On continuum model analog to zig-zag optical lattice in quantum optics

Tantawy, M.; Abdel‐Gawad, H. I.

doi:10.1007/s00340-021-07669-8

Cited by 12 publications

(1 citation statement)

References 27 publications

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“…Different methods and techniques were used to solve nonlinear evolution equations; T anh and Exp-function [25,26], G G expansion [27], Darboux transformation [28], Kyrdiashov method, [29], Hirota-bilinear transformation [30], Lie symmetries of NLPDEs [31].Very recently different methods and techniques were introduced, among them, the first integral method, the improved q-homotopy perturbation method and the unified algebraic and auxiliary equation expansion methods [32][33][34][35][36][37][38][39][40][41]. Here, the unified method (UM) [42][43][44][45][46][47] is used in this paper.…”

Section: Introductionmentioning

confidence: 99%

Continuum Soliton Chain Analog to Heisenberg Spin Chain System. Modulation Stability and Spectral Characteristics

Abdel‐Gawad

2022

Int J Theor Phys

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The one-dimensional (1D) Heisenberg spin chain system (HSCS) allows to investigate anomalous features originating from strong quantum fluctuations, which become more significant than those in higher dimensions. A continuum model equation of the HSCS, based on the discrete model, was constructed in the literature. It is nonlinear Schrodinger equation (NLSE) with biquadratic dispersion and fifth degree nonlinearity. Rare research works were done in this area. Notably for deriving the exact solutions and investigate the physical phenomena produced. Our objective, here, is to obtain these solutions, which we think they are new. Further, an analog of the different geometric solutions structures to the known characteristics of HSCS is performed. The unified method is implemented to find the exact solutions of the continuum model equation. A variety of solutions are obtained where they are evaluated numerically and represented in graphs. In these graphs, it is remarked that the solutions exhibit soliton chain (or dense soliton chain).in an analog to spin chain. In the contour plots, they show different shapes of super lattices. Furthermore, complex chirped waves are observed. A significant result is that these solutions are bounded by −1/4 and 1/4, which can be relevant to the spins−1/2 and 1/2. The analysis of modulation stability is carried and it is found that there is a critical value for the dominant parameters, where below this value, modulation instability holds otherwise modulation stability occurs.For the spectral characteristics, it is shown that the wave number increases abruptly and decreases to an asymptotic state, while the frequency is monotonic increasing. The spectrum is periodic wave away from the origin, but near the origin it is soliton.

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Section: Introductionmentioning

confidence: 99%

Continuum Soliton Chain Analog to Heisenberg Spin Chain System. Modulation Stability and Spectral Characteristics

Abdel‐Gawad

2022

Int J Theor Phys

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Analytical insights into cold bosonic atoms in a zigzag optical lattice: Invariant analysis, exact solutions, and bifurcation analysis using phase portraits

Yadav,

Kumar Gupta

2024

Math Methods in App Sciences

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This work delves into the behavior of cold bosonic atoms within a zigzag optical lattice which offers a rich platform for studying quantum many‐body physics and has potential applications in various fields including quantum simulation, quantum computing, and precision measurement. The study aims to derive exact solutions and explore qualitative properties through dynamical analysis. Exact solutions of the nonlinear differential equation model can provide insights into the behavior of bosonic atoms in complex optical lattice configurations, allowing researchers to simulate and study phenomena such as quantum phase transitions, many‐body localization, and exotic states of matter. The invariant analysis has been performed for the first time on the specified equation, yielding a reduced system of equations with forms that are easier to handle. Novel techniques are applied to extract solutions from the reduced system of equations obtained via invariant analysis. The results reveal a rich set of solutions, including various traveling wave solutions and doubly periodic functions in the form of Jacobian elliptic functions. Bifurcation analysis, conducted through phase portraits, provides insights into the system's long‐term behavior under different parameter values. This work contributes to a deeper understanding of the dynamics of cold bosonic atoms in optical lattices via 2D and 3D plots of obtained solutions depicting the change in the behavior of soliton solutions with fractional derivative parameter.

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Longitudinal-transverse soliton chains analog to heisenberg ferromagnetic spin chains in (2+1) dimensional with biquadrant interactions

Abdel‐Gawad

2022

Opt Quant Electron

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In quantum spin systems, lower energy state for a nuclear spin in an external field is spin ($$+1/2$$ + 1 / 2 ) while the higher energy state is spin ($$-1/2$$ - 1 / 2 ). A continuum analog to the discrete model led to nonlinear Schrodinger equation (NLSE) with higher order dispersion. Recently, a model equation with bilinear and biquadratic interactions in the (2 + 1) dimensional Heisenberg ferromagnetic spin chain (HFSC), was derived in the literature. This model equation is a NLSE with quartic dispersion and fifth degree nonlinearity, and it was rarely considered in the literature. The, only, work done has shown solitons solutions. This motivated us to consider this problem for inspecting the multiple characteristics of the HFSC in space-time. Here, the exact solutions of the HFSC biquadratic model equation are obtained by using the unified method (UM). In applications, it is found the UM is of low time cost in symbolic computations. So, we think that it prevails the known methods. The solutions obtained are evaluated numerically and shown in figures.These figures reveal that the solutions exhibit longitudinal-transverse (L-T) solitons chains (SCs) with presence (or absence) of tunneling, depending on the values of the parameters of high dispersivity and high nonlinearity. Also, L-T zig-zag SCs are observed, where pulses with higher amplitudes (or gaps) occur along a characteristic line. Furthermore, L-TSCs modulation along the space axes is shown. It is remarked that the contour plots show complex lattice waves, which is relevant to the spin chain system. So, the present work reveals new solitons structures induced by bilinear-biquadratic interactions of HFSC.

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On continuum model analog to zig-zag optical lattice in quantum optics

Cited by 12 publications

References 27 publications

Continuum Soliton Chain Analog to Heisenberg Spin Chain System. Modulation Stability and Spectral Characteristics

Continuum Soliton Chain Analog to Heisenberg Spin Chain System. Modulation Stability and Spectral Characteristics

Analytical insights into cold bosonic atoms in a zigzag optical lattice: Invariant analysis, exact solutions, and bifurcation analysis using phase portraits

Longitudinal-transverse soliton chains analog to heisenberg ferromagnetic spin chains in (2+1) dimensional with biquadrant interactions

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