Abstract
The theme of this piece of research is to investigate the collective variable (CV) as well as semi-inverse techniques to explore a significant model of cold bosonic atoms in a zig-zag optical lattice. The system is reduced to an important equation by utilizing the continuum approximation and explains the soliton’s dynamics in sense of pulse variables. These parameters are amplitude, temporal position, chirp, width, frequency, and phase which are termed as collective variables (CVs). The proposed methods are more straightforward, succinct, accurate, and simple to calculate. Furthermore, to employ the computational counterfeit on the system of six ordinary differential equations that denote all the CVs incorporated in the supposed ansatz, a well-established computational method which is the Runge–Kutta scheme of order four is applied. The CV method is exerted to resolve the evolution of pulse parameters with the propagation distance and graphical illustrations which are also given. Moreover, figures reveal the fascinating periodic oscillations of frequency, width, amplitude, and chirp of soliton. Solitons and their numerical behavior to interpret fluctuations in CVs are presented for several values of super-Gaussian pulse parameters. Also, the results for the semi-inverse method which are bright solitons are provided in the form of 2D, 3D, and density plots for the distinct values of the fractal parameter to understand their physical significance. This scheme is effective in finding variational principles of various nonlinear evolution equations. Some compelling characteristics pertaining to the current scrutiny are also deduced.
1. Introduction
Nonlinearity is a fascinating aspect of nature and numerous scientists believe nonlinear science to be the fundamental domain to understand the nature of physical processes. These physical phenomena are often modeled by nonlinear equations. Once these equations are obtained, it is necessary to solve them in order to obtain a good understanding of the dynamics of the system under consideration. Plasma physics, optics, and nonlinear quantum field theory are just a few of the disciplines in which these equations have been employed [1–7]. Quantum physics, in particular, has grown in importance in recent decades and continues to do so. There are many structures that portray the physical significance of quantum field theory. For instance, the Bose–Hubbard model is employed to characterize the quantum lattice gas model [8]. In the optical lattice, this quantum state can be examined utilizing the system of ultra-cold bosonic atoms [9]. Different experiments have been performed by Fermi-Pasta-Ulam on the atoms when the localized excitations were found in the monatomic chain. Quantum breathers related to modulation instability have been investigated by Djoufack et al. recently in 1D ultra-cold boson involving next-nearest neighbor interactions in optical lattice [10–12]. They have attained the wave function of single-boson using the time-dependent Hartree approximation combined with the semidiscrete multiple-scale technique and also obtained the constraint for the stability or instability of the wave. Similarly, they also studied the intrinsic localized excitation and quantum breathers in 1D Bose-Hubbard chain associated with the modulation instability [13]. With the help of Bose-Hubbard structure, they investigated the intrinsic localized modes and energy spectrum related to the modulation instability of boson chains and also other considerable attempts have been adopted on atomic chains [14, 15]. The authors have utilized an oblique magnetic field in Heisenberg ferromagnetic spin chains to control quantum breathers in [14] and examined localized modes as well as with single-ion easy-axis anisotropy modulation instability in Heisenberg ferromagnetic chains in [15]. In 2021, Tantawy and Abdel-Gawad [16] studied the continuum model analog to zig-zag optical lattice in quantum optics. A collection of polynomial and rational solutions of the model equation have been produced by employing the unified and generalized unified methods.
In this article, our main focus is on the cold bosonic atoms in a zig-zag optical lattice [17]. To understand the quantum phase evolution, Bose-Einstein contraction and quantum magnetism experiments with ultra-cold atoms play a very crucial role. The aim of this manuscript is to extract the solutions of the proposed model. Panoply of techniques are existed in the literature to solve these types of models [18–28]. In this paper, two of them will be considered to extract the soliton solutions of the governing model that are CV and semi-inverse methods. This article provides the Ritz-like approach combined with the variational method known as He’s variational method [29] to find the soliton solution of the system under consideration which may help physicists to recognize the physical significance of this fractal model. The dynamics of bright solitons can be addressed using approaches [30, 31]. Traveling wave solutions that are developed from existing techniques may also be practiced by determining the involved parameters in the methods of specific values. Many nonlinear mathematical physical models can be constructed using fractal calculus. The CV approach shows equations of motion for both nonconservative and conservative systems, irrespective of dissipative components or nonlinearities. The fourth-order Runge–Kutta approach for integration of the related system of differential equations is also used for mathematical approximation. This method divides the supposed solution of the governing equation into two factors, that is, soliton and residual parts. The physical system under consideration dictates the number of CVs that can be implemented. The implementation of a transformation converts the original field equation into CVs. Moreover, soliton solutions are affected by components such as amplitude, width, chirp, phase, temporal position, and frequency. These methods are more meaningful and up-to-date for locating solutions to various nonlinear models.
The following is how this article is planned: the governing model is presented in Section 2, and the semi-inverse method is demonstrated in Section 3. Section 4 presents the dynamics of the soliton parameter using the CV technique, and Subsection 4.1.1 provides a graphical interpretation. The discussion is shown in Section 5, and the review is presented in Section 6.
2. Governing Model
The model under consideration is given as
Reference [32] includes detail derivation of equation (1). In equation (1), , are the first and second nearest neighbor hoppings, with and , where is denoting the interaction of boson-boson.
Here, we consider an adequately large wavelength in comparison to the length of one part, also using the continuum approximation by assuming that nonlinearity is weak. Thus, we have the following equation from equation (1),where and stand for the temporal and spatial coordinates and is the space between two adjoining sections.
Consider the transformation as
Substituting equation (3) into (2), we get the following: Real part: Imaginary part:
From the above equation, we have
3. Semi-Inverse Technique
By implementing the He’s variational principle for innovative solitary wave solutions of equation (4), in view of [28, 33], a fractal form of the given model can be written aswhere is the fractal dimension value and denotes the fractal derivative, narrated as
By variational principle [29], the following trial-functional can be constructed as
The variational formulation of equation (7) is given aswhere is the potential energy and is the kinetic energy.
Also,are the Hamiltonian and Lagrangian. Using the two-scale transformation,
Equation (10) can be written as
Using Ritz approach, the solitary wave solution can be assumed aswhere and are constants to be further calculated. Putting equation (15) into (14), we have
Letting stationary in relation to and yields
From equations (17) and (18), we have
Equation (15) becomes
The solution for equation (2) becomes
Next, we consider another solitary wave solution aswhere and are constants to be further calculated. Putting equation (23) into (14),, we have
Making stationary with respect to and yields
From equations (26) and (27), we have
Equation (23) becomes
The solution of solitary wave for equation (2) is
4. Collective Variable Method
The solution of the provided equation is expected to split into two sections: residual radiation and soliton, according to the method of the considered approach. The idea is that solitons are reliant on CVs, which can indicate the amplitude, phase, chirp, frequency, width, temporal position, and so on. Moreover, the onset of CVs is accompanied by an increase in the phase space of the dynamical organization of the soliton borders. The solution’s residual factor is approximated to zero. The constraints develop a nonlinear dynamical arrangement of the collective parameters, which is mathematically investigated.
In addition, the soliton field is followed by which represents the soliton part and which denotes the residual factor and also and show the temporal and spatial components, respectively. Takingin equation (32), is supposed to be . It is crucial to note that the existence of variables in increases the problem’s degree of freedom. As a result, we change equation (32) in the presence of these variables as follows:
The residual free energy is calculated as follows:
Taking partial derivatives of equation (34) w.r.t. and denoting by ,
The standard inner product is defined as
Now, by utilizing this definition, equation (35) can be written as
In the above equation, Re denotes the real part, further taking equal to zero. Now, utilizing equation (33), becomes
This results in the following relationships:
Utilizing these relations and with the help of equation (35), we haveor rather
Again, we also getand
Plugging equation (43) in (42), one can have
Furthermore, Dirac’s postulate states that if all of the variations in variables are not equal to zero, the function is nearly zero.
Thus, be minimized ifwith
Utilizing equations (2) and (32), we have
Moreover, from equation (34), we haveand also by combining equation (47) with equation (48), we getwith
Similarly, we know that
Its compact form iswithwherewhile
Thus, finally is as follows:for which .
4.1. Soliton Parameter Dynamics
We use the constraint equations previously acquired by constructing the function to derive the system of equations of motion for the CVs. In addition, using the Gaussian ansatz, we suppose that the soliton function has six CVs of the formwhere denotes the width, temporal position, amplitude, chirp, phase, and frequency. The basic approximation, often known as the lowest order CV hypothesis, is used to create differential equations for CVs. The residuary component is set to zero in this idea, . We find and as follows:
For equation (2), are calculated as
Using equation (54) and putting equations (59) and (60)–(65), the following system of ODEs is generated:
4.1.1. Graphical Interpretation
5. Results and Discussion
This section is devoted to the discussion of the outcomes, we have obtained using CV and semi-inverse methods. The first one is examined for equation (2) and the latter one is used to solve the proposed model in the presence of fractal parameter . In the semi-inverse approach, bright solitons are achieved, and to understand the physical significance of the obtained outcomes, we have provided 2D, 3D, and density plots for different values of the fractal parameter in Figures 1 and 2. Fractal parameter induces the irregularity in the surface of the graphs. For the CV technique, we investigated numerical simulations for the pulse propagation parameters that are seen graphically to depict the complete achievement of the CV technique. Also, standard RK4-method is utilized to solve the ODEs that we attained as a result of our method. For the following parametric values, , , , , , and , we have presented a graphical interpretation of the given system for the mentioned initial guess.
(a)
(b)
(c)
(a)
(b)
(c)
It is important to mention here that the pulse propagates , , and , that is, the soliton’s chirp, width, and amplitude shows periodic behavior with uniform wavelengths but center position and phase are increasing with propagation distance in Figure 3. Figure 4 shows, however, that , , and are the soliton’s amplitude; chirp and width vary on a regular basis with the same wavelengths whereas temporal position and frequency are behaving constant for all the values of propagation distance and the phase of pulse is almost linear which also behaves periodically having a negligible amplitude and showing increasing behavior. As a consequence, it is recognized that the energy firmly depends upon the amplitude , width , and chirp .
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
From all the above discussion, we are unable to conclude that the recommended solutions are not identical to those available in the literature. It might be helpful to understand the behavior of governing model and can be a beneficial contribution to future study. Hence, the CV strategy offers an easy to understand analysis for the attained equations and characterizes the combined influence of numerous variables on the considered equation.
6. Conclusion
The current work shows the study of relatively new equation by continuum approximation, describing the dynamics of the cold bosonic atoms in a zig-zag optical lattice. This piece of research discusses the CV and semi-inverse methods to explore the solitons of the mentioned equation. New and bright solitons are attained using the semi-inverse method in the presence of fractal effect. For dynamics of wave propagation in optical fibers, the CV strategy regulates higher-order nonlinearities in a system of ODEs. This scrutiny discloses solitons as both analytical and numerical effects and causes a distinctive and enormous set of pulse parameters that happens for their various values. The implementation of condition on the CVs , which provides them significant physical relevance by building the Gaussian ansatz type function, is the foundation of the CV method. The main benefit of this method is that it makes deriving the equations of motion for CVs more easier. We can learn a lot about the dynamics of cold bosonic atoms in a zig-zag optical lattice by looking at the graphical depictions of obtained solutions. These results may be helpful in fiber optics and play an important part in optical communications. These approaches evolve into standardized processes that may be used to a wide range of physical models. The acquired solitons may be beneficial to mathematicians, physicists, and engineers in grasping the physical importance of governing model.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project No. R-2021-325.