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

Abstract

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.

1. 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 et al. [8] obtained the exact solutions to NLS using what they called the subequation method. They got four kinds of exact solutions of 𝑖𝜕𝑢+1𝜕𝑡2𝜕²𝑢𝜕𝑥²+𝛼|𝑢|^𝑝𝑢+𝛽|𝑢|^2𝑝𝑢=0, (1.1) for which no sign to the initial or boundary conditions type is made. Xu and Zhang [9] followed the same previous technique in solving the higher-order NLS: 𝑖𝜕𝑢−1𝜕𝑥2𝛼𝜕²𝑢𝜕𝑡²+𝛽|𝑢|²𝜕𝑢+𝑖𝜀³𝑢𝜕𝑡³+𝑖𝛿|𝑢|²𝜕𝑢𝜕𝑡+𝑖𝛾𝑢²𝜕𝑢^∗𝜕𝑡=0. (1.2) Sweilam [10] solved 𝑖𝜕𝑢+𝜕𝜕𝑡²𝑢𝜕𝑥²+𝑞|𝑢|²𝑢=0,𝑡>0,𝐿₀<𝑥<𝐿₁, (1.3)

with initial condition 𝑢(𝑥,0)=𝑔(𝑥) and boundary conditions 𝑢_𝑥(𝐿₀,𝑡)=𝑢_𝑥(𝐿₁,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 higher-order NLS: 𝑖𝑞_𝑧−𝛽₁2𝑞_𝑡𝑡+𝛾₁|𝑞|²𝛽𝑞=𝑖²6𝑞_𝑡𝑡𝑡+𝛽₃𝑞24^{𝑡𝑡𝑡𝑡}−𝛾₂|𝑞|⁴𝑞, (1.4) without any conditions. Sun et al. [12] solved the NLS: 𝑖𝜕𝜓+𝜕𝜕𝑡²𝜓𝜕𝑥²+𝑎|𝜓|²𝜓=0, (1.5) with the initial condition 𝜓(𝑥,0)=𝜓₀(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.

2. The Linear Case

Consider the nonhomogeneous linear Schrodinger equation: 𝑖𝜕𝑢(𝑡,𝑧)𝜕𝜕𝑧+𝛼²𝑢(𝑡,𝑧)𝜕𝑡²=𝐹₁(𝑡,𝑧)+𝑖𝐹₂(𝑡,𝑧),(𝑡,𝑧)∈(0,𝑇)×(0,∞), (2.1) where 𝑢(𝑡,𝑧) is a complex valued function which is subjected to I.C.∶𝑢(𝑡,0)=𝑓₁(𝑡)+𝑖𝑓₂(𝑡),acomplexvaluedfunction,B.C.∶𝑢_𝑧(0,𝑧)=0,𝑢_𝑧(𝑇,𝑧)=0. (2.2) Let 𝑢(𝑡,𝑧)=𝜓(𝑡,𝑧)+𝑖𝜙(𝑡,𝑧), 𝜓,𝜙: real valued functions. Substituting (2.2) in (2.1), the following coupled equations are got as follows: 𝜕𝜙(𝑡,𝑧)𝜕𝜕𝑧=𝛼²𝜓(𝑡,𝑧)𝜕𝑡²+𝐺₁(𝑡,𝑧),𝜕𝜓(𝑡,𝑧)𝜕𝜕𝑧=𝛼²𝜙(𝑡,𝑧)𝜕𝑡²+𝐺₂(𝑡,𝑧), (2.3) where 𝜓(𝑡,0)=𝑓₁(𝑡), 𝜙(𝑡,0)=𝑓₂(𝑡), 𝐺₁(𝑡,𝑧)=−𝐹₁(𝑡,𝑧), 𝐺₂(𝑡,𝑧)=𝐹₂(𝑡,𝑧), and all corresponding other I.C. and B.C. are zeros.

Eliminating one of the variables in (2.3), one can get the following independent equations: 𝜕⁴𝜓(𝑡,𝑧)𝜕𝑡⁴+1𝛼²𝜕²𝜓(𝑡,𝑧)𝜕𝑧²=1𝛼^2∼𝜓₁𝜕(𝑡,𝑧),₄𝜙(𝑡,𝑧)𝜕𝑡⁴+1𝛼²𝜕²𝜙(𝑡,𝑧)𝜕𝑧²=1𝛼^2∼𝜓₂(𝑡,𝑧), (2.4) where 𝜓^∼₁(𝑡,𝑧)=𝜕𝐺₂𝜕𝜕𝑧−𝛼²𝐺₁𝜕𝑡²_∼,𝜓₂(𝑡,𝑧)=𝛼𝜕𝐺₂𝜕𝑡²+𝜕𝐺₁.𝜕𝑧 (2.5) Using the eigenfunction expansion technique [20], the following solutions for (2.4) are obtained: 𝜓(𝑡,𝑧)=^∞_𝑛=0𝑇_𝑛(𝑧)sin(𝑛𝜋𝑇)𝑡,𝜙(𝑡,𝑧)=^∞_𝑛=0𝜏_𝑛(𝑧)sin(𝑛𝜋𝑇)𝑡, (2.6) 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 parameter [21]. The final expressions can be got as follows 𝑇_𝑛(𝑧)=(𝐶₁+𝐴₁(𝑧))sin𝛽_𝑛𝑧+(𝐶₂+𝐵₁(𝑧))cos𝛽_𝑛𝜏𝑧,_𝑛(𝑧)=(𝐶₃+𝐴₂(𝑧))sin𝛽_𝑛𝑧+(𝐶₄+𝐵₂(𝑧))cos𝛽_𝑛𝑧, (2.7) where 𝛽_𝑛=𝛼(𝑛𝜋𝑇)²,𝐴₁1(𝑧)=𝛽_𝑛^∼𝜓_1𝑛(𝑧;𝑛)cos(𝛽_𝑛𝐵)𝑧𝑑𝑧,₁(𝑧)=−1𝛽_𝑛^∼𝜓_1𝑛(𝑧;𝑛)sin(𝛽_𝑛𝐴)𝑧𝑑𝑧,₂1(𝑧)=𝛽_𝑛^∼𝜓_2𝑛(𝑧;𝑛)cos(𝛽_𝑛𝐵)𝑧𝑑𝑧,₂(𝑧)=−1𝛽_𝑛^∼𝜓_2𝑛(𝑧;𝑛)sin(𝛽_𝑛)𝑧𝑑𝑧, (2.8) in which 𝜓^∼_1𝑛2(𝑧;𝑛)=𝑇^𝑇₀^∼𝜓₁(𝑡,𝑧)sin(𝑛𝜋𝑇𝑡)𝑑𝑡,_∼𝜓_2𝑛2(𝑧;𝑛)=𝑇^𝑇₀^∼𝜓₂(𝑡,𝑧)sin(𝑛𝜋𝑇𝑡)𝑑𝑡. (2.9) The following conditions should also be satisfied: 𝐶₂=2𝑇^𝑇₀𝑓₁(𝑡)sin(𝑛𝜋𝑇)𝑑𝑡−𝐵₁𝐶(0),₄=2𝑇^𝑇₀𝑓₂(𝑡)sin(𝑛𝜋𝑇)𝑑𝑡−𝐵₂(0). (2.10) Finally, the following solution is obtained: 𝑢(𝑡,𝑧)=(𝜓(𝑡,𝑧)+𝑖𝜙(𝑡,𝑧)), (2.11) or |𝑢(𝑡,𝑧)|²=(𝜓²(𝑡,𝑧)+𝜙²(𝑡,𝑧)). (2.12)

