Abundant soliton-type solutions to the new generalized KdV equation via auto-Bäcklund transformations and extended transformed rational function technique

Jannat, Nahal; Kaplan, Melike; Raza, Nauman

doi:10.1007/s11082-022-03862-x

Cited by 13 publications

(5 citation statements)

References 18 publications

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“…The values of A, B, and 𝜌, as determined by Equations ( 10), (11), and (18), respectively, and the value of v from Equation (8), are incorporated in the following solutions. These solutions will be explained while considering the constraint condition described by Equation (18).…”

Section: Application Of Ehfmmentioning

confidence: 99%

“…where A, B, and 𝜌 are given by Equations ( 10), (11), and (29), respectively, and v is given by Equation (8). These soliton will be specified under the constraint condition Equation (31).…”

Section: Application Of Ehfmmentioning

confidence: 99%

“…Nonlinear evolution equations (NLEEs) are crucial in the field of material and applied sciences, as they provide explicit and exact soliton solutions [1][2][3][4][5][6][7][8][9]. Over the past few decades, several techniques have been introduced for this purpose such as Darboux transformation [10], Riemann-Hilbert technique [11], modified Kudryashov's method [12,13], first integral method [14,15], Sardar sub-equation approach [16][17][18], extended hyperbolic tangent function method [19], extended rational sinh-cosh method [20], fractional sub-equation method [21], ansatz method [22,23], modified simple equation method [24], Hirota's method [25], sine-Gordon method [26], and many more [27][28][29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

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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: Application Of Ehfmmentioning

confidence: 99%

Section: Application Of Ehfmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…where, a 1 , a 2 , b, c, and K are arbitrary constants. Putting Eq (33) or Eq (34) into Eq (2) we get the solution ψ(x, y, z, t) which is then represented in Figs 6 and 7.…”

Section: Exploring Solutions With the Extended Transformed Rational F...mentioning

confidence: 99%

“…This method complements the Bäcklund transformation by offering an alternative approach to tackle nonlinear partial differential equations. By employing rational function solutions and extended transform techniques, we can effectively capture the intricate dynamics of these equations and derive analytical solutions that exhibit complex behaviors [ 32 , 33 ]. In this study, we will utilize the bilinear form of the given model to explore various ansatz and apply the Bäcklund transformation to derive different solutions in the form of traveling waves.…”

Section: Introductionmentioning

confidence: 99%

An exploration of the (3+1)-dimensional negative order KdV-CBS model: Wave solutions, Bäcklund transformation, and complexiton dynamics

Vivas-Cortez,

Rani,

Raza

et al. 2024

PLoS ONE

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This research paper focuses on the study of the (3+1)-dimensional negative order KdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation, an important nonlinear partial differential equation in oceanography. The primary objective is to explore various solution techniques and analyze their graphical representations. Initially, two wave, three wave, and multi-wave solutions of the negative order KdV CBS equation are derived using its bilinear form. This analysis shed light on the behavior and characteristics of the equation’s wave solutions. Furthermore, a bilinear Bäcklund transform is employed by utilizing the Hirota bilinear form. This transformation yields exponential and rational function solutions, contributing to a more comprehensive understanding of the equation. The resulting solutions are accompanied by graphical representations, providing visual insights into their structures. Moreover, the extended transformed rational function method is applied to obtain complexiton solutions. This approach, executed through the bilinear form, facilitated the discovery of additional solutions with intriguing properties. The graphical representations, spanning 2D, 3D, and contour plots, serve as valuable visual aids for understanding the complex dynamics and behaviors exhibited by the equation’s solutions.

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A novel investigation of dark, bright, and periodic soliton solutions for the Kadomtsev-Petviashvili equation via different techniques

2023

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Abundant soliton-type solutions to the new generalized KdV equation via auto-Bäcklund transformations and extended transformed rational function technique

Cited by 13 publications

References 18 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

An exploration of the (3+1)-dimensional negative order KdV-CBS model: Wave solutions, Bäcklund transformation, and complexiton dynamics

A novel investigation of dark, bright, and periodic soliton solutions for the Kadomtsev-Petviashvili equation via different techniques

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