The Zakharov system consists of a complex field u and a real field n which evolve according to the equations
thus evolves according to a coupled Schrodinger equation, while evolves according to a coupled wave equation. We usually place the initial data , the initial position , and the initial velocity for some real .
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 . 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
and the energy
The non-quadratic term 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 .