3. The Nonlinear Case

Consider the homogeneous nonlinear Schrodinger equation: 𝑖𝜕𝑢(𝑡,𝑧)𝜕𝜕𝑧+𝛼²𝑢(𝑡,𝑧)𝜕𝑡²+𝜀|𝑢(𝑡,𝑧)|²𝑢(𝑡,𝑧)+𝑖𝛾𝑢(𝑡,𝑧)=𝐹₁(𝑡,𝑧)+𝑖𝐹₂(𝑡,𝑧),(𝑡,𝑧)∈(0,𝑇)×(0,∞), (3.1) 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: 𝑖𝜕𝑢(𝑡,𝑧)𝜕𝜕𝑧+𝛼²𝑢(𝑡,𝑧)𝜕𝑡²+𝑖𝛾𝑢(𝑡,𝑧)=𝐹(𝑡,𝑧),(𝑡,𝑧)∈(0,𝑇)×(0,∞), (3.2) which has the solution, see the previous section, 𝑢₀(𝑡,𝑧)=𝑒^−𝛾𝑧(𝑤₀(𝑡,𝑧)+𝑖𝑣₀(𝑡,𝑧)). (3.3) Following Pickard approximation, (3.13) can be rewritten as 𝑖𝜕𝑢_𝑛(𝑡,𝑧)𝜕𝜕𝑧+𝛼²𝑢_𝑛(𝑡,𝑧)𝜕𝑡²+𝑖𝛾𝑢_𝑛(𝑡,𝑧)=𝐹(𝑡,𝑧)−𝜀|𝑢_𝑛−1(𝑡,𝑧)|²𝑢_𝑛−1(𝑡,𝑧),𝑛≥1. (3.4) At 𝑛=1, the iterative equation takes the following form: 𝑖𝜕𝑢₁(𝑡,𝑧)𝜕𝜕𝑧+𝛼²𝑢₁(𝑡,𝑧)𝜕𝑡²+𝑖𝛾𝑢₁(𝑡,𝑧)=𝐹(𝑡,𝑧)−𝜀|𝑢₀(𝑡,𝑧)|²𝑢₀(𝑡,𝑧)=𝐹(𝑡,𝑧)+𝜀ℎ₁(𝑡,𝑧), (3.5) which can be solved as a linear case with zero initial and boundary conditions. The following general solution can be obtained: 𝑤₁(𝑡,𝑧)=^∞_𝑛=0(𝑇_0𝑛+𝜀𝑇_1𝑛)sin(𝑛𝜋𝑇𝑣)𝑡,₁(𝑡,𝑧)=^∞_𝑛=0(𝜏_0𝑛+𝜀𝜏_1𝑛)sin(𝑛𝜋𝑇𝑢)𝑡,₁(𝑡,𝑧)=𝑒^−𝛾𝑧(𝑤₁(𝑡,𝑧)+𝑖𝑣₁(𝑡,𝑧))=𝑢₁⁽⁰⁾+𝜀𝑢₁⁽¹⁾. (3.6) At 𝑛=2, the following equation is obtained: 𝑖𝜕𝑢₂(𝑡,𝑧)𝜕𝜕𝑧+𝛼²𝑢₂(𝑡,𝑧)𝜕𝑡²+𝑖𝛾𝑢₂(𝑡,𝑧)=𝐹(𝑡,𝑧)−𝜀|𝑢₁(𝑡,𝑧)|²𝑢₁(𝑡,𝑧)=𝐹(𝑡,𝑧)+𝜀ℎ₂(𝑡,𝑧), (3.7) which can be solved as a linear case with zero initial and boundary conditions. The following general solution can be obtained: 𝑢₂(𝑡,𝑧)=𝑢₂⁽⁰⁾+𝜀𝑢₂⁽¹⁾+𝜀²𝑢₂⁽²⁾+𝜀³𝑢₂⁽³⁾+𝜀⁴𝑢₂⁽⁴⁾. (3.8) Continuing like this, one can get 𝑢_𝑛(𝑡,𝑧)=𝑢_𝑛⁽⁰⁾+𝜀𝑢_𝑛⁽¹⁾+𝜀²𝑢_𝑛⁽²⁾+𝜀³𝑢_𝑛⁽³⁾+⋯+𝜀^(𝑛+𝑚)𝑢_𝑛^(𝑛+𝑚). (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: 𝑢(𝑡,𝑧)=^∞_𝑛=0𝜀^𝑛𝑢_𝑛. (3.10) Let 𝑢(𝑡,𝑧)=𝜓(𝑡,𝑧)+𝑖𝜙(𝑡,𝑧), 𝜓,𝜙: real valued functions. The following coupled equations are got: 𝜕𝜙(𝑡,𝑧)𝜕𝜕𝑧=𝛼²𝜓(𝑡,𝑧)𝜕𝑡²+𝜀(𝜓²+𝜙²)𝜓−𝛾𝜙−𝐹₁,𝜕𝜓(𝑡,𝑧)𝜕𝜕𝑧=−𝛼²𝜙(𝑡,𝑧)𝜕𝑡²−𝜀(𝜓²+𝜙²)𝜙−𝛾𝜓+𝐹₂, (3.11) where 𝜓(𝑡,0)=𝑓₁(𝑡), 𝜙(𝑡,0)=𝑓₂(𝑡), and all corresponding other I.C. and B.C. are zeros.

