Instabilities in Zakharov equations for laser propagation in a plasma. (English) Zbl 1133.35303

Colin, Thierry; Métivier, Guy

Bove, Antonio (ed.) et al., Phase space analysis of partial differential equations. Basel: Birkhäuser (ISBN 978-0-8176-4511-3/hbk; 978-0-8176-4521-2/e-book). Progress in Nonlinear Differential Equations and Their Applications 69, 63-81 (2006).

Summary: F. Linares, G. Ponce and J.-C. Saut [Bull. Braz. Math. Soc. (N.S.) 36, No. 1, 1–23 (2005; Zbl 1070.35087)] have proved that a non-fully dispersive Zakharov system arising in the study of Laser-plasma interaction, is locally well posed in the whole space, for fields vanishing at infinity. Here we show that in the periodic case, seen as a model for fields non-vanishing at infinity, the system develops strong instabilities of Hadamard’s type, implying that the Cauchy problem is strongly ill-posed.
For the entire collection see [Zbl 1105.35001].

MSC:

35A07	Local existence and uniqueness theorems (PDE) (MSC2000)
35Q53	KdV equations (Korteweg-de Vries equations)
78A60	Lasers, masers, optical bistability, nonlinear optics
82D10	Statistical mechanical studies of plasmas
35R25	Ill-posed problems for PDEs

Keywords:

non-fully dispersive Zakharov system; periodic case