Solitary Waves, Homoclinic Orbits, and Nonlinear Oscillations Within the Non-dissipative Lorenz Model, the Inviscid Pedlosky Model, and the KdV Equation

2021 ◽  
pp. 827-851 ◽  
Author(s):  
Bo-Wen Shen
Keyword(s):  
Solitary Waves ◽  
Nonlinear Oscillations ◽  
Kdv Equation ◽  
Homoclinic Orbits ◽  
Lorenz Model ◽  
The Kdv Equation

scholarly journals Homoclinic Orbits and Solitary Waves within the Nondissipative Lorenz Model and KdV Equation

2020 ◽  
Vol 30 (14) ◽  
pp. 2050257
Author(s):  
Bo-Wen Shen
Keyword(s):  
Growth Model ◽  
Solitary Waves ◽  
Homoclinic Orbit ◽  
Kdv Equation ◽  
Homoclinic Orbits ◽  
Mathematical Form ◽  
Error Growth ◽  
Lorenz Model ◽  
Solitary Solutions ◽  
Text Component

Recent studies using the classical Lorenz model and generalized Lorenz models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the [Formula: see text], [Formula: see text], and [Formula: see text] state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The [Formula: see text] and [Formula: see text] solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function ([Formula: see text]) and a hyperbolic secant squared function ([Formula: see text]), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the [Formula: see text] component and the KdV equation were obtained. Additionally, the ODE for the [Formula: see text] component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth model, and an improved error growth model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth model and the 3D-NLM, as well as other Lorenz models, to propose an improved error growth model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions.

Dust acoustic solitary waves in non-thermal plasmas consisting of negatively charged dust grains and isothermal electrons

2009 ◽  
Vol 75 (4) ◽  
pp. 455-474 ◽  
Author(s):  
ANIMESH DAS ◽  
ANUP BANDYOPADHYAY
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Acoustic Waves ◽  
Kdv Equation ◽  
Thermal Parameter ◽  
Charged Dust ◽  
Average Temperature ◽  
The Kdv Equation ◽  
Dust Acoustic Solitary Waves ◽  
Dust Acoustic

AbstractA Korteweg–de Vries (KdV) equation is derived here, that describes the nonlinear behaviour of long-wavelength weakly nonlinear dust acoustic waves propagating in an arbitrary direction in a plasma consisting of static negatively charged dust grains, non-thermal ions and isothermal electrons. It is found that the rarefactive or compressive nature of the dust acoustic solitary wave solution of the KdV equation does not depend on the dust temperature if σdc < 0 or σdc > σd*, where σdc is a function of β1, α and μ only, and σd*(<1) is the upper limit (upper bound) of σd. This β1 is the non-thermal parameter associated with the non-thermal velocity distribution of ions, α is the ratio of the average temperature of the non-thermal ions to that of the isothermal electrons, μ is the ratio of the unperturbed number density of isothermal electrons to that of the non-thermal ions, Zdσd is the ratio of the average temperature of the dust particles to that of the ions and Zd is the number of electrons residing on the dust grain surface. The KdV equation describes the rarefactive or the compressive dust acoustic solitary waves according to whether σdc < 0 or σdc > σd*. When 0 ≤ σdc ≤ σd*, the KdV equation describes the rarefactive or the compressive dust acoustic solitary waves according to whether σd > σdc or σd < σdc. If σd takes the value σdc with 0 ≤ σdc ≤ σd*, the coefficient of the nonlinear term of the KdV equation vanishes and, for this case, the nonlinear evolution equation of the dust acoustic waves is derived, which is a modified KdV (MKdV) equation. A theoretical investigation of the nature (rarefactive or compressive) of the dust acoustic solitary wave solutions of the evolution equations (KdV and MKdV) is presented with respect to the non-thermal parameter β1. For any given values of α and μ, it is found that the value of σdc completely defines the nature of the dust acoustic solitary waves except for a small portion of the entire range of the non-thermal parameter β1.

scholarly journals ON THE SHOALING OF SOLITARY WAVES IN THE KDV EQUATION

2014 ◽  
Vol 1 (34) ◽  
pp. 44 ◽  
Author(s):  
Zahra Khorsand ◽  
Henrik Kalisch
Keyword(s):  
Solitary Waves ◽  
Kdv Equation ◽  
The Kdv Equation

scholarly journals Two‐level solitary waves as generalized solutions of the KdV equation

1995 ◽  
Vol 7 (5) ◽  
pp. 1056-1062 ◽  
Author(s):  
B. Izrar ◽  
F. Lusseyran ◽  
V. Miroshnikov
Keyword(s):  
Solitary Waves ◽  
Kdv Equation ◽  
Generalized Solutions ◽  
The Kdv Equation

Interaction of Fully-Nonlinear Internal Solitary Waves of Opposite Polarity

2021 ◽  
Author(s):  
Kevin Lamb
Keyword(s):  
Wave Field ◽  
Solitary Waves ◽  
Kdv Equation ◽  
Nonlinear Coefficient ◽  
Opposite Polarity ◽  
Internal Solitary Waves ◽  
Linear Wave ◽  
Gardner Equation ◽  
Fully Nonlinear ◽  
The Kdv Equation

&lt;p&gt;Previous studies have suggested that fully nonlinear internal solitary waves (ISWs) are very soliton-like as the interaction of two ISWs results in only very small changes in amplitude of the interacting ISWs and in the production of a very small amplitude wave train. Previous studies have, however, considered ISWs with the polarity predicted by the sign of the quadratic nonlinear coefficient of the KdV equation. The Gardner equation, which is an extension of the KdV equation that includes a cubic nonlinear term, has ISWs of two polarities (i.e., waves of depression and elevation) when the cubic coefficient of the Gardner equation is positive. These waves are soliton solutions of the Gardner equations. &amp;#160;In this talk I will discuss the interaction of ISWs of opposite polarity in continuous asymmetric three layer stratifications. Regions in parameter space where ISWs of opposite polarity exist will be discussed and I will demonstrate via fully nonlinear numerical simulations that the interaction of ISWs of opposite polarity waves are far from soliton-like: their interaction can result in very large changes in wave amplitude and may produce a very complicated wave field with multiple large ISWs, a large linear wave field and breather-like waves.&lt;span&gt;&amp;#160;&lt;/span&gt;&lt;/p&gt;

scholarly journals Multiscaled Solitary Waves

2017 ◽  
Author(s):  
Oleg G. Derzho
Keyword(s):  
Solitary Waves ◽  
Multiple Scales ◽  
Kdv Equation ◽  
Core Area ◽  
Internal Solitary Waves ◽  
Density Profiles ◽  
The Core ◽  
The Kdv Equation

Abstract. It is analytically shown how competing nonlinearities yield new multiscaled (multi humped) structures for internal solitary waves in shallow fluids. These solitary waves only exist for large amplitudes beyond the limit of applicability of the KdV equation or its usual extensions. Multiscaling phenomenon exists or do not exist for almost identical density profiles. Trapped core inside the wave prevents appearance of such multiple scales within the core area. It is anticipated that multiscaling phenomena exist for solitary waves in various physical origins.

scholarly journals Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns

2012 ◽  
Vol 468 (2148) ◽  
pp. 3784-3803 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Solitary Waves ◽  
Space Dimension ◽  
Kdv Equation ◽  
Wave Action ◽  
Periodic Patterns ◽  
Periodic State ◽  
The Kdv Equation ◽  
Spatially Periodic ◽  
Natural Mechanism ◽  
Hamiltonian Pdes

A dark solitary wave, in one space dimension and time, is a wave that is bi-asymptotic to a periodic state, with a phase shift, and with localized modulation in between. The most well-known case of dark solitary waves is the exact solution of the defocusing nonlinear Schrödinger equation. In this paper, our interest is in developing a mechanism for the emergence of dark solitary waves in general, and not necessarily integrable, Hamiltonian PDEs. The focus is on the periodic state at infinity as the generator. It is shown that a natural mechanism for the emergence is a transition between one periodic state that is (spatially) elliptic and another one that is (spatially) hyperbolic. It is shown that the emergence is governed by a Korteweg–de Vries (KdV) equation for the perturbation wavenumber on a periodic background. A novelty in the result is that the three coefficients in the KdV equation are determined by the Krein signature of the elliptic periodic orbit, the curvature of the wave action flux and the slope of the wave action, with the last two evaluated at the critical periodic state.

scholarly journals A universal form for the emergence of the Korteweg–de Vries equation

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120707 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Dispersion Relation ◽  
Solitary Waves ◽  
Vries Equation ◽  
Kdv Equation ◽  
Universal Form ◽  
Conventional View ◽  
The Kdv Equation ◽  
De Vries ◽  
New Perspective ◽  
Background State

A new perspective on the emergence of the Korteweg–de Vries (KdV) equation is presented. The conventional view is that the KdV equation arises as a model when the dispersion relation of the linearization of some system of partial differential equations has the appropriate form, and the nonlinearity is quadratic. The assumptions of this paper imply that the usual spectral and nonlinearity assumptions for the derivation of the KdV equation are met. In addition to a new mechanism, the theory shows that the emergence of the KdV equation always takes a universal form, where the coefficients in the KdV equation are completely determined from the properties of the background state—even an apparently trivial background state. Moreover, the mechanism for the emergence of the KdV equation is simplified, reducing it to a single condition. Well-known examples, such as the KdV equation in shallow-water hydrodynamics and in the emergence of dark solitary waves, are predicted by the new theory for emergence of KdV.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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.

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