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Wellposedness for Zakharov systems with generalized nonlinearity. (English) Zbl 0921.35162
The paper is devoted to the asymptotic behaviour of solutions of the Cauchy problem $i\partial u/\partial t+\Delta u=nf\bigl(| u|^2 \bigr)u,\;\square n=\Delta \biggl(F\bigl(| u|^2\bigr)\biggr)$ $(u,n, \partial n/\partial t)(0)= (\varphi,a,b)$ where $$F(s)=s^m$$, $$f(s)=ms^{m-1}$$ $$(m \geq 2$$, $$u:\mathbb{R}^d \times\mathbb{R} \to\mathbb{C},n: \mathbb{R}^d\times \mathbb{R}\to \mathbb{R})$$. Certain modification of the methods by J. Bourgain and J. Colliander [Int. Math. Res. Not. 1996, No. 11, 515-546 (1996; Zbl 0909.35125)] are introduced to establish the following result: The problem is locally well-posed in $$H^s \times L^2\times \widehat H^{-1}$$ with $$d=1,2$$ and $$s<1$$. If moreover $$(\varphi, a,b) \in\cap_{\sigma <s}H^\sigma \times H^{\sigma-1} \times\widehat H^{\sigma-2}$$ where $$s\geq 1$$, then the solution $$(u(t),n(t),n(t))$$ belongs to the same space for fixed $$t$$.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
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