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

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 s0,s1.

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^2_x norm of u

\int |u|^2 dx

and the energy

\int |\nabla u|^2 + \frac{|n|^2}{2}  + \frac{|D^{-1}_x \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 (s0,s1) = ((d − 3) / 2,(d − 2) / 2).

Specific dimensions

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