As a perturbation solution, one can assume that 𝜓(𝑡,𝑧)=𝜓₀+𝜀𝜓₁+𝜀²𝜓₂+⋯,𝜙(𝑡,𝑧)=𝜙₀+𝜀𝜙₁+𝜀²𝜙₂+⋯, (3.12) where 𝜓₀(𝑡,0)=𝑓₁(𝑡), 𝜙₀(𝑡,0)=𝑓₂(𝑡), and all corresponding other I.C. and B.C. are zeros.

Substituting (3.12) into (3.11) and then equating the equal powers of 𝜀, one can get the following set of coupled equations: 𝜕𝜙₀(𝑡,𝑧)𝜕𝜕𝑧=𝛼²𝜓₀(𝑡,𝑧)𝜕𝑡²−𝛾𝜙₀−𝐹₁,𝜕𝜓₀(𝑡,𝑧)𝜕𝜕𝑧=−𝛼²𝜙₀(𝑡,𝑧)𝜕𝑡²−𝛾𝜓₀+𝐹₂,𝜕𝜙₁(𝑡,𝑧)𝜕𝜕𝑧=𝛼²𝜓₁(𝑡,𝑧)𝜕𝑡²−𝛾𝜙₁+(𝜓³₀+𝜓₀𝜙²₀),𝜕𝜓₁(𝑡,𝑧)𝜕𝜕𝑧=−𝛼²𝜙₁(𝑡,𝑧)𝜕𝑡²−𝛾𝜓₁−(𝜙³₀+𝜙₀𝜓²₀),𝜕𝜙₂(𝑡,𝑧)𝜕𝜕𝑧=𝛼²𝜓₂(𝑡,𝑧)𝜕𝑡²−𝛾𝜙₂+(3𝜓²₀𝜓₁+2𝜓₀𝜙₀𝜙₁+𝜓₁𝜙²₀),𝜕𝜓₂(𝑡,𝑧)𝜕𝜕𝑧=−𝛼²𝜙₂(𝑡,𝑧)𝜕𝑡²−𝛾𝜓₂−(3𝜙²₀𝜙₁+2𝜙₀𝜓₀𝜓₁+𝜙₁𝜓²₀), (3.13)(3.14)(3.15)(3.16)(3.17)(3.18) and so on. The prototype equations to be solved are 𝜕𝜙_𝑖(𝑡,𝑧)𝜕𝜕𝑧=𝛼²𝜓_𝑖(𝑡,𝑧)𝜕𝑡²−𝛾𝜙_𝑖+𝐺_𝑖⁽¹⁾,𝑖≥1,𝜕𝜓_𝑖(𝑡,𝑧)𝜕𝜕𝑧=−𝛼²𝜙_𝑖(𝑡,𝑧)𝜕𝑡²−𝛾𝜓_𝑖+𝐺_𝑖⁽²⁾,𝑖≥1, (3.19) where 𝜓_𝑖(𝑡,0)=𝛿_𝑖,0𝑓₁(𝑡), 𝜙_𝑖(𝑡,0)=𝛿_𝑖,0𝑓₂(𝑡) and all other corresponding conditions are zeros. The nonhomogeneity functions 𝐺_𝑖⁽¹⁾ and 𝐺_𝑖⁽²⁾ 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.

3.1. The Zero-Order Approximation

In this case, 𝑢⁽⁰⁾(𝑡,𝑧)=(𝜓₀+𝑖𝜙₀), (3.20) where 𝜓₀(𝑡,𝑧)=𝑒^{∞−𝛾𝑧}_𝑛=0𝑇_0𝑛sin(𝑛𝜋𝑇𝜙)𝑡,₀(𝑡,𝑧)=𝑒^{∞−𝛾𝑧}_𝑛=0𝜏_0𝑛sin(𝑛𝜋𝑇)𝑡, (3.21) in which 𝑇_0𝑛(𝑧)=𝐴₀₁(𝑧)sin𝛽_𝑛𝑧+(𝐶₀₂+𝐵₀₁(𝑧))cos𝛽_𝑛𝜏𝑧,_0𝑛(𝑧)=𝐴₀₂(𝑧)sin𝛽_𝑛𝑧+(𝐶^∼₀₂+𝐵₀₂(𝑧))cos𝛽_𝑛𝑧, (3.22) where the constants and variables 𝐴₀₁(𝑧),𝐶₀₂,𝐵₀₁(𝑧), 𝐴₀₂(𝑧),𝐶^∼₀₂, and 𝐵₀₂(𝑧) can be got by the aid of Section 2.

