level and its higher variability.

Case Study 4
Taking f 1 t 0, f 2 t 1, F 1 t, z 0, F 2 t, z 0 and following the solution algorithm, it has been noticed that the same results for the case study 3 are got:

Case Study 5
Taking f 1 t 0, f 2 t 0, F 1 t, z 1, F 2 t, z 1 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 7, 8, and 9.
One can notice that the solution level becomes a little bit higher than that of case study 2.

Case Study 6
Taking f 1 t 1, f 2 t 1, F 1 t, z 0, F 2 t, z 0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 10,11,and 12.One can notice the little increase of the solution level than that of case studies 3 and 4.

Case Study 7
Taking f 1 t 1, f 2 t 0, F 1 t, z 1, F 2 t, z 0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 13,14,and 15.One can notice the small perturbations at small values of z.

Three Inputs Are On
Case Study 8 Taking f 1 t 1, f 2 t 1, F 1 t, z 1, F 2 t, z 0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 16,17,and 18.One can notice the increase of the depth of the perturbations.

Case Study 9
Taking

Case Study 10
Taking the case of , the following final results for the zero-order approximation are obtained in Figures 22, 23, and 24.
One can notice the little increase in the solution level.

Case Study 11
Taking the case of f 1 t 1, f 2 t 0, F 1 t, z e −t , F 2 t, z 0, the following final results for the zero-order approximation are obtained in Figures 25, 26, and 27.
One can notice the low solution level and high perturbations.
Differential Equations and Nonlinear Mechanics

Case Study 12
Taking the case of f 1 t e −t , f 2 t 0, F 1 t, z 1, F 2 t, z 0, the following final results for the zero-order approximation are obtained in Figures 28, 29, and 30.One can notice that a higher solution level is got compared with case study 11 and less perturbations are got at small values of z.

Case Study 13
Taking the case of f 1 t 1, f 2 t 0, F 1 t, z 1, F 2 t, z 0, the following final results for the zero and first-order approximations are obtained in Figures 31-38.
One can notice the oscillations of the solution level compared with case 7.
One can notice that the solution level increases with the increase of ε.

Figure 1 :
Figure 1: The zero-order approximation of |u 0 | at α 1, T 10, γ 02 with considering only one term in the series M 1 .

10 Figure 2 :
Figure 2: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 3 :Figure 4 :
Figure 3: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

10 Figure 5 :
Figure 5: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 6 :
Figure 6: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

Figure 7 :
Figure 7: The zero-order approximation of |u 0 | at α 1, T 10, γ 02 with considering only one term in the series M 1 .

10 Figure 8 :
Figure 8: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 9 :
Figure 9: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

Figure 10 :
Figure 10: The zero-order approximation of |u 0 | at α 1, T 10, γ 02 with considering only one term in the series M 1 .

10 Figure 11 :
Figure 11: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 12 :
Figure 12: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

Figure 13 :
Figure 13: The zero-order approximation of |u 0 | at α 1, T 10, γ 02 with considering only one term in the series M 1 .

4 Figure 14 :
Figure 14: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

4 Figure 17 :
Figure 17: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 18 :Figure 19 :
Figure 18: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

4 Figure 20 :
Figure 20: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 21 :Figure 22 :
Figure 21: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

Figure 23 :
Figure 23: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 24 :Figure 25 :
Figure 24: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

10 Figure 26 :
Figure 26: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 27 :
Figure 27: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

Figure 28 :
Figure 28: The zero-order approximation of |u 0 | at α 1, T 10, γ 02 with considering only one term in the series M 1 .

4 Figure 29 :
Figure 29: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of z, considering only one term in the series M 1 .

5 Figure 30 :
Figure 30: The zero-order approximation |u 0 | at α 1, T 10, γ 02 for different values of t, considering only one term in the series M 1 .

Figure 31 :
Figure 31: The zero-order approximation of |u 0 | at α 1, T 10, γ 0 with considering only one term in the series M 1 .

4 Figure 32 :
Figure 32: The zero-order approximation |u 0 | at α 1, T 10, γ 0 for different values of z, considering only one term in the series M 1 .

Figure 39 :
Figure 39: The general solution algorithm.
et al. 8 obtained the exact solutions to NLS using what they called the subequation method.They got four kinds of exact solutions of Equations and Nonlinear Mechanics followed the same previous technique in solving the higher-order NLS: for which no sign to the initial or boundary conditions type is made.Xu and Zhang 9 2 D i fferential and all corresponding other I.C. and B.C. are zeros.As a perturbation solution, one can assume that The zero-order approximation |u 0 | at α 1, T 10, γ 0 for different values of t, considering only one term in the series M 1 .