Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times

Başhan, Ali; Esen, Alaattin

doi:10.1002/num.22597

Cited by 14 publications

(2 citation statements)

References 39 publications

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“…Looking at the study examining the two-soliton, rogue structure [26,28,29] which are the rare types in the literature, it will be seen that the error obtained is similar to the error in this study. It can be seen that the classical PINN method gives poor results compared to advanced numerical methods such as B-Spline and the modified Laplace decomposition method in layered and complex soliton types [30,31].…”

Section: Tablementioning

confidence: 99%

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

Onder,

Sahiner,

Secer

et al. 2024

AIA

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In this study, we aim to obtain numerical results of the modified Benjamin-Bona-Mahony equation, Ostrovsky-Benjamin-Bona-Mahony equation and Mikhailov-Novikov-Wang equation via the physics-informed neural networks (PINN) method. The equations are modeled for shallow and long water waves, as well as fundamental and phenomenonal models in ocean engineering. According to the implementation, we obtained the PINN solutions of kink, bright, multisoliton (two-soliton) and mixed dark-bright soliton solutions. According to the inference from the obtained results, we achieved good results in some cases compared to other approximate solution methods in the literature. However, it was also observed that the best possible results could not be obtained in cases where the soliton type was intricate and layered. While the results were obtained, the number of hidden layers and the number of neural networks in the layers also varied. These results are shown in tables. Since it is known that the aforementioned models are not solved by the PINN method, we anticipate that the study will lead to other studies in the field of ocean engineering.

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Section: Tablementioning

confidence: 99%

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

Onder,

Sahiner,

Secer

et al. 2024

AIA

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“…Its coefficient d plays an important role in the form of the rNLSE, as it determines solutions with different behavior. Many studies on Schrödinger and rNLSE are made by various scientists [19][20][21][22][23][24]. For instance, Williams et al [17] argued the stability and dynamical properties of soliton waves in rLNSE.…”

Section: Introductionmentioning

confidence: 99%

Generalized Sub-Equation Method for the (1+1)-Dimensional Resonant Nonlinear Schrodinger’s Equation

Taşbozan

Tozar

Kurt

2021

Bilecik Şeyh Edebali Üniversitesi Fen Bilimleri Dergisi

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Interest in studying nonlinear models has been increasing in recent years. Dynamical systems, in which the state of the system changes continuously over time, have nonlinear interactions. The use of unique nonlinear differential equations is inescapable in the evaluation of such systems. In mathematical point of view, for obtaining analytical solutions of nonlinear differential equations, it must be fully integrable. Consequently, the importance of fully integrable nonlinear differential equations for nonlinear science has become indisputable. Among these equations, one of the most studied by physicists and mathematicians is the nonlinear Schrödinger equation. This equation has undergone many modifications to evaluate different phenomena. In this study, the resonant nonlinear Schrödinger equation, which is the most important of these physical equations in terms of explaining many physical phenomena, is solved analytically with the generalized sub-equation method.

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Study of dynamical behavior of Burgers' equation in quantum modeling

Gupta

Shukla

Singh

2023

Math Methods in App Sciences

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Our main goal is to investigate the dynamical behavior of the quantum model of Bateman–Burgers equation utilizing a new hybrid semidiscretization technique: nCB‐DQT‐RK scheme. The nCB‐DQT‐RK is traditional differential quadrature technique (DQT) with newly modified cubic B‐splines (nCB), as a basis, which have been utilized for the discretization of spatial derivatives, and the time integration scheme SSP‐RK for the temporal discretization. The efficiency/effectiveness of the reported nCB‐DQT‐RK results, computationally, are illustrated not only numerically but also graphically for better cognizance and reliable performance of the nCB‐DQT‐RK. This is validated based on the estimation of evaluated accuracy by means of stability and convergence analysis, which infer that the proposed nCB‐DQT‐RK scheme is capable of producing stable results with spatial convergence of order four, and the use of nCB in DQT is capable to improve performance (in terms of accuracy and spatial rate of convergence as well) as compared to existing techniques (collocation/quadrature) with traditional cubic B‐splines and its existing modifications, and so, nCB is capable to surpass cubic B‐splines and its existing modification.

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Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times

Cited by 14 publications

References 39 publications

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

Generalized Sub-Equation Method for the (1+1)-Dimensional Resonant Nonlinear Schrodinger’s Equation

Study of dynamical behavior of Burgers' equation in quantum modeling

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