We study the direct and inverse scattering problem for the semilinear Schrödinger equation \(\Delta u+a(x,u)+k^2u=0\) in \(\mathbb {R}^d\). We show well-posedness in the direct problem for small solutions based on the Banach fixