View source for Zakharov system

From DispersiveWiki

← Zakharov system

You do not have permission to edit this page, for the following reasons:

• The action you have requested is limited to users in the group: confirmed.
• You must confirm your email address before editing pages. Please set and validate your email address through your user preferences.

---

You can view and copy the source of this page.

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]]

Return to Zakharov system.