Periodic structures described by the perturbed Burgers–Korteweg–de Vries equation

Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I.

doi:10.1016/j.ijnonlinmec.2015.02.008

Cited by 4 publications

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References 34 publications

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“…It simplifies the process of further investigation. We apply the Painlevé test for equation (15) using three steps (see for example [18]). On the first step we have to determine the order of the pole and the first term in expansion of the solution in the Laurent series.…”

Section: Introductionmentioning

confidence: 99%

The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

Kudryashov

Volkov

2017

Communications in Nonlinear Science and Numerical Simulation

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We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi-Pasta-Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlevé approach. We show that the equation does not pass the Painlevé test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth-order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.set of approaches was used to explain it (see for example [2][3][4][5][6][7]). The main ideas of these approaches and their results can be found in work [8].Usually α or β FPU model is considered in literature, but in paper [9] the α + β FPU model is investigated. In this paper, we also use the α + β model but we use the continuous limit approximation for the investigation.This approach was used for the first time by N. Zabusky and M. Kruskal in work [10], where the authors derived the Korteweg-de Vries (KdV) equation for the description of the α FPU model. This equation is integrable by the inverse scattering transform and has been widely studied by now ( [11, 12]). Zabusky and Kruskal explained the FPU paradox (recurrence of initial conditions) using the numerical simulation of the KdV equation. Another result of work [10] was an introduction of soliton. In article [13] the author took into account high order (in comparison with [10]) terms in the Taylor series expansions for the continuous limit approximation of the α FPU model. The generalized KdV equation was obtained for a more accurate description of wave processes in the FPU model. Some exact solutions of the derived equation were found . It is shown [13] that if one takes into account more terms in the Taylor series and acts in analogy to paper [10] then he can not derive an integrable equation for the description of wave processes in the FPU mass chain. Recurrence of initial statement is also unrepresentative for wave processes, described by the fifth-order equation with arbitrary initial conditions. In paper [14] wave processes for the generalized KdV equation are simulated numerically.This brings up the question, whether the situation is similar for the β and the α + β FPU models? It is known [11] that one can obtain the Gardner equation for the description of the α + β FPU model using the continuous limit. The Gardner equation is integrable by the inverse scatterring transform and can be reduced to the modified KdV equation [15]. If we take more terms in the Taylor series expansion, we can derive the fifth-order partial differential ...

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Section: Introductionmentioning

confidence: 99%

The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

Kudryashov

Volkov

2017

Communications in Nonlinear Science and Numerical Simulation

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Effects of viscosity and induced magnetic fields on weakly nonlinear wave transmission in a viscoelastic tube using physics-informed neural networks

Bhaumik,

Changdar,

Chakraverty

et al. 2024

Physics of Fluids

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This study presents an advanced deep learning methodology that utilizes physics-informed neural networks (PINNs), to analyze the transmission of weakly nonlinear waves in a prestressed viscoelastic arterial tube. Using the long wave approximation, a mathematical model is constructed to replicate the propagation of weakly nonlinear waves in a viscoelastic arterial tube filled with viscous nanofluid, taking into account the influence of an induced magnetic field. The perturbed Burger, perturbed Korteweg–de Vries, and perturbed Korteweg–de Vries-Burgers equations are formulated based on the combined effects of nanofluid viscosity and the applied magnetic field using the reductive perturbation technique. Semi-supervised physics-informed neural network models are utilized to solve these perturbed evolutionary equations, trained on a limited dataset within their rectangular domain of definition. Gaussian process-based Bayesian optimization is used to determine the hyperparameters of the neural network, ensuring optimal model is performance. The effectiveness of the optimal models is evaluated by calculating the residual losses associated with the perturbed partial differential equations (PDEs). Visual representations of weakly non-linear wave propagation, considering nanofluid viscosity and induced magnetic fields, enhance the comprehension of dissipative effects in the cardiovascular system. These insights aid in obtaining precise measurements of pulse wave velocity for cardiovascular health monitoring. Consequently, the application of PINN proves to be a valuable tool for solving real-world PDEs and highlights its importance in advancing medical machine learning fields.

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Approximate analytical solutions for a class of generalized perturbed KdV-burgers equation

Deng

2023

THERM SCI

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Periodic structures described by the perturbed Burgers–Korteweg–de Vries equation

Cited by 4 publications

References 34 publications

The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

The fifth-order partial differential equation for the description of theα+βFermi–Pasta–Ulam model

Effects of viscosity and induced magnetic fields on weakly nonlinear wave transmission in a viscoelastic tube using physics-informed neural networks

Approximate analytical solutions for a class of generalized perturbed KdV-burgers equation

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