Reductions of Kinetic Equations to Finite Component Systems
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Reductions of Kinetic Equations to Finite Component Systems A.A. Chesnokov · M.V. Pavlov
Received: 15 January 2012 / Accepted: 15 March 2012 / Published online: 24 May 2012 © Springer Science+Business Media B.V. 2012
Abstract We consider two distinguish approaches for extraction of finite component systems from kinetic equations. The first method is based on the theory of generalized functions, which in simplest case is nothing but the so called multi flow hydrodynamics well known in plasma physics. An alternative is the so called the moment decomposition method successfully utilized for hydrodynamic chains. The method of hydrodynamic reductions successfully utilized in the theory of integrable hydrodynamic chains is applied to the local and nonlocal kinetic equations. N component reductions parameterized by N − 1 arbitrary constants for non-hydrodynamic chain arising in the theory of high frequency nonlinear waves in electron plasma are found. These evolution dispersive systems equipped by a local Hamiltonian structure possess periodic solutions. Keywords Plasma physics · Collisionless kinetic equations · Reductions · Hamiltonian structures · Periodic solutions Mathematics Subject Classification 35L40 · 35L65 · 37K10
1 Introduction This paper is dedicated to a well-known problem: extraction of finite component reductions of kinetic equations. In the so called integrable case, this problem was successfully solved recently (see, for instance, a set of papers in the historical order: [3, 11–17, 21–29, 35, 43, 44]).
A.A. Chesnokov Lavrent’ev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, Novosibirsk, 630090, Russia M.V. Pavlov () Department of Mathematical Physics, P.N. Lebedev Physical Institute of Russian Academy of Sciences, Moscow, Leninskij Prospekt, 53, Russia e-mail: [email protected]
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A.A. Chesnokov, M.V. Pavlov
Let us consider the remarkable Vlasov kinetic (collisionless Boltzmann) equation (see, for instance, [10, 43, 44]) λt + pλx − λp ux = 0,
(1)
where (λ is a distribution function, x is a space coordinate, t is a time variable, p is a momentum) ∞ λdp. u= −∞
Following the approach developed in plasma physics, one can introduce an infinite set of moments ∞ p k λdp. Ak = −∞
Then multiplying (1) on p k and integrating in p by parts, we obtain the Benney hydrodynamic chain (see [2]) + kAk−1 A0x = 0, Akt + Ak+1 x
k = 0, 1, 2, . . . ,
(2)
which was derived by D. Benney in 1973 for description of long waves of finite depth fluid. An alternative formal construction is based on an asymptotic expansion λ=p+
A0 A1 A2 + 2 + 3 + ··· p p p
(3)
Substitution of this series directly to (1) leads to (2) again. In this paper we do not discuss the concept of “integrability” of the Benney hydrodynamic chain, which was extensively and deeply investigated in other articles. However, we would like emphasize that just integrable hydrodynamic chains are connected with corresponding Vlasov type kinetic equations by two above so different approaches. The Benney hydrodynami
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