Dynamics of a modified Davey–Stewartson system in $\mathbb {R}^3$
Volume 145 / 2016
Colloquium Mathematicum 145 (2016), 69-87
MSC: Primary 35Q53; Secondary 47J35.
DOI: 10.4064/cm6608-10-2015
Published online: 14 April 2016
Abstract
We study the Cauchy problem in $\mathbb {R}^3$ for the modified Davey–Stewartson system $$ i\partial _t u + \varDelta u =\lambda _1|u|^{4}u+\lambda _2b_1uv_{x_1},\hskip 1em -\varDelta v=b_2(|u|^2)_{x_1}. $$ Under certain conditions on $\lambda _1$ and $\lambda _2$, we provide a complete picture of the local and global well-posedness, scattering and blow-up of the solutions in the energy space. Methods used in the paper are based upon the perturbation theory from [Tao et al., Comm. Partial Differential Equations 32 (2007), 1281–1343] and the convexity method from [Glassey, J. Math. Phys. 18 (1977), 1794–1797].
