# Zakharov system

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

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

thus ${\displaystyle u}$ evolves according to a coupled Schrodinger equation, while ${\displaystyle n}$ evolves according to a coupled wave equation. We usually place the initial data ${\displaystyle u(0)\in H^{s_{0}}}$, the initial position ${\displaystyle n(0)\in H^{s_{1}}}$, and the initial velocity ${\displaystyle \partial _{t}n(0)\in H^{s_{1}-1}}$ for some real ${\displaystyle s_{0},s_{1}}$.

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 ${\displaystyle n}$. To recover this large loss of derivatives one needs to use the separation between the paraboloid ${\displaystyle t=x2\,}$ and the light cone ${\displaystyle |t|=|x|\,}$.

There are two conserved quantities: the ${\displaystyle L_{x}^{2}}$ norm of ${\displaystyle u}$

${\displaystyle \int |u|^{2}dx}$

and the energy

${\displaystyle \int |\nabla u|^{2}+{\frac {|n|^{2}}{2}}+{\frac {|D_{x}^{-1}\partial _{t}n|^{2}}{2}}+n|u|^{2}dx.}$

The non-quadratic term ${\displaystyle n|u|^{2}}$ 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 ${\displaystyle (s_{0},s_{1})=((d-3)/2,(d-2)/2)}$.