Symmetries and Conservation Laws of Kadomtsev—Pogutse Equations

Gusyatnikova, V. N.; Samokhin, Alexey; Titov, V. S.; Vinogradov, A. M.; Yumaguzhin, V. A.

doi:10.1007/978-94-009-1948-8_2

Cited by 4 publications

(10 citation statements)

References 5 publications

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“…or ∇ 2 u + exp(−2u) = 0 slightly different from (1), several different representations are known of its general solutions [11,13], but not all admit an analogous expression for our equation (1). For instance, equation (6) For a full discussion about this (not marginal) point and a comparison between the possible expressions representing the various solutions of these Liouville equations, see [20].…”

Section: Plasma Kinetic Equilibria Described By Liouville Equationmentioning

confidence: 99%

“…Actually, it can be shown [11] that any solution to (1) can be obtained in this way; this means in particular that any solution can be transformed (locally) into the 1-dimensional solution (2) by means of a suitable holomorphic transformation z → ψ(z). The close relationship between the generating function γ(z) of any solution of (1) and symmetry properties, can be also emphasized by the following result [10].…”

Section: Plasma Kinetic Equilibria Described By Liouville Equationmentioning

confidence: 99%

“…It is interesting to observe that the system (8) can be seen as a generalization of the twodimensional limit (i.e., the case of solutions independent of the coordinate z orthogonal to the plane (x, y)), of the so-called Reduced Magnetohydrodynamic equations (Kadomtsev-Pogutse equations) [48], see also [11]. The symmetry properties of these equations, that are frequently used to describe plasmas embedded in a very strong magnetic field, have been studied in [26], see also [6, Vol.…”

Section: General Remarksmentioning

confidence: 99%

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Applications of Symmetry Methods to the Theory of Plasma Physics

Cicogna¹

2006

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Abstract. The theory of plasma physics offers a number of nontrivial examples of partial differential equations, which can be successfully treated with symmetry methods. We propose three different examples which may illustrate the reciprocal advantage of this "interaction" between plasma physics and symmetry techniques. The examples include, in particular, the complete symmetry analysis of system of two PDE's, with the determination of some conditional and partial symmetries, the construction of group-invariant solutions, and the symmetry classification of a nonlinear PDE.

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Section: Plasma Kinetic Equilibria Described By Liouville Equationmentioning

confidence: 99%

Section: Plasma Kinetic Equilibria Described By Liouville Equationmentioning

confidence: 99%

Section: General Remarksmentioning

confidence: 99%

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Applications of Symmetry Methods to the Theory of Plasma Physics

Cicogna¹

2006

SIGMA

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“…In Sec. 11, we consider the problem of constructing the pairs of solutions to the hyperbolic Liouville equation (33) and the wave equation…”

Section: Then We Havementioning

confidence: 99%

“…The Liouville equation appeares in the study of the Kadomtsev-Pogutse equations. The latter equation is a reduction of the general magneto-hydrodynamic system, where some detailes that are inessential in the stability problem for high-temperature plasma in TOKAMAK are omitted ( [33]). For any solution of the elliptic Liouville equation (37) there is a Lobachevsky plane model that is conformally equivalent to the Euclidean plane with the diagonal metric g ij = δ ij .…”

Section: The Toda Equationmentioning

confidence: 99%

Methods of geometry of differential equations in analysis of integrable models of field theory

Kiselev

2006

J Math Sci

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In this paper, we investigate the algebraic and geometric properties of the hyperbolic Toda equations u xy = exp(Ku) associated with nondegenerate symmetrizable matrices K. A hierarchy of analogs to the potential modified Korteweg-de Vriesx is constructed, and its relation with the hierarchy for the Korteweg-de Vries equation T t = T xxx +T T x is established. Group-theoretic structures for the dispersionless (2+1)dimensional Toda equation u xy = exp(−u zz ) are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems Ψ t = i Ψ xx + i f (|Ψ|) Ψ (multi-soliton complexes) are described.

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Generalized self-similarity of intermittent plasma turbulence in space and laboratory plasmas

2015

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Statistical characteristics of plasma fluctuations in the solar wind (SW), the Earth's magnetosphere and fusion devices are reviewed. The turbulence in all these media has a complicated multiscale structure and exhibits a generalized self-similarity in an extended scale range. The anomalous transport of mass and momentum is intermittent and is carried by sporadic plasma flux bursts with non-Gaussian statistics, long-range correlation and multifractality. Intermittent turbulent transport is characterized by superdiffusion with power law δx 2 ∝ τ α , α ≈ 1.2-1.8. The structure functions in all these plasma environments are well fitted by the log-Poisson model of turbulence. Intermittent plasma turbulence displays universal properties and consists of quasi-1-D singular dissipative structures.

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Symmetries and Conservation Laws of Kadomtsev—Pogutse Equations

Cited by 4 publications

References 5 publications

Applications of Symmetry Methods to the Theory of Plasma Physics

Applications of Symmetry Methods to the Theory of Plasma Physics

Methods of geometry of differential equations in analysis of integrable models of field theory

Generalized self-similarity of intermittent plasma turbulence in space and laboratory plasmas

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