scholarly journals Genuine nonlinearity and its connection to the modified Korteweg–de Vries equation in phase dynamics

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 30-65
Author(s):  
D J Ratliff
Keyword(s):  
Optical Model ◽  
Vries Equation ◽  
Linear Dispersion ◽  
Phase Dynamics ◽  
Contact Discontinuities ◽  
De Vries ◽  
Genuine Nonlinearity ◽  
Hyperbolic Waves ◽  
The Waves ◽  
Insight Into

Abstract The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of genuine nonlinearity, namely the ability for shocks and rarefactions to form instead of contact discontinuities. In the context of the Whitham modulation equations, this paper demonstrate that a loss of genuine nonlinearity leads to the appearance of a dispersive set of dynamics in the form of the modified Korteweg de-Vries equation governing the evolution of the waves instead. Its form is universal in the sense that its coefficients can be written entirely using linear properties of the underlying waves such as the conservation laws and linear dispersion relation. This insight is applied to two systems of physical interest, one an optical model and the other a stratified hydrodynamics experiment, to demonstrate how it can be used to provide insight into how waves in these systems evolve when genuine nonlinearity is lost.

scholarly journals Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

2021 ◽  
Vol 7 (2) ◽  
pp. 1580-1602
Author(s):  
Khalid Khan ◽  
◽  
Amir Ali ◽  
Manuel De la Sen ◽  
Muhammad Irfan ◽  
...  
Keyword(s):  
Decomposition Method ◽  
Vries Equation ◽  
Coupled System ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Double Laplace Transform ◽  
Proposed Model ◽  
De Vries ◽  
Singular Kernels ◽  
The Waves

<abstract><p>In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 &lt; \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.</p></abstract>

scholarly journals NUMERICAL SIMULATION OF IRREGULAR BIMODAL WAVE SYSTEM DYNAMICS WITHIN THE FRAMEWORK OF KORTEWEG-DE VRIES EQUATION

2019 ◽  
Vol 47 (1) ◽  
pp. 38-40
Author(s):  
E.G. Didenkulova ◽  
A.V. Slunyaev ◽  
E.N. Pelinovsky
Keyword(s):  
Numerical Simulation ◽  
Shallow Water ◽  
Vries Equation ◽  
Kdv Equation ◽  
Wave Spectrum ◽  
Wave System ◽  
Quasi Equilibrium ◽  
Long Wave ◽  
De Vries ◽  
The Waves

The dynamics of wave ensembles in shallow water is studied within the framework of the nonlinear dispersive Korteweg – de Vries (KdV) equation by numerical simulation. Bimodal wave systems whose energy is distributed over two spectral domains are considered: the “additional” lobe which corresponds to the system of longer or shorter waves is added to the “main” spectral peak. The concerned problem describes, for example, the interaction between wind waves and swell in shallow water. The case of the unimodal waves (considered in (Pelinovsky, Sergeeva, 2006) is used as the reference. The limitations of the implied assumptions and the relationship of the idealized model to the realistic conditions in the ocean were discussed in the recent paper (Wang et al, 2018). Based on the detailed consideration of the 6 simulated cases, the following general conclusions may be formulated. The transition from the initial state to the quasi-equilibrium one is accompanied by strong variations of the wave characteristics, when the waves exhibit the most extreme features. In particular, the wave kurtosis grows suddenly and the abnormal heavy tails in the wave amplitude probability distributions appear. These processes are observed in all the cases of the bimodal spectra and are quite similar to the single-mode regime. The coexistence of a long-wave system smoothens the rapid oscillations of the wave extremes and kurtosis which take place during the transition stage. The presence of a short-wave system makes the waves on average more symmetric. Skewness attains the minimum value compared to the other cases. The co-existence of shorter waves practically does not change the wave kurtosis or the probability of the wave heights. In contrast, the presence of a long-wave system makes the waves more asymmetric and more extreme. The probability of large waves increases in the bimodal systems with a low-frequency component. The initial wave spectrum expands as a result of the wave interaction and tends to a quasistationary state. One may anticipate that the formulated conclusions are applicable beyond the limits of the Korteweg-de Vries equation to other kindred frameworks and corresponding phenomena. This work was supported by the Russian Science Foundation (project No. 18-77-00063).

Integration of the nonlinear modified Korteweg-de Vries equation with a loaded term

2020 ◽  
Vol 2020 (2) ◽  
pp. 85-98
Author(s):  
A.B. Khasanov ◽  
T.J. Allanazarova
Keyword(s):  
Vries Equation ◽  
De Vries ◽  
Loaded Term

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

scholarly journals A flux reconstruction method for the Korteweg-de Vries equation

2021 ◽  
Vol 1978 (1) ◽  
pp. 012031
Author(s):  
Ningbo Guo ◽  
Yaming Chen ◽  
Xiaogang Deng
Keyword(s):  
Vries Equation ◽  
Flux Reconstruction ◽  
Reconstruction Method ◽  
De Vries

scholarly journals Wave Breaking in Undular Bores with Shear Flows

Water Waves ◽  
2021 ◽  
Author(s):  
Maria Bjørnestad ◽  
Henrik Kalisch ◽  
Malek Abid ◽  
Christian Kharif ◽  
Mats Brun
Keyword(s):  
Turbulent Flows ◽  
Wave Breaking ◽  
Vries Equation ◽  
Kdv Equation ◽  
Flow Depth ◽  
Present Contribution ◽  
Undular Bore ◽  
Wave Flume ◽  
The Kdv Equation ◽  
The Waves

AbstractIt is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

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