Exact solutions of (2 + 1)‐dimensional Schrödinger's hyperbolic equation using different techniques

Rehman, Hamood Ur; Imran, Muhammad; Ullah, Naeem; Akgül, Ali

doi:10.1002/num.22644

Cited by 15 publications

(7 citation statements)

References 45 publications

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“…This section gives the steps of EHFM [73][74][75][76][77] in detail to obtain the periodic-singular solutions, dark, singular and bright soliton solutions to Equation (1). Equation ( 12) has the following solution by using homogeneous rule:…”

Section: Narrative Of Ehfmmentioning

confidence: 99%

Dynamics of optical solitons in the extended (3+1)‐dimensional nonlinear conformable Kudryashov equation with generalized anti‐cubic nonlinearity

Mirzazadeh,

Hashemi,

Akbulu

et al. 2024

Math Methods in App Sciences

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The nonlinear Schrödinger equation (NLSE) is a fundamental equation in the field of nonlinear optics and plays an important role in the study of many physical phenomena. The present study introduces a new model that demonstrates the novelty of the paper and provides the advancement of knowledge in the area of nonlinear optics by solving a challenging problem known as the extended ‐dimensional nonlinear conformable Kudryashov's equation (CKE) with generalized anti‐cubic nonlinearity, which is a generalization of the NLSE to three spatial dimension and one temporal dimension for the first time. This work is significant because it advances our understanding of nonlinear optics and its applications to solve complex equations in physics and related disciplines. The extended hyperbolic function method (EHFM) and Nucci's reduction method are applied to the extended ‐dimensional nonlinear CKE with generalized anti‐cubic nonlinearity. The equation is solved by using the concept of conformable derivative, a recently developed operator in fractional calculus, which has advantages over other fractional derivatives in terms of accuracy and flexibility. The attained solutions include periodic singular, dark 1‐soliton, singular 1‐soliton, and bright 1‐soliton which are visualized using 3D and contour plots. This study highlights the potential of using conformable derivative and the applied techniques to solve complex nonlinear differential equations in various fields. The obtained solutions and analysis will be useful in the design and analysis of optical communication systems and other related fields. Overall, this study contributes for the understanding of the dynamics of the extended ‐dimensional nonlinear CKE and offers new insights into the use of mathematical techniques to tackle complex problems in physics and related fields.

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Section: Narrative Of Ehfmmentioning

confidence: 99%

Dynamics of optical solitons in the extended (3+1)‐dimensional nonlinear conformable Kudryashov equation with generalized anti‐cubic nonlinearity

Mirzazadeh,

Hashemi,

Akbulu

et al. 2024

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…In spite of the fact that the symmetries of the Schrödinger model are of great importance in solving several problems of quantum mechanics, it remains indubitable that they are most beneficial in constructing their exact solutions [25]. Due to the potential applications of the NLSEs, the study of soliton solutions has been performed from different perspectives [26][27][28][29][30][31]. The investigation of chiral nonlinear NLSEs in two dimensions are of great importance [32].…”

Section: Introductionmentioning

confidence: 99%

The new stochastic solutions for three models of non-linear Schrödinger’s equations in optical fiber communications via Itô sense

Alkhidhr

2023

Front. Phys.

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In this paper, we consider three models of non-linear Schrödinger’s equations (NLSEs) via It\^{o} sense. Specifically, we study these equations forced by multiplicative noise via the Brownian motion process. There are numerous approaches for converting non-linear partial differential equations (NPDEs) into ordinary differential equations (ODEs) to extract wave solutions. The majority of these methods are a type of symmetry reduction known as non-classical symmetry. We apply the unified technique based on symmetry reduction to produce some new optical soliton solutions for the proposed equations. The obtained stochastic solutions depict the propagation of waves in optical fiber communications. The theoretical analysis and proposed results clarify that the presented technique is sturdy, appropriate, and efficacious. Some graphs of selected solutions are also depicted with the help of the MATLAB packet program. Indeed, the structure, bandwidth, amplitude, and phase shift are controlled by the influences of physical parameters in the presence of noise term via It\^{o} sense. Our results show that the proposed technique is better suited for solving many other complex models arising in real-life problems.

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“…The propagation of electrostatic nonlinear dissipative envelope structures including dissipative rogue waves and dissipative breathers in nonlinear and dispersive mediums, such as unmagnetized collisional pairing plasmas composed of warm positive and negative ions, has been investigated analytically and numerically in [11]. In [12,13] the 2D-chiral nonlinear Schrödinger equation is effectively analyzed using the extended direct algebraic method and the extended hyperbolic function method. Using these two techniques, a variety of soliton and some other solutions to this model are achieved.…”

Section: Introductionmentioning

confidence: 99%

Crank-Nicolson/finite element approximation for the Schrödinger equation in the de Sitter spacetime

Selvi̇topi̇¹,

Zaky²,

Hendy³

2021

Phys. Scr.

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Central to much of science, engineering, and society today is the building of mathematical models to represent complex processes. Recently, the non-relativistic limit of nonlinear Klein-Gordon equations in de Sitter spacetime has been used to derive Schrödinger equations with weighted nonlinear terms In this paper, numerical simulations are constructed to clarify the behavior of the solution in both one- and two-dimensions. These simulations are constructed based on the Crank-Nicolson scheme in the time direction and the Galerkin finite element in the spatial direction. The nonlinear system of algebraic equations resulting from the constructed scheme is solved using Newton’s method. It is also demonstrated that the numerical approximation converges to the exact one.

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Exact solutions of (2 + 1)‐dimensional Schrödinger's hyperbolic equation using different techniques

Cited by 15 publications

References 45 publications

Dynamics of optical solitons in the extended (3+1)‐dimensional nonlinear conformable Kudryashov equation with generalized anti‐cubic nonlinearity

Dynamics of optical solitons in the extended (3+1)‐dimensional nonlinear conformable Kudryashov equation with generalized anti‐cubic nonlinearity

The new stochastic solutions for three models of non-linear Schrödinger’s equations in optical fiber communications via Itô sense

Crank-Nicolson/finite element approximation for the Schrödinger equation in the de Sitter spacetime

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