# New Optical Solitons with Variational Principle and Collective Variable Strategy for Cold Bosons in Zig-Zag Optical Lattices

Authors: Ziyad A. Alhussain; Nauman Raza
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 Section2, 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(1)ιdφkdτ=ς1φk−ς2φk+1+φk−1−ς3ςk+2+ςk−2+ς4N−1φk2φk.Reference [32] includes detail derivation of equation (1). In equation (1), ς2, ς3 are the first and second nearest neighbor hoppings, with ς1>0 and ς2>0, where ς4 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),(2)ιdφdτ=ς1−2ς2−2ς3φ−δ2ς2+4ς3d2φdy2+ς4N−1φ2φ,where τ and y stand for the temporal and spatial coordinates and δ is the space between two adjoining sections.Consider the transformation as(3)φy,τ=gαeιky+ωτ+β,withα=λy−ντ.Substituting equation (3) into (2), we get the following:Real part:(4)k2δ2ς2+4ς3+ω+ς1−2ς2+ς3g+ς4N−1g3−λ2δ2ς2+4ς3g″=0.Imaginary part:(5)−ν+2kλδ2ς2+4ς3g′=0.From the above equation, we have(6)ν=2kλδ2ς2+4ς3.


## 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 as(7)k2δ2ς2+4ς3+ω+ς1−2ς2+ς3g+ς4N−1g3−λ2δ2ς2+4ς3ddαχdgdαχ=0,where χ is the fractal dimension value and dg/dαχ denotes the fractal derivative, narrated as(8)dgdαχ=Γ1+αlimα−αo⟶Δαgα−gαoα−αoχ,Δα≠0.By variational principle [29], the following trial-functional can be constructed as(9)J=∫Ldα=∫K−Edα.The variational formulation of equation (7) is given as(10)J=∫0∞−12λ2δ2ς2+4ς3dgdαχ2−k2δ2ς2+4ς3+ω+ς1−2ς2+ς3g22−ς4N−1g44dαχ,where E=k2δ2ς2+4ς3+ω+ς1−2ς2+ς3g2/2+ς4N−1g4/4 is the potential energy and K=−1/2λ2δ2ς2+4ς3dg/dαχ2 is the kinetic energy.Also,(11)H=−12λ2δ2ς2+4ς3dgdαχ2+k2δ2ς2+4ς3+ω+ς1−2ς2+ς3g22+ς4N−1g44,(12)L=−12λ2δ2ς2+4ς3dgdαχ2−k2δ2ς2+4ς3+ω+ς1−2ς2+ς3g22−ς4N−1g44,are the Hamiltonian and Lagrangian. Using the two-scale transformation,(13)f=αχ.Equation (10) can be written as(14)J=∫0∞−12λ2δ2ς2+4ς3dgdf2−k2δ2ς2+4ς3+ω+ς1−2ς2+ς3g22−ς4N−1g44df.Using