On nonlinear superposition of the KdV–Burgers shock waves and the behavior of solitons in a layered medium

Samokhin, Alexey

doi:10.1016/j.difgeo.2017.03.001

Cited by 12 publications

(6 citation statements)

References 5 publications

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“…Discussion. The present paper as well as our previous research of the KdV solitons in inhomogeneous media ( [4,5,6,7]) persuades that a distorted by inhomogeneity compact impulse getting into homogeneous region behaves according the same scenario: it became a soliton or scatter into two or more of them. Usually, but not necessarily, the obstacle generates a reflected wave.…”

Section: 1supporting

confidence: 65%

The KdV soliton crosses a dissipative and dispersive border

Samokhin¹

2020

Preprint

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We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The passed wave either retains the form of a soliton (though of different parameters) or becomes a bi-soliton. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapour saturation/condensation area. Some rough estimations for a prediction of an output are given using relative decay of the KdV conserved quantities are given.

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Section: 1supporting

confidence: 65%

The KdV soliton crosses a dissipative and dispersive border

Samokhin¹

2020

Preprint

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“…After the oscillations cease, the wave develops as a constant height and velocity shock. It almost coincides with a traveling wave solution (TWS) of the Burgers equation [8,9].…”

Section: Introductionmentioning

confidence: 66%

On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

Samokhin

2021

Symmetry

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We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found.

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“…The KdV-Burgers equation is a nonlinear partial differential equation containing higher-order derivatives that has been of interest to many researchers [56,57]. Today, the KdV-Burgers equation is widely studied and applied in many fields, such as the study of the flow of liquids containing bubbles, the flow of liquids in elastic tubes, and other problems [58,59]. In mathematical form, the KdV-Burgers equation is defined, as follows:…”

Section: Kdv-burgers Equationmentioning

confidence: 99%

Solving Partial Differential Equations Using Deep Learning and Physical Constraints

et al. 2020

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The various studies of partial differential equations (PDEs) are hot topics of mathematical research. Among them, solving PDEs is a very important and difficult task and, since many partial differential equations do not have analytical solutions, numerical methods are widely used to solve PDEs. Although numerical methods have been widely used with good performance, researchers are still searching for new methods for solving partial differential equations. In recent years, deep learning has achieved great success in many fields, such as image classification and natural language processing. Studies have shown that deep neural networks have powerful function-fitting capabilities and have great potential in the study of partial differential equations. In this paper, we introduce an improved Physics Informed Neural Network (PINN) for solving partial differential equations. PINN takes the physical information that is contained in partial differential equations as a regularization term, which improves the performance of neural networks. In this study, we use the method to study the wave equation, the KdV–Burgers equation, and the KdV equation. The experimental results show that PINN is effective in solving partial differential equations and deserves further research.

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On nonlinear superposition of the KdV–Burgers shock waves and the behavior of solitons in a layered medium

Cited by 12 publications

References 5 publications

The KdV soliton crosses a dissipative and dispersive border

The KdV soliton crosses a dissipative and dispersive border

On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

Solving Partial Differential Equations Using Deep Learning and Physical Constraints

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