# Zakharov system

### From DispersiveWiki

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 *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 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 (*s*_{0},*s*_{1}) = ((*d* − 3) / 2,(*d* − 2) / 2).

## 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.