Ritz approach, the solitary wave solution can be assumed as(15)g=σ1sechσ2f,where σ1 and σ2 are constants to be further calculated. Putting equation (15) into (14), we have(16)J=−λ2δ2σ12σ2ς26+2ς33−δ2k2σ12σ2ς22+2ς3−σ122σ2ω+ς1+σ12σ2ς2+ς3−σ14ς46σ2N−1.LettingJ stationary in relation to σ1 and σ2 yields(17)∂J∂σ1=−λ2δ2σ1σ2ς23+4ς33−δ2k2σ1σ2ς2+4ς3−σ1σ2ω+ς1+2σ1σ2ς2+ς3−2σ13ς43σ2N−1=0,(18)∂J∂σ2=−λ2δ2σ12ς26+2ς33+δ2k2σ12σ22ς22+2ς3+σ122σ22ω+ς1−σ12σ22ς2+ς3+σ14ς46σ22N−1=0.From equations (17) and (18), we have(19)σ1=±2δ2k2ς2+8δ2k2ς3+2ω+2ς1−4ς2−4ς3ς4−Nς4,(20)σ2=±1λδδ2k2ς2+4δ2k2ς3+ω+ς1−2ς2−2ς3ς2+4ς3.Equation (15) becomes(21)g=±2δ2k2ς2+8δδ2k2ς3+2ω+2ς1−4ς2−4ς3ς4−Nς4sech±1λδδ2k2ς2+4δ2k2ς3+ω+ς1−2ς2−2ς3ς2+4ς3f.The solution for equation (2) becomes(22)φy,τ=±2δ2k2ς2+8δδ2k2ς3+2ω+2ς1−4ς2−4ς3ς4−Nς4eιky+ωτ+βsech±1λδδ2k2ς2+4δ2k2ς3+ω+ς1−2ς2−2ς3ς2+4ς3αχ,withα=λy−ντ.Next, we consider another solitary wave solution as(23)g=σ3sech4σ4f,where σ3 and σ4 are constants to be further calculated. Putting equation (23) into (14),, we have(24)J=−λ2δ2σ32σ4128ς2315+512ς3315−δ2k2σ32σ48ς235+32ς335−8σ3235σ4ω+ς1+16σ3235σ4ς2+ς3−512σ34ς46435σ4N−1.MakingJ stationary with respect to σ3 and σ4 yields(25)∂J∂σ3=−λ2δ2σ3σ4256ς2315+1024ς3315−δ2k2σ3σ416ς235+64ς335−16σ335σ4ω+ς1+32σ335σ4ς2+ς3−2048σ33ς46435σ4N−1=0,(26)∂J∂σ4=−λ2δ2σ32128ς2315+512ς3315+δ2k2σ32σ428ς235+32ς335+8σ3235σ42ω+ς1,(27)−16σ3235σ42ς2+ς3+512σ34ς46435σ42N−1=0.From equations (26) and (27), we have(28)σ3=±429δ2k2ς2+1716δ2k2ς3+429ω+429ς1−858ς2−858ς3224ς4−224Nς4,(29)σ4=±1λδ3δ2k2ς2+12δ2k2ς3+3ω+3ς1−6ς2−6ς316ς2+64ς3.Equation (23) becomes(30)g=±429δ2k2ς2+1716δδ2k2ς3+429ω+429ς1−858ς2−858ς3224ς4−224Nς4sech4±1λδ3δ2k2ς2+12δ2k2ς3+3ω+3ς1−6ς2−6ς316ς2+64ς3f.The solution of solitary wave for equation (2) is(31)φy,τ=±429δ2k2ς2+1716δδ2k2ς3+429ω+429ς1−858ς2−858ς3224ς4−224Nς4eιky+ωτ+βsech4±1λδ3δ2k2ς2+12δ2k2ς3+3ω+3ς1−6ς2−6ς316ς2+64ς3αχ,withα=λy−ντ.


