Four-Wave Interaction of Positive and Negative Energy Waves in Plasmas

Turner, John G.

doi:10.1088/0031-8949/21/2/015

Cited by 7 publications

(13 citation statements)

References 15 publications

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“…(2.5) Furthermore, four-wave interactions governed by (1.1) are bounded since from (2.4) (conditions for 'explosive' four-wave interactions are detailed in Turner (1980), Verheest (1982) and Safdi & Segur (2007); none of these are fulfilled here)…”

Section: Bretherton Equationmentioning

confidence: 99%

“…This was achieved by Inoue (1975), Boyd & Turner (1978), Turner (1980), Chen & Snyder (1989) and Stiassnie & Shemer (2005) using elliptic functions. Some of the 'exceptional cases' mentioned by Bretherton correspond to solutions now called 'pump' and 'breathers'; some of these were given by Inoue (1975) and Turner (1980), but some others were missing.…”

Section: Introductionmentioning

confidence: 99%

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On elementary four-wave interactions in dispersive media

Leblanc

2024

J. Fluid Mech.

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The cubic interactions in a discrete system of four weakly nonlinear waves propagating in a conservative dispersive medium are studied. By reducing the problem to a single ordinary differential equation governing the motion of a classical particle in a quartic potential, the complete explicit branches of solutions are presented, either steady, periodic, breather or pump, thus recovering or generalizing some already published results in hydrodynamics, nonlinear optics and plasma physics, and presenting some new ones. Various stability criteria are also formulated for steady equilibria. Theory is applied to deep-water gravity waves for which models of isolated quartets are described, including bidirectional standing waves and quadri-directional travelling waves, steady or not, resonant or not.

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Section: Bretherton Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On elementary four-wave interactions in dispersive media

Leblanc

2024

J. Fluid Mech.

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“…18 Such a reduction process is very useful for obtaining a finite set of dynamical equations from an infinite-dimensional system, while retaining the Hamiltonian properties of the latter. Potential uses for this Hamiltonian truncation procedure include constructing low-dimensional models for describing specific physical mechanisms [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] and obtaining semi-discrete schemes for numerical integration of partial differential equations, as an alternative to techniques used or derived, for example, in references [43][44][45][46][47][48][49][50]. Second, as a collateral effect of beatification, all Casimir invariants ii of a system become linear in the dynamical variables.…”

Section: Introductionmentioning

confidence: 99%

Beatification: Flattening the Poisson bracket for two-dimensional fluid and plasma theories

2017

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A perturbative method called beatification is presented for a class of two-dimensional fluid and plasma theories. The Hamiltonian systems considered, namely the Euler, Vlasov-Poisson, Hasegawa-Mima, and modified Hasegawa-Mima equations, are naturally described in terms of noncanonical variables. The beatification procedure amounts to finding the correct transformation that removes the explicit variable dependence from a noncanonical Poisson bracket and replaces it with a fixed dependence on a chosen state in phase space. As such, beatification is a major step toward casting the Hamiltonian system in its canonical form, thus enabling or facilitating the use of analytical and numerical techniques that require or favor a representation in terms of canonical, or beatified, Hamiltonian variables.

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“…Similarly, four-wave models have been widely derived and studied (e.g. [11][12][13][14][15][16][17]). These reductions are obtained from a parent model, a nonlinear partial differential equation, by linearizing about an equilibrium state and analyzing the dispersion relation for the possibilities of three or four-wave resonances between linear eigenmodes.…”

Section: Introductionmentioning

confidence: 99%

A method for Hamiltonian truncation: a four-wave example

Viscondi¹,

Caldas²,

Morrison³

2016

J. Phys. A: Math. Theor.

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A method for extracting finite-dimensional Hamiltonian systems from a class of 2+1 Hamiltonian mean field theories is presented. These theories possess noncanonical Poisson brackets, which normally resist Hamiltonian truncation, but a process of beatification by coordinate transformation near a reference state is described in order to perturbatively overcome this difficulty. Two examples of four-wave truncation of Euler's equation for scalar vortex dynamics are given and compared: one a direct non-Hamiltonian truncation of the equations of motion, the other obtained by beatifying the Poisson bracket and then truncating. arXiv:1509.09247v2 [physics.flu-dyn]

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Four-Wave Interaction of Positive and Negative Energy Waves in Plasmas

Abstract: Fourwave interaction of positive and negative energy waves in plasmas.

Cited by 7 publications

References 15 publications

On elementary four-wave interactions in dispersive media

On elementary four-wave interactions in dispersive media

Beatification: Flattening the Poisson bracket for two-dimensional fluid and plasma theories

A method for Hamiltonian truncation: a four-wave example

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