Zakharov system

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

$i \partial_t^{} u + \Delta u = un$
$\Box n = -\Delta (|u|^2_{})$

thus u evolves according to a coupled Schrodinger equation, while n evolves according to a coupled wave equation. We usually place the initial data $u(0) \in H^{s_0}$, the initial position $n(0) \in H^{s_1}$, and the initial velocity $\partial_t n(0) \in H^{s_1 -1}$ 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 to 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 $L^2_x$ norm of u

$\int |u|^2 dx$

and the energy

$\int |\nabla u|^2 + \frac{|n|^2}{2} + \frac{|D^{-1}_x \partial_t n|^2}{2} + n |u|^2 dx.$

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).