## 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 fieldφy,τ is followed by oy,τ which represents the soliton part and py,τ which denotes the residual factor and also τ and y show the temporal and spatial components, respectively. Taking(32)φy,τ=oy,τ+py,τ,in equation (32), y is supposed to be y=yw1,w2,…,wn. It is crucial to note that the existence of variables in o increases the problem’s degree of freedom. As a result, we change equation (32) in the presence of these variables as follows:(33)φy,τ=ow1y,w2y,…,wny,τ+py,τ.The residual free energy is calculated as follows:(34)E=∫−∞∞p2dτ=∫−∞∞φ−ow1y,w2y,…,wny,τ2dτ.Taking partial derivatives of equation (34) w.r.t. wi and denoting by Qi,(35)Qi=∂E∂wi=∂∂wi∫−∞∞p2dτ=∂∂wi∫−∞∞pp∗dτ=∫−∞∞∂∂wipp∗dτ=∫−∞∞∂p∂wip∗+∂p∗∂wipdτ.The standard inner product is defined as(36)α1,α2=∫−∞∞α1τα2τdτ.Now, by utilizing this definition, equation (35) can be written as(37)Qi=∂p∂wi,p∗+p,∂p∗∂wi=∂p∂wi,p∗+∂p∂wi,p∗=∂p∗∂wi,p+∂p∗∂wi,p=2Re∂p∗∂wi,p=2Re∂φy,τ−ow1,w2,…,wn,τ∗∂wi,p=2Re∂φ∗y,τ∂wi−∂o∗w1,w2,…,wn,τ∂wi,p=−2Re∂o∗∂wi,p=−2Re∫−∞∞∂o∗∂wipdτ.In the above equation, Re denotes the real part, further taking∂φ∗y,τ/∂wi equal to zero. Now, utilizing equation (33), py,τ becomes(38)py,τ=φy,τ−ow1,w2,…,wn,τ.This results in the following relationships:(39)∂p∂wi=−∂o∂wi,∂p∗∂wi=−∂o∗∂wi.Utilizing these relations and with the help of equation (35), we have(40)Qi=∂E∂wi=∫−∞∞−∂o∂wip∗−∂o∂wipdτ=−∫−∞∞∂o∂wip∗+∂o∂wipdτ,or rather(41)Qi=−2Re∫−∞∞∂o∗∂wipdτ.Again, we also get(42)Q˙i=dQidy=−2Re∫−∞∞ddy∂o∗∂wipdτ=−2Re∫−∞∞∂o∗∂wi∂p∂y+p∂∂y∂o∗∂widτ,and(43)∂o∗∂y=∑j=1n∂o∗∂wj∂wj∂y.Plugging equation (43) in (42), one can have(44)Q˙i=−2Re∫−∞∞∂o∗∂wi∂o∂y+p∂∂wi∑j=1n∂o∗∂wj∂wj∂ydτ=−2Re∫−∞∞∂o∗∂wi∂p∂ydτ+∑j=1n∫−∞∞∂2o∗∂wi∂wj∂wj∂ypdτ.Furthermore, Dirac’s postulate states that if all of the variations in variables are not equal to zero, the function is nearly zero.Thus,Qi be minimized if(45)Qi≈0,with(46)Qi.≈0.Utilizing equations (2) and (32), we have(47)ιδ2ς2+4ς3o+pττ−ις4N−1o+p2o+p−ις1−2ς2−2ς3o+p=0.Moreover, from equation (34), we have(48)φy=∂ow1y,w2y,…,wny∂y+∂py,τ∂y,and also by combining equation (47) with equation (48), we get(49)∂p∂y=−∑i=1n∂o∂widwidy+η,with(50)η=ιδ2ς2+4ς3o+pττ−ις4N−1o+p2o+p−ις1−2ς2−2ς3o+p.Similarly, we know that(51)Q˙i=−2Re∫−∞∞∂o∗∂wi∂o∂ydτ+∑j=1n∫−∞∞∂2o∗∂wi∂wjdwjdypdτ,(52)Q˙i=2Re∑j=1n∫−∞∞∂o∗∂wi∂o∂wj−∂2o∗∂wk∂wjodτdwjdy−2Re∫−∞∞∂o∗∂wiηdτ.Its compact form is(53)Q˙i=∂Q∂wW˙+Z,with(54)W˙=−∂Q∂W−1Z,where(55)∂Q∂w=∂Q1∂w1∂Q1∂w2∂Q1∂w3∂Q1∂w4∂Q1∂w5∂Q1∂w6∂Q2∂w1∂Q2∂w2∂Q2∂w3∂Q2∂w4∂Q2∂w5∂Q2∂w6∂Q3∂w1∂Q3∂w2∂Q3∂w3∂Q3∂w4∂Q3∂w5∂Q3∂w6∂Q4∂w1∂Q4∂w2∂Q4∂w3∂Q4∂w4∂Q4∂w5∂Q4∂w6∂Q5∂w1∂Q5∂w2∂Q5∂w3∂Q5∂w4∂Q5∂w5∂Q5∂w6∂Q6∂w1∂Q6∂w2∂Q6∂w3∂Q6∂w4∂Q6∂w5∂Q6∂w6,while(56)W˙=w˙1w˙2w˙3w˙4w˙5w˙6andZ=Z1Z2Z3Z4Z5Z6.Thus, finallyZi is as follows:(57)Zi=−2Re∫−∞∞∂o∗∂wiηdτ=−2Re∫−∞∞ιδ2ς2+4ς3∂o∗∂wioττdτ−2Re∫−∞∞ιδ2ς2+4ς3∂o∗∂wipττdτ−2Re∫−∞∞−ις4N−1o+p2∂o∗∂wiodτ−2Re∫−∞∞−ις4N−1o+p2∂o∗∂wipdτ−2Re∫−∞∞−ις1−2ς2−2ς3∂o∗∂wiodτ−2Re∫−∞∞−ις1−2ς2−2ς3∂o∗∂wipdτ,for which i∈1,2,…n.


### 4.1. Soliton Parameter Dynamics

We use the constraint equations previously acquired by constructing the functionow1,w2,…,wn,τ to derive the system of equations of motion for the CVs. In addition, using the Gaussian ansatz, we suppose that the soliton function o has six CVs of the form(58)o=w1expι2w4τ−w22−τ−w22w32+ιw5τ−w2+ιw6,where wi,i=1,2,…6 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, bigpy,tau=0big. We find ∂Q/∂w and Z as follows:(59)∂Q∂w−1=322πw30−12πw10000w32πw1200w3w42πw12w3w52πw12−12πw102/πw3w12000000162/πw12w350−22/πw12w330w3w42πw120032π5/2w18/82w42w39/4+w35π3w110w38w3w4w52πw120w3w52πw120−22/πw12w33w3w4w52πw1232π5/2w18/82w42w39/4+w35π3w110w38.For equation (2), Zii=1,2,…,6 are calculated as(60)Z1=0,(61)Z2=2πw12−3δ2ς2w33w42w54−3δ2ς3w33w42w5−δ2ς2w3w53−4δ2ς3w3w53−3δ2ς2w5w3−12δ2ς3w5w3−ς4N−1w12w3w52−ς1−2ς2−2ς3w3w5,(62)Z3=2πw12−δ2ς2w4−4δ2ς3w4,(63)Z4=2πw123δ2ς2w35w4232+3δ2ς3w35w428+δ2ς2w33w528+δ2ς3w33w522−δ2ς2w38−δ2ς3w32−ς4N−1w12v33162+ς1−2ς2−2ς3w338,(64)Z5=2πw12δ2ς2w33w4w52+2δ2ς3w33w4w5,(65)Z6=2πw12δ2ς2w33w424+δ2ς3w33w42+δ2ς2w3w52+4δ2ς3w3w52+δ2ς2w3+4δ2ς3w3+ς4N−1w12w32+ς1+2ς2+2ς3w3.Using equation (54) and putting equations (59) and (60)–(65), the following system of ODEs is generated:(66)w˙1=−δ2ς2w1w4+4ς3w1w4,w˙2=2δ2ς2w5+4ς3w5,w˙3=2δ2ς2w3w4+4ς3w3w4,w˙4=−1w342δ2ς2w34w42+8δ2ς3w34w42−8δ2ς2−32δ2ς3−2N−1ς4w12w32,w˙5=0,w˙6=18w328δ2ς2w32w52+32δ2ς3w32w52+16ς2w32+16ς3w32−16δ2ς2−8ς1w32−64δ2ς3−52N−1ς4w12w32.


#### 4.1.1. Graphical Interpretation

## 4.1. Soliton Parameter Dynamics

We use the constraint equations previously acquired by constructing the functionow1,w2,…,wn,τ to derive the system of equations of motion for the CVs. In addition, using the Gaussian ansatz, we suppose that the soliton function o has six CVs of the form(58)o=w1expι2w4τ−w22−τ−w22w32+ιw5τ−w2+ιw6,where wi,i=1,2,…6 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, bigpy,tau=0big. We find ∂Q/∂w and Z as follows:(59)∂Q∂w−1=322πw30−12πw10000w32πw1200w3w42πw12w3w52πw12−12πw102/πw3w12000000162/πw12w350−22/πw12w330w3w42πw120032π5/2w18/82w42w39/4+w35π3w110w38w3w4w52πw120w3w52πw120−22/πw12w33w3w4w52πw1232π5/2w18/82w42w39/4+w35π3w110w38.For equation (2), Zii=1,2,…,6 are calculated as(60)Z1=0,(61)Z2=2πw12−3δ2ς2w33w42w54−3δ2ς3w33w42w5−δ2ς2w3w53−4δ2ς3w3w53−3δ2ς2w5w3−12δ2ς3w5w3−ς4N−1w12w3w52−ς1−2ς2−2ς3w3w5,(62)Z3=2πw12−δ2ς2w4−4δ2ς3w4,(63)Z4=2πw123δ2ς2w35w4232+3δ2ς3w35w428+δ2ς2w33w528+δ2ς3w33w522−δ2ς2w38−δ2ς3w32−ς4N−1w12v33162+ς1−2ς2−2ς3w338,(64)Z5=2πw12δ2ς2w33w4w52+2δ2ς3w33w4w5,(65)Z6=2πw12δ2ς2w33w424+δ2ς3w33w42+δ2ς2w3w52+4δ2ς3w3w52+δ2ς2w3+4δ2ς3w3+ς4N−1w12w32+ς1+2ς2+2ς3w3.Using equation (54) and putting equations (59) and (60)–(65), the following system of ODEs is generated:(66)w˙1=−δ2ς2w1w4+4ς3w1w4,w˙2=2δ2ς2w5+4ς3w5,w˙3=2δ2ς2w3w4+4ς3w3w4,w˙4=−1w342δ2ς2w34w42+8δ2ς3w34w42−8δ2ς2−32δ2ς3−2N−1ς4w12w32,w˙5=0,w˙6=18w328δ2ς2w32w52+32δ2ς3w32w52+16ς2w32+16ς3w32−16δ2ς2−8ς1w32−64δ2ς3−52N−1ς4w12w32.


### 4.1.1. Graphical Interpretation

## 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 χ. 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, ς1=1, ς2=1/30, ς3=1, ς4=−1/30, δ=1/15, and N=30, we have presented a graphical interpretation of the given system for the mentioned initial guess.Figure 1
In these figures, we takeς1=ς4=ω=1, ς2=0.5=δ, ς3=0.4, N=31, λ=5, k=0.1, β=0, and χ=0.2,0.4,0.6,0.8. For the solution φ2, (a) displays the 3D-plot of φ2 with χ=1, (b) displays the density plot of the solution φ2, and (c) displays the 2D-plot of φ2 with different values of χ.
(a)(b)(c)Figure 2
In these figures, we takeς1=ς4=ω=1, ς2=0.5=δ, ς3=0.4, N=31, λ=5, k=0.4, β=0, and χ=0.2,0.4,0.6,0.8. For the solution φ2, (a) displays the 3D-plot of φ2 with χ=1, (b) displays the density plot of the solution φ2, and (c) displays the 2D-plot of φ2 with different values of χ.
(a)(b)(c)It is important to mention here that the pulse propagatesw1, w3, and w4, that is, the soliton’s chirp, width, and amplitude shows periodic behavior with uniform wavelengths but center position w2 and phase w6 are increasing with propagation distance in Figure 3. Figure 4 shows, however, that w1, w3, and w4 are the soliton’s amplitude; chirp and width vary on a regular basis with the same wavelengths whereas temporal position w2 and frequency w5 are behaving constant for all the values of propagation distance y and the phase of pulse w6 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 w1, width w3, and chirp w4.Figure 3
2D analysis of CVswi,i=1,2,…,6 with distance of propagation y taking initial guess w1=w2=w3=w5=1,w4=w6=0.
(a)(b)(c)(d)(e)(f)Figure 4
2D analysis of CVswi,i=1,2,…,6 with distance of propagation y taking initial values w1=w2=w3=1,w4=w5=w6=0.
(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 conditionQi≈0 on the CVs wii=1,…,n, 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.



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