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Smoothing effects for some derivative nonlinear Schrödinger equations. (English) Zbl 0951.35122
Summary: We study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of derivative type: \[ iu_t+ u_{xx}= {\mathcal N}(u,\overline u, u_x,\overline u_x),\quad t\in\mathbb{R},\quad x\in\mathbb{R};\quad u(0,x)= u_0(x),\quad x\in\mathbb{R},\tag{1} \] where \({\mathcal N}(u,\overline u, u_x,\overline u_x)= K_1|u|^2 u+ K_2|u|^2 u_x+ K_3 u^2\overline u_x+ K_4|u_x|^2 u+ K_5\overline uu^2_x+ K_6|u_x|^2 u_x\), the functions \(K_j= K_j(|u|^2)\), \(K_j(z)\in\mathbb{C}^\infty([0, \infty))\). If the nonlinear terms \({\mathcal N}= {\overline uu^2_x\over 1+|u|^2}\) then equation (1) appears in the classical pseudospin magnet model. Our purpose in this paper is to consider the case when the nonlinearity \({\mathcal N}\) depends both on \(u_x\) and \(\overline u_x\). We prove that if the initial data \(u_0\in \mathbb{H}^{3, \infty}\) and the norms \(\|u_0\|_{3,l}\) are sufficiently small for any \(l\in\mathbb{N}\) (when \({\mathcal N}\) depends on \(\overline u_x\)), then for some time \(T>0\) there exists a unique solution \(u\in \mathbb{C}^\infty([- T,T]\setminus\{0\}; \mathbb{C}^\infty(\mathbb{R}))\) of the Cauchy problem (1). Here \(\mathbb{H}^{m,s}= \{\varphi\in \mathbb{L}^2;\|\varphi\|_{m, s}< \infty\}\), \(\|\varphi\|_{m, s}= \|(1+ x^2)^{s/2}(1- \partial^2_x)^{m/2} \varphi\|_{\mathbb{L}^2}\), \(\mathbb{H}^{m,\infty}= \bigcap_{s\geq 1}\mathbb{H}^{m, s}\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B65 Smoothness and regularity of solutions to PDEs
46N20 Applications of functional analysis to differential and integral equations
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