Abstract
This paper addresses the (3 + 1)-dimensional nonlinear Schrödinger equation, which could be utilized to express many physical media the envelope of the wave amplitude. With the help of He’s semi-inverse method, the solitary wave solutions are explicitly constructed to the equation. The dark soliton solutions of the equation are also strictly constructed by making use of the solitary ansatz method; in order to guarantee the existence of solitons, some conditions are given. Furthermore, by employing the method, we also present complexitons of the equation. Finally, with the aid of linear stability analysis, an effective and straightforward method is presented to analyze modulation instability of the equation.
1. Introduction
As everyone knows, the nonlinear Schrödinger equations (NLSEs) have been attractive, which has the vital role in various fields, such as applied mathematics [1], theoretical physics [2], and engineering [3] [4]. NLSEs arises from the research of plasma physics, fluid dynamics, elastic media, modelling of deep water, nonlinear optics, and many others. In the field of nonlinear optics, optical soliton is one of the important research areas. In the past many years, great progress has been made in the area of research, and the optical solitons are successfully used in the long distance telecommunication. As mathematical models for this phenomena, the study of exact solutions of NLSEs can help us better understand this phenomena. In past years, to construct the exact solutions of these equations, various straightforward and effective approaches have been used, for example, the sine-cosine method [5, 6], the -expansion method [7–9], the first integral method [10], and the soliton ansatz method [11–14].
In the present paper, we study the (3 + 1)-dimensional nonlinear Schrödinger equation [15–18]:
where is the envelope amplitude and , , and are real constants. When , it can be reduced to (3 + 1)-dimensional time dependent NLSE [19], which occurs in lots of areas of electromagnetic wave propagation, physics quantum mechanics, and optoelectronic devices [20]. In [19], Gagnon and Winternitz studied symmetry reductions of the (3 + 1)-dimensional nonlinear Schrödinger equation.
To the best of our knowledge, optical soliton solutions by applying the He’s semi-inverse method and the solitary ansatz method have not been discussed of Equation (1). The main result of this paper is to derive the solitary wave solitons, dark soliton solutions, and complexitons; the modulation instability analysis of the model is also studied.
The structure of the present paper is given as follows. In Section 2, a variable transformation could be applied to obtain the solitary wave solutions of Equation (1). Its dark soliton solutions are constructed on the basis of the solitary ansatz method [21–23] and symbol calculation methods [24, 25] in Section 3. In Section 4, according to the method, its complexitons are also provided in a rational method. In Section 5, modulation instability of the equation is also studied.
2. Solitary Wave Solution
We will employ He’s semi-inverse method [26, 27] to construct solitary wave solutions for Equation (1). Firstly, we seek solution of the form:
where is constant wave speed to be decided later and , , and , are also constants. Thus, one obtains
If we substitute these into Equation (1), then from the imaginary part of the obtained equation, we have
Therefore, we arrive at the following ordinary differential equation from the obtained equation
According to [26–28], we could establish the following variational formulation:
Based on the Ritz method, we consider the solitary wave solution in the form
where represents the amplitude of the soliton and is the inverse width of the soliton. Putting (7) into (6), we can get
For guaranteeing stationary with respect to and leads to the following formulation: which is equivalent to
Solving the above equation about and , we have
Therefore, we can get the solitary wave solution
In terms of (2), it is straightforward to find out the solitary wave solution of the (3 + 1)-dimensional nonlinear Schrödinger equation:
With the help of software Maple, we present Figures 1 and 2 by taking suitable parameters.