The absolute value of the zero-order approximation is got from |𝑢⁽⁰⁾(𝑡,𝑧)|²=𝜓₀²+𝜙₀². (3.23)

3.2. The First-Order Approximation

𝑢⁽¹⁾(𝑡,𝑧)=𝑢⁽⁰⁾+𝜀(𝜓₁+𝑖𝜙₁), (3.24) where 𝜓₁(𝑡,𝑧)=𝑒^{∞−𝛾𝑧}_𝑛=0𝑇_1𝑛(𝑧)sin(𝑛𝜋𝑇𝜙)𝑡,₁(𝑡,𝑧)=𝑒^{∞−𝛾𝑧}_𝑛=0𝜏_1𝑛(𝑧)sin(𝑛𝜋𝑇)𝑡, (3.25) in which 𝑇_1𝑛(𝑧)=𝐴₁₁(𝑧)sin𝛽_𝑛𝑧+(𝐶₁₂+𝐵₁₁(𝑧))cos𝛽_𝑛𝜏𝑧,_1𝑛(𝑧)=𝐴₁₂(𝑧)sin𝛽_𝑛𝑧+(𝐶^∼₁₂+𝐵₁₂(𝑧))cos𝛽_𝑛𝑧, (3.26) where the constants and variables 𝐴₁₁(𝑧), 𝐵₁₁(𝑧), 𝐴₁₂(𝑧), and 𝐵₁₂(𝑧) can be evaluated in a similar manner as the zero-order approximation whereas 𝐶^∼₁₂=−𝐵₁₂(0) and 𝐶₁₂=−𝐵₁₁(0).

The absolute value of the first-order approximation can be got using |𝑢⁽¹⁾(𝑡,𝑧)|²=|𝑢⁽⁰⁾(𝑡,𝑧)|²+2𝜀(𝜓₀𝜓₁+𝜙₀𝜙₁)+𝜀²(𝜓₁²+𝜙₁²). (3.27)

3.3. The Second-Order Approximation

𝑢⁽²⁾(𝑡,𝑧)=𝑢⁽¹⁾(𝑡,𝑧)+𝜀²(𝜓₂+𝑖𝜙₂), (3.28) where 𝜓₂(𝑡,𝑧)=𝑒^{∞−𝛾𝑧}_𝑛=0𝑇_2𝑛(𝑧)sin(𝑛𝜋𝑇𝜙)𝑡,₂(𝑡,𝑧)=𝑒^{∞−𝛾𝑧}_𝑛=0𝜏_2𝑛(𝑧)sin(𝑛𝜋𝑇)𝑡, (3.29) in which 𝑇_2𝑛(𝑧)=𝐴₂₁(𝑧)sin𝛽_𝑛𝑧+(𝐶₂₂+𝐵₂₁(𝑧))cos𝛽_𝑛𝜏𝑧,_2𝑛(𝑧)=𝐴₂₂(𝑧)sin𝛽_𝑛𝑧+(𝐶^∼₂₂+𝐵₂₂(𝑧))cos𝛽_𝑛𝑧, (3.30) where the constants and variables 𝐴₂₁(𝑧),𝐶₂₂,𝐵₂₁(𝑧), 𝐴₂₂(𝑧),𝐶^∼₂₂, and 𝐵₂₂(𝑧) can be evaluated similarly as the previous approximation.

The absolute value of the second-order approximation can be got using |𝑢⁽²⁾(𝑡,𝑧)|²=|𝑢⁽¹⁾(𝑡,𝑧)|²+2𝜀²(𝜓₀𝜓₂+𝜙₀𝜙₂)+2𝜀³(𝜓₁𝜓₂+𝜙₁𝜙₂)+𝜀⁴(𝜓₂²+𝜙₂²). (3.31)

4. Case Studies

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

4.1. One Input Is On

Case Study 1
Taking 𝑓₁(𝑡)=0, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=1, 𝐹₂(𝑡,𝑧)=0, and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 1, 2, and 3.

Figure 1 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 2 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 3 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

Case Study 2
Taking 𝑓₁(𝑡)=0, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=0, 𝐹₂(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 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=0, 𝐹₂(𝑡,𝑧)=0 and following the solution algorithm, the selective results for the first-zero approximation are got in Figures 4, 5, and 6.

Figure 4 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 5 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 6 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice the decrease of the solution level and its higher variability.

Case Study 4
Taking 𝑓₁(𝑡)=0, 𝑓₂(𝑡)=1, 𝐹₁(𝑡,𝑧)=0, 𝐹₂(𝑡,𝑧)=0 and following the solution algorithm, it has been noticed that the same results for the case study 3 are got:

4.2. Two Inputs Are On

Case Study 5
Taking 𝑓₁(𝑡)=0, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=1, 𝐹₂(𝑡,𝑧)=1 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 7, 8, and 9.

Figure 7 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 8 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 9 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice that the solution level becomes a little bit higher than that of case study 2.

Case Study 6
Taking 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=1, 𝐹₁(𝑡,𝑧)=0, 𝐹₂(𝑡,𝑧)=0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 10, 11, and 12.

Figure 10 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 11 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 12 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice the little increase of the solution level than that of case studies 3 and 4.

Case Study 7
Taking 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=1, 𝐹₂(𝑡,𝑧)=0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 13, 14, and 15.

Figure 13 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 14 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 15 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice the small perturbations at small values of 𝑧.

4.3. Three Inputs Are On

Case Study 8
Taking 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=1, 𝐹₁(𝑡,𝑧)=1, 𝐹₂(𝑡,𝑧)=0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 16, 17, and 18.

Figure 16 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 17 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 18 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice the increase of the depth of the perturbations.

Case Study 9
Taking 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=1, 𝐹₂(𝑡,𝑧)=1 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 19, 20, and 21.

Figure 19 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 20 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 21 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice that the perturbations become smaller than that of case study 8.

4.4. Four Inputs Are On

Case Study 10
Taking the case of 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=1, 𝐹₁=1, 𝐹₂=1, the following final results for the zero-order approximation are obtained in Figures 22, 23, and 24.

Figure 22 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 23 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 24 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice the little increase in the solution level.

4.5. Exponential Nonhomogeneity

Case Study 11
Taking the case of 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=𝑒^−𝑡, 𝐹₂(𝑡,𝑧)=0, the following final results for the zero-order approximation are obtained in Figures 25, 26, and 27.

Figure 25 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 26 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 27 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

One can notice the low solution level and high perturbations.

4.6. Exponential Initial Condition

Case Study 12
Taking the case of 𝑓₁(𝑡)=𝑒^−𝑡, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=1, 𝐹₂(𝑡,𝑧)=0, the following final results for the zero-order approximation are obtained in Figures 28, 29, and 30.

Figure 28 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ with considering only one term in the series ($M=1$).

Figure 29 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 30 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =02$ for different values of $t$, considering only one term in the series ($M=1$).

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.

4.7. First-Order Approximation

Case Study 13
Taking the case of 𝑓₁(𝑡)=1, 𝑓₂(𝑡)=0, 𝐹₁(𝑡,𝑧)=1, 𝐹₂(𝑡,𝑧)=0, the following final results for the zero and first-order approximations are obtained in Figures 31, 32, and 33.

Figure 31 

The zero-order approximation of $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0$ with considering only one term in the series ($M=1$).

Figure 32 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 33 

The zero-order approximation $|u^{(0)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0$ for different values of $t$, considering only one term in the series ($M=1$).

Figure 34 

The first-order approximation of $|u^{(1)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0\text{,}$$\epsilon =01$ with considering only one term in the series ($M=1$).

Figure 35 

The first-order approximation of $|u^{(1)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0\text{,}$$\epsilon =05$ with considering only one term in the series ($M=1$).

Figure 36 

The first-order approximation of $|u^{(1)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0\text{,}$$\epsilon =1$ with considering only one term in the series ($M=1$).

Figure 37 

The first-order approximation $|u^{(1)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0\text{,}$$\epsilon =05$ for different values of $z$, considering only one term in the series ($M=1$).

Figure 38 

The first-order approximation $|u^{(1)}|$ at $\alpha =1\text{,}$$T=10\text{,}$$\gamma =0\text{,}$$\epsilon =05$ for different values of $t$, considering only one term in the series ($M=1$).

Figure 39 

The general solution algorithm.

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.

5. Conclusions

The perturbation technique introduces an approximate solution to the NLS equation with a perturbative nonlinear term for a finite time interval. Using mathematica, the difficult and huge computations problems were fronted to some extent for limited series terms. To get more improved orders, it is expected to face a problem of computation. With respect to the solution level, the effect of the nonhomogeneity is higher than the effect of the initial condition. The initial conditions also cause perturbations for the solution at small values of the space variable. The solution level increases with the increase of 𝜀.