Zakharov system

The 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 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 $(s_{0},s_{1})=((d-3)/2,(d-2)/2)$.

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].