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3. Dark Soliton Solution
We utilize the solitary ansatz method to Equation (1) for constructing the dark soliton solution in this section. To begin with, we consider the following hypothesis [21, 29]:
where denotes the shape of the pulse and represents the phase portion of the soliton; , and are the wave numbers; is the frequency of the soliton; and denotes the center of the phase. From (14), we can get the following form:
Then, substituting (15) into Equation (1) and separating real and imaginary parts of the obtained equation yields
For getting the dark soliton solution of Equation (1), we assume
in which ; , , , and the exponent are unknown free parameters; and is the velocity of the soliton. Therefore, one obtains
The substitution of (18) into (16) yields
According to (20), it is obvious that
from (19), equating the exponents of and yields so that . Then, setting their respective coefficients to zero results in
After some calculation with (22), we get
Finally, the dark soliton solution to Equation (1) is presented by
Here, and , , and are restricted by the first expression in (23); is given by (21); and the second expression of (23) provides . All of these conditions guarantee the existence of the dark soliton solutions. In the following, we also present Figure 3 of the dark soliton solution of Equation (1) by selecting suitable parameters.
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4. Complexitons
In this part, the method will be used to construct the complexitons [29] of Equation (1). Meanwhile, it can be utilized to carry out the integration to Equation (1). Firstly, consider the following hypothesis [30]:
where ; is a real valued function; ; and the constants , , , and are to be determined later. By putting (25) into Equation (1), the following equation is revealed:
Here, , and . According to balancing principle, the linear term of highest order derivative is balanced with the highest order nonlinear terms . Then by the method, we suppose that Equation (26) has the following solution:
where ; substituting (27) into (26) yields the following algebra equations:
From (28), it is easy to get the following results:
Finally, Equation (1) admits the following complexitons:
5. Modulation Instability Analysis
In the previous sections, solitary wave solutions, dark soliton solutions, and complexitons of Equation (1) have been investigated. Now, in this section, we will employ linear stability analysis to study the modulation instability of Equation (1). Modulation instability analysis can be utilized to analyze whether the modulated envelopes are modulationally stable or not [31–33]. Firstly, we suppose that Equation (1) has the stationary solutions of the following form:
where , , and are the real constants. Substituting (31) into Equation (1), we can get
Based on the concept of linear stability analysis, we set the stationary solutions (31) with a small perturbation as follows:
where is a perturbation parameter; as for , we consider it has the following form:
where , (j =1,2,3) and are the normalized wave numbers and real disturbance frequency, respectively. Taking Equations (33) and (34) into Equation (1) yields two linear homogeneous equations about and as follows: with
For making (35) has the nontrivial solutions: the determinant of coefficients of Equation (35) need to satisfy
Substituting (36) into (37), we get the following dispersion relation: with
If , is always real, then the steady state for Equation (1) is stable against the small perturbation. On the other hand, the steady-state solution becomes unstable in the case of due to the value of will be complex.
6. Conclusions and Discussions
In this paper, we have investigated the (3 + 1)-dimensional nonlinear Schrödinger equation. With the help of the He’s semi-inverse method, we have constructed a solitary wave solution of Equation (1). It could be seen that He’s semi-inverse method is straightforward. The profile of the solitary wave solution that was obtained numerically has been shown in Figures 1 and 2. Subsequently, based on the solitary ansatz method, we have established dark soliton solution of Equation (1). In Figure 3, we have shown the dynamical characteristics of the dark soliton solution, in which amplitude and shape of dark soliton solution maintain unchange during the propagation. It needs to be noted that the (3 + 1)-dimensional nonlinear Schrödinger equation can be integrated by the inverse scattering transform method that is one of the most powerful methods to carry out the integration of NLEEs. However, this paper has carried out the integration of the (3 + 1)-dimensional nonlinear Schrödinger equation by using a couple of simpler methods that are known as the ansatz method and the tanh method. These two methods are a few of the several methods of integrability that was developed in the past decade. What’s more, the method have been applied to study the complexitons solutions of Equation (1). Finally, the modulation instability analysis for the (3 + 1)-dimensional nonlinear Schrödinger equation has been presented. We hope that the obtained solutions will be indeed valuable for the future research, and our results are going to be helpful to a variety of fields of applied mathematics.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by Foundation of Anhui Provincial Education Department (Grant Nos. 2020jxtd286, 2020xsxxkc488, KJ2021A1102, and KJ2021ZD0136) and the Foundation of Suzhou University (Grant Nos. 2020BS011, 2020yzd06, szxy2020xxkc03, szxy2021jyxm16, and 2021XJPT40).