On Schrödinger-Boussinesq equations. (English) Zbl 1103.35094

Linares, F.; Navas, A.

Adv. Differ. Equ. 9, No. 1-2, 159-176 (2004).

Summary: We study local and global well-posedness for the initial-value problem associated to the one-dimensional Schrödinger-Boussinesq equations \[ i\partial_tu+\partial^2_xu=uv+\alpha|u|^2u,\quad x\in \mathbb{R},\;t>0, \]
\[ \partial^2_tv-\partial^2_xv+\partial^4_xv= \partial^2_x \bigl(\beta|v|^{p-1}v+|u|^2\bigr), \] in low regularity spaces. To establish these results we use sharp \(L^p\)-\(L^q\) estimates.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
78A60	Lasers, masers, optical bistability, nonlinear optics
35Q35	PDEs in connection with fluid mechanics
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

interaction of solitons in optics; nonlinear Schrödinger equation; local and global well-posedness