Analytical soliton solutions for cold bosonic atoms (CBA) in a zigzag optical lattice model employing efficient methods

Abstract

This research finds an equation in a continuous domain and a discrete equation governing the system of cold bosonic atoms (CBA) in a zigzag optical lattice using a continuum approximation. Many solutions to the equation were obtained using two distinct methods: the three-wave approach (multi-wave interaction, rational solutions, and rational solution interaction) and the extended sub-equation method. These analytical approaches are more effective, consistent, and comprehensive mathematical tools for obtaining various exact closed-form solutions for a wide range of fractional space-time nonlinear evolution equations encountered in optical physics, condensed matter physics, and plasma physics. The solutions generated are in the form of hyperbolic and trigonometric solutions, and other-form solutions are obtained. Three-dimensional graphics and contour plots are often used to depict the graphical representations of the combined soliton solutions. These findings will aid our understanding of the dynamics of the zigzag optical grids and many other structures formed by colder bosonic atoms. The applied approaches are more simple, efficient, and straightforward to obtain the closed-form solutions for various nonlinear evolution equations in the fields of nonlinear sciences and physical engineering.

Abundance of Exact Solutions of a Nonlinear Forced (2 + 1)-Dimensional Zakharov–Kuznetsov Equation for Rossby Waves

“…Also, the exact solutions of these equations are crucial to studying the propagation of Rossby waves [8][9][10][11]. So, many methods have been proposed on how to solve the exact solutions to nonlinear equations, for instance, the Hirota method [12][13][14], the Jacobi elliptic function expansion method [15][16][17][18], the G′/G-expansion method [19][20][21][22][23], the Exp (− Φ(ξ))-expansion method [24,25], the generalised exponential rational function method [26][27][28], the negative power expansion method [29], the hyperbolic function expansion method [30][31][32][33], the extended sub-equation method [34], (ω/g)-expansion method [35], the improved sub-ODE method [36], the Riccati-Bernoulli sub-ODE method [37][38][39][40], the Lie symmetry technique [41][42][43][44][45][46][47], the fractional sub-equation [48] etc. Tese are valid methods and tools for computing nonlinear equations.…”

“…Zkan et al used the improved tan (φ/2) expansion method to obtain the exact solution to the (2 + 1)-dimensional KdV equation and explained the infuence of diferent parameters on wave propagation through three-dimensional graphs and tables [69]. Compared to the negative power expansion method [29], the extended subequation method [34], and the improved sub-ODE method [36], we can obtain more formal solutions using the improved tan (φ/2) expansion method, which is one of the efcient mathematical methods and tools that are widely used and easy to implement.…”

Analytical Soliton Solutions of the Coupled Radhakrishnan‐Kundu‐Lakshmanan Equation via Three Techniques

“…The theory and investigation of soliton solutions is one of the important research fields relating to nonlinear partial differential equations ascending in telecommunication engineering, optics, mathematical physics, and other domains of nonlinear sciences. Therefore, diverse academics and researchers developed a number of numerical and analytical techniques, namely, the ðm + 1/G ′ Þ-expansion technique [1], the truncated M-fractional derivative scheme [2], the q-homotopy analysis technique [3], Atangana-Baleanu operator scheme [4], the improved Bernoulli subequation function process [5], the sine-Gordon expansion approach [6], the Haar wavelet technique [7], the biframelet systems process [8], the Lie symmetry technique [9], the generalized exponential rational function mode [10], the Painlevé analysis [11], the extended subequation method [12], the improved ðG′/GÞ-expansion scheme [13], the Hirota simplified method [14], the onedimensional subalgebra system [15], Painlevé analysis and multi-soliton solutions technique [16], the one-parameter Lie group of transformations approach [17], etc.…”

New Fractal Soliton Solutions and Sensitivity Visualization for Double-Chain DNA Model

“…Many physicists and mathematicians have worked hard to develop further precise alternatives to NLPDEs for a better understanding of these processes. Therefore, exact solutions of NLPDEs are essential for exploring physical explanations and qualitative aspects of different mechanisms [8][9][10][11][12][13][14][15][16][17][18]. These solutions demonstrate the dynamics of several nonlinear complex models symbolically and physically.…”