The Zakharov system consists of a complex field u and a real field n which evolve according to the equations
thus u evolves according to a coupled Schrodinger equation, while n evolves according to a coupled wave equation. We usually place the initial data , the initial position , and the initial velocity 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 and the light cone .
There are two conserved quantities: the norm of u
and the energy
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).