scholarly journals Two-Wave Interactions for Weakly Nonlinear Hyperbolic Waves

1994 ◽  
Vol 91 (3) ◽  
pp. 275-277
Author(s):  
Yuanping He ◽  
T. Bryant Moodie
Keyword(s):  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Hyperbolic Waves

Two-Wave Interactions for Weakly Nonlinear Hyperbolic Waves

1993 ◽  
Vol 88 (3) ◽  
pp. 241-268 ◽  
Author(s):  
Yuanping He ◽  
T. Bryant Moodie
Keyword(s):  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Hyperbolic Waves

scholarly journals Linear and Weakly Nonlinear Energetics of Global Nonhydrostatic Normal Modes

2019 ◽  
Vol 76 (12) ◽  
pp. 3831-3846 ◽  
Author(s):  
Carlos F. M. Raupp ◽  
André S. W. Teruya ◽  
Pedro L. Silva Dias
Keyword(s):  
Normal Modes ◽  
Physical Space ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Vertical Acceleration ◽  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Resonant Wave ◽  
Zonal Wavenumber ◽  
Resonant Triads

Abstract Here the theory of global nonhydrostatic normal modes has been further developed with the analysis of both linear and weakly nonlinear energetics of inertia–acoustic (IA) and inertia–gravity (IG) modes. These energetics are analyzed in the context of a shallow global nonhydrostatic model governing finite-amplitude perturbations around a resting, hydrostatic, and isothermal background state. For the linear case, the energy as a function of the zonal wavenumber of the IA and IG modes is analyzed, and the nonhydrostatic effect of vertical acceleration on the IG waves is highlighted. For the nonlinear energetics analysis, the reduced equations of a single resonant wave triad interaction are obtained by using a pseudoenergy orthogonality relation. Integration of the triad equations for a resonance involving a short harmonic of an IG wave, a planetary-scale IA mode, and a short IA wave mode shows that an IG mode can allow two IA modes to exchange energy in specific resonant triads. These wave interactions can yield significant modulations in the dynamical fields associated with the physical-space solution with periods varying from a daily time scale to almost a month long.

Nonlinear and three-wave resonant interactions in magnetohydrodynamics

2000 ◽  
Vol 63 (5) ◽  
pp. 393-445 ◽  
Author(s):  
G. M. WEBB ◽  
A. R. ZAKHARIAN ◽  
M. BRIO ◽  
G. P. ZANK
Keyword(s):  
Wave Field ◽  
Wave Interaction ◽  
Resonant Interaction ◽  
Alfven Wave ◽  
Magnetoacoustic Wave ◽  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Resonant Interactions ◽  
Resonant Triads ◽  
High Frequency Waves

Hamiltonian and variational formulations of equations describing weakly nonlinear magnetohydrodynamic (MHD) wave interactions in one Cartesian space dimension are discussed. For wave propagation in uniform media, the wave interactions of interest consist of (a) three-wave resonant interactions in which high-frequency waves may evolve on long space and time scales if the wave phases satisfy the resonance conditions; (b) Burgers self-wave steepening for the magnetoacoustic waves, and (c) mean wave field effects, in which a particular wave interacts with the mean wave field of the other waves. The equations describe four types of resonant triads: slow–fast magnetoacoustic wave interaction, Alfvén–entropy wave interaction, Alfvén–magnetoacoustic wave interaction, and magnetoacoustic–entropy wave interaction. The formalism is restricted to coherent wave interactions. The equations are used to investigate the Alfvén-wave decay instability in which a large-amplitude forward propagating Alfvén wave decays owing to three-wave resonant interaction with a backward-propagating Alfvén wave and a forward-propagating slow magnetoacoustic wave. Exact solutions of the equations for Alfvén–entropy wave interactions are also discussed.

A REVIEW OF NUMERICAL ASYMPTOTIC AVERAGING FOR WEAKLY NONLINEAR HYPERBOLIC WAVES

2004 ◽  
Vol 9 (3) ◽  
pp. 209-222 ◽  
Author(s):  
A. Krylovas ◽  
R. Čiegis
Keyword(s):  
Elastic Waves ◽  
Water Waves ◽  
Gas Dynamics ◽  
Averaging Method ◽  
Hyperbolic Systems ◽  
Finite Difference Schemes ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Averaged Systems ◽  
Hyperbolic Waves

We present an overview of averaging method for solving weakly nonlinear hyperbolic systems. An asymptotic solution is constructed, which is uniformly valid in the “large” domain of variables t + |x| ∼ O(ϵ –1). Using this method we obtain the averaged system, which disintegrates into independent equations for the nonresonant systems. A scheme for theoretical justification of such algorithms is given and examples are presented. The averaged systems with periodic solutions are investigated for the following problems of mathematical physics: shallow water waves, gas dynamics and elastic waves. In the resonant case the averaged systems must be solved numerically. They are approximated by the finite difference schemes and the results of numerical experiments are presented.

scholarly journals Route to thermalization in the α-Fermi–Pasta–Ulam system

2015 ◽  
Vol 112 (14) ◽  
pp. 4208-4213 ◽  
Author(s):  
Miguel Onorato ◽  
Lara Vozella ◽  
Davide Proment ◽  
Yuri V. Lvov
Keyword(s):  
Wave Interaction ◽  
Discrete Systems ◽  
Interaction Theory ◽  
Random Waves ◽  
Wave Interactions ◽  
Weakly Nonlinear ◽  
Time Dynamics ◽  
Resonant Wave ◽  
Resonant Interactions ◽  
The One

We study the original α-Fermi–Pasta–Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave–wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ8, where ϵ is the small parameter in the system. The wave–wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

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