# Difference between revisions of "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 $s_{0},s_{1}$ .

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 $L_{x}^{2}$ norm of $u$ $\int |u|^{2}dx$ and the energy

$\int |\nabla u|^{2}+{\frac {|n|^{2}}{2}}+{\frac {|D_{x}^{-1}\partial _{t}n|^{2}}{2}}+n|u|^{2}dx.$ 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 $(s_{0},s_{1})=((d-3)/2,(d-2)/2)$ .