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The Cauchy problem for Davey-Stewartson systems. (English) Zbl 1026.35096
From the text: The authors study the Cauchy problem of the following generalized Davey-Stewartson systems: $$\cases iu_t+\Delta u=a |u|^\alpha u+b_1uv_{x_1}\\ -\Delta v=b_2(|u|^2)_{x_1}\\ u(0,x) =u_0(x)\endcases$$ where $u(t,x)$ and $v(t,x)$ $(x=(x_1,\dots,x_n))$ are complex- and real-valued functions of $(t,x)\in\bbfR_+ \times\bbfR^n$, respectively, $\Delta$ is the Laplace operator on $\bbfR^n$, and $a,b_1$, and $b_2$ are real constants. They study the local and global existence of solutions in $H^s$ $(1\le s\le 2)$. The main tools used are time-space $L^p-L^{p'}$ estimates for solutions of linear Schrödinger equations in Lebesgue-Besov spaces; these estimates are usually named generalized Strichartz inequalities. The method of the proof of the main results is a contraction mapping argument.

35Q55NLS-like (nonlinear Schrödinger) equations
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