scholarly journals Whitham modulation theory for the Ostrovsky equation

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160709 ◽  
Author(s):  
A. J. Whitfield ◽  
E. R. Johnson
Keyword(s):  
Analytical Solution ◽  
Solitary Wave ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Anomalous Dispersion ◽  
Normal Dispersion ◽  
Modulation Equations ◽  
Outer Solution ◽  
Physical Systems ◽  
Ostrovsky Equation

This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localized cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg–de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localized, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.

scholarly journals On sound waves of finite amplitude

1953 ◽  
Vol 216 (1127) ◽  
pp. 539-547 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Arbitrary Function ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Sound Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Integral ◽  
Gas Cloud

An analytical solution of Riemann’s equations for the one-dimensional propagation of sound waves of finite amplitude in a gas obeying the adiabatic law p = k ρ γ is obtained for any value of the parameter γ. The solution is in the form of a complex integral involving an arbitrary function which is found from the initial conditions by solving a generalization of Abel’s integral equation. The results are applied to the problem of the expansion of a gas cloud into a vacuum.

scholarly journals Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150416 ◽  
Author(s):  
Roger Grimshaw ◽  
Yury Stepanyants ◽  
Azwani Alias
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Anomalous Dispersion ◽  
Essential Difference ◽  
De Vries ◽  
Long Time ◽  
Ostrovsky Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg–de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg–de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg–de Vries solitary wave.

scholarly journals Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1608
Author(s):  
Igor Andrianov ◽  
Aleksandr Zemlyanukhin ◽  
Andrey Bochkarev ◽  
Vladimir Erofeev
Keyword(s):  
Numerical Simulation ◽  
Solitary Wave ◽  
Periodic Solutions ◽  
Initial Conditions ◽  
Periodic Waves ◽  
Two Stage ◽  
Second Stage ◽  
Ostrovsky Equation ◽  
Discrete Analogs

In this paper, stationary solitary and periodic waves of a nonlinear nonintegrable lattice are numerically constructed using a two-stage approach. First, as a result of continualization, a nonintegrable generalized Boussinesq—Ostrovsky equation is obtained, for which the solitary-wave and periodic solutions are numerically found by the Petviashvili method. In the second stage, discrete analogs of the obtained solutions are used as initial conditions in the numerical simulation of the original lattice. It is shown that the initial perturbations constructed in this way propagate along the lattice without changing their shape.

scholarly journals Transport fluctuations in integrable models out of equilibrium

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Jason Myers ◽  
Joe Bhaseen ◽  
Rosemary J. Harris ◽  
Benjamin Doyon
Keyword(s):  
Initial Conditions ◽  
Ballistic Transport ◽  
Integrable Models ◽  
Exact Results ◽  
Heisenberg Chain ◽  
Physical Systems ◽  
Fluctuation Relations ◽  
Full Counting Statistics ◽  
Counting Statistics ◽  
Non Equilibrium

We propose exact results for the full counting statistics, or the scaled cumulant generating function, pertaining to the transfer of arbitrary conserved quantities across an interface in homogeneous integrable models out of equilibrium. We do this by combining insights from generalised hydrodynamics with a theory of large deviations in ballistic transport. The results are applicable to a wide variety of physical systems, including the Lieb-Liniger gas and the Heisenberg chain. We confirm the predictions in non-equilibrium steady states obtained by the partitioning protocol, by comparing with Monte Carlo simulations of this protocol in the classical hard rod gas. We verify numerically that the exact results obey the correct non-equilibrium fluctuation relations with the appropriate initial conditions.

scholarly journals A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Alvaro H. Salas S ◽  
Jairo E. Castillo H ◽  
Darin J. Mosquera P
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytical Solution ◽  
Initial Conditions ◽  
Negative Ions ◽  
Nonlinear Problems ◽  
New Approach ◽  
Weierstrass Elliptic Function ◽  
Electronegative Plasma ◽  
The Given

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.

scholarly journals Analytical Solution of Heat Conduction in a Symmetrical Cylinder Using the Solution Structure Theorem and Superposition Technique

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1522 ◽  
Author(s):  
Rasool Kalbasi ◽  
Seyed Mohammadhadi Alaeddin ◽  
Mohammad Akbari ◽  
Masoud Afrand
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Solution Structure ◽  
Structure Theorem ◽  
Initial Conditions ◽  
Hyperbolic Heat Conduction ◽  
Fourier Heat Conduction ◽  
Superposition Technique ◽  
Complex Origin

In this paper, non-Fourier heat conduction in a cylinder with non-homogeneous boundary conditions is analytically studied. A superposition approach combining with the solution structure theorems is used to get a solution for equation of hyperbolic heat conduction. In this solution, a complex origin problem is divided into, different, easier subproblems which can actually be integrated to take the solution of the first problem. The first problem is split into three sub-problems by setting the term of heat generation, the initial conditions, and the boundary condition with specified value in each sub-problem. This method provides a precise and convenient solution to the equation of non-Fourier heat conduction. The results show that at low times (t = 0.1) up to about r = 0.4, the contribution of T1 and T3 dominate compared to T2 contributing little to the overall temperature. But at r > 0.4, all three temperature components will have the same role and less impact on the overall temperature (T).

scholarly journals The Propagation of Nonlinear Internal Waves under the Influence of Variable Topography and Earth’s Rotation in a Two-Layer Fluid

Fluids ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 140
Author(s):  
Nik Nur Amiza Nik Ismail ◽  
Azwani Alias ◽  
Fatimah N. Harun
Keyword(s):  
Solitary Wave ◽  
Wave Packet ◽  
Internal Waves ◽  
Wave Solution ◽  
Variable Coefficient ◽  
Solitary Wave Solution ◽  
Additional Term ◽  
Earth’S Rotation ◽  
Ostrovsky Equation ◽  
Variable Topography

A nonlinear equation of the Korteweg–de Vries equation usually describes internal solitary waves in the coastal ocean that lead to an exact solitary wave solution. However, in any real application, there exists the Earth’s rotation. Thus, an additional term is required, and consequently, the Ostrovsky equation is developed. This additional term is believed to destroy the solitary wave solution and form a nonlinear envelope wave packet instead. In addition, an internal solitary wave is commonly disseminated over the variable topography in the ocean. Because of these effects, the Ostrovsky equation is retrieved by a variable-coefficient Ostrovsky equation. In this study, the combined effects of both background rotation and variable topography on a solitary wave in a two-layer fluid is studied since internal waves typically happen here. A numerical simulation for the variable-coefficient Ostrovsky equation with a variable topography is presented. Two basic examples of the depth profile are considered in detail and sustained by numerical results. The first one is the constant-slope bottom, and the second one is the specific bottom profile following the previous studies. These indicate that the combination of variable topography and rotation induces a secondary trailing wave packet.

Wave Motion of a Nonlinear Elastic Bar Subjected to Axial Excitation

2004 ◽  
Author(s):  
Liming Dai ◽  
Qiang Han
Keyword(s):  
Wave Propagation ◽  
Solitary Wave ◽  
Elastic Wave ◽  
Nonlinear Wave ◽  
Wave Motion ◽  
Initial Conditions ◽  
System Parameters ◽  
Elastic Bar ◽  
Nonlinear Elastic ◽  
Nonlinear Elastic Wave

This research intends to investigate the wave motion in a nonlinear elastic bar with large deflection subjected to an axial external exertion. A nonlinear elastic constitutive relation governs the material of the bar. General form of the nonlinear wave equations governing the wave motion in the bar is derived. With a modified complete approximate method, the asymptotic solution of solitary wave is developed for theoretical and numerical analyses of the wave motion. Various initial conditions and system parameters are considered for investigating the shape and propagation of the nonlinear elastic wave. With the governing equation of the wave motion of the bar and the solution developed, the characteristics of the nonlinear elastic wave of the bar are analyzed theoretically and numerically. Properties of the wave propagation and the effects of the system parameters of the bar and the influences of the initial conditions to the characteristics of the wave motion are investigated in details. Based on the theoretical analysis as well as the numerical simulations, it is found that the nonlinearity of the elastic bar may cause solitary wave in the bar. The velocity of the solitary wave propagating in the bar is related to the initial condition of the wave motion. This exhibits an obvious different characteristic between the nonlinear wave and that of the linear wave of an elastic bar. It is also found in the research that the solitary wave is a pulse wave with stable propagation. If the stability of the wave propagation is destroyed, the solitary wave will no longer exist. The results of the present research may provide guidelines for the wave motion analysis of nonlinear elastic solid elements.

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