Zakharov system

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The Zakharov system

The Zakharov system consists of a complex field u and a real field n which evolve according to the equations

i u_t + D u = un Box n = -(|u|2)xx

thus u evolves according to a coupled Schrodinger equation, while n evolves according to a coupled wave equation. We usually place the initial data u(0) in H^{s0}, the initial position n(0) in H^{s1}, and the initial velocity nt(0) in H^{s1-1} for some real s0, s1.

This system is a model for the propagation of Langmuir turbulence waves in an unmagnetized ionized plasma [Zk1972]. Heuristically, u behaves like a solution cubic NLS, smoothed by 1/2 a derivative. If one sends the speed of light in Box to infinity, one formally recovers the cubic nonlinear Schrodinger equation. Local existence for smooth data – uniformly in the speed of light! - was established in [KnPoVe1995b] by energy and gauge transform methods; this was generalized to non-scalar situations in [Lau-p], [KeWg1998].

An obvious difficulty here is the presence of two derivatives in the non-linearity for n. To recover this large loss of derivatives one needs to use the separation between the paraboloid t = x2 and the light cone |t| = |x|.

There are two conserved quantities: the L2 norm of u

ò |u|2

and the energy

ò |ux|2 + |n|2/2 + |D-1x nt|2/2 + n |u|2.

The non-quadratic term n|u|2 in the energy becomes difficult to control in three and higher dimensions. Ignoring this part, one needs regularity in (1,0) to control the energy.

Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = ((d-3)/2, (d-2)/2). In dimensions d³4 LWP is known on Rd with an e of this value [GiTsVl1997]. For the lower dimensional cases, see below.