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

<center><math>i \partial_t^{} u +  \Delta u = un</math> </center>
<center><math>\Box n = -\Delta (|u|^2_{})</math></center>

thus <math>u</math> evolves according to a coupled Schrodinger equation, while <math>n</math> evolves according to a coupled wave equation.  We usually place the initial data <math>u(0) \in H^{s_0}</math>, the initial position <math>n(0) \in H^{s_1}</math>, and the initial velocity <math>\partial_t n(0) \in H^{s_1 -1}</math> for some real <math>s_0, s_1</math>. 

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 <math>n</math>.  To recover this large loss of derivatives one needs to use the separation between the paraboloid <math>t = x2\,</math> and the light cone <math>|t| = |x|\,</math>. 

There are two conserved quantities: the <math>L^2_x</math> norm of <math>u</math> 

<center><math>\int |u|^2 dx </math></center>

and the energy 

<center><math>\int |\nabla u|^2 + \frac{|n|^2}{2}  + \frac{|D^{-1}_x \partial_t n|^2}{2} + n |u|^2 dx.</math></center>

The non-quadratic term <math>n|u|^2</math> 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 <math>(s_0,s_1) = ((d-3)/2, (d-2)/2)</math>.  

== Specific dimensions ==

* [[Zakharov system on R]]
* [[Zakharov system on T]]
* [[Zakharov system on R^2]]
* [[Zakharov system on R^3]]
* In dimensions d>4 LWP is known on R^d within an epsilon of the critical regularity [[GiTsVl1997]].

[[Category:Equations]]
[[Category:Wave]]
[[Category:Schrodinger]]