Affiliations: Department of Applied Mathematics, Science University of Tokyo, Tokyo 162‐8601, Japan E‐mail: [email protected] | Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China | Instituto de Física y Matemáticas, Universidad Michoacana, AP 2‐82, CP 58040, Morelia, Michoacan, Mexico E‐mail: [email protected]
Note: [] Corresponding author.
Abstract: We study the global in time existence of small solutions to the generalized derivative nonlinear Schrödinger equations of the form \begin{equation} \cases {\mathrm{i}\curpartial_t u + (1/2)\Delta u = \mathcal{N}(u,\nabla u,\overline u,\nabla \overline u), &$(t,x) \in \mathbf{R}\times {\mathbf{R}}^n$,\cr \noalign{\vskip4pt} u(0,x) = u_0 (x),\quad x\in\mathbf{R}^n,\cr} \end{equation}
where the space dimension n \geqslant 3 , the initial data u_0 are sufficiently small, \bar u is the complex conjugate of u and the nonlinear term \mathcal{N} is a smooth complex valued function \mathbf{C}\times \mathbf{C}^n\times \mathbf{C}\times \mathbf{C}^n\rightarrow \mathbf{C}. We assume that {\mathcal N} is a quadratic function in the neighborhood of the origin and always includes at least one derivative, that is, |{\mathcal N}(u,w,\bar u,\bar w)| \leqslant C|w|(|u|+|w|), for small u and w in the case of space dimensions n = 3, 4. As a typical example we consider the case of the polynomial type nonlinearity of the form {\mathcal N}(u,w,\bar u,\bar w) = \sum_{\tiny{\matrix{2 \leqslant |\alpha| + |\beta| + |\gamma| \leqslant l\cr \noalign{\vskip2pt} m \leqslant |\beta| + |\gamma| \leqslant l}}} \normalsize\lambda_{\alpha\beta\gamma} u^{\alpha_1}\bar u^{\alpha_2} w^{\beta}\bar w^{\gamma}
with w = (w_j)_{1\leqslant j\leqslant n}, \lambda_{\alpha\beta\gamma} \in \mathbf{C} , l\geqslant 2, m \geqslant 1 for n=3,4, and m \geqslant 0 for n\geqslant 5 . We prove the global existence of solutions to the Cauchy problem (A) under the condition that the initial data u_0 \in \mathbf{H}^{[n/2]+5,0} \cap \mathbf{H}^{[n/2]+3,2}, where \mathbf{H}^{m,s} = \{ \phi \in \mathbf{L}^2;\ \|\phi\|_{m,s} = \|(1+x^2)^{s/2} (1-\Delta)^{m/2}\phi\|_{\mathbf{L}^2} <\infty\} is the weighted Sobolev space. We also show the existence of the usual scattering states. Our result for n=3,4 is an improvement of Hayashi and Hirata, Nonlinear Anal. 31 (1998), 671–685.
Keywords: Derivative nonlinear Schrödinger equations, global small solutions, general space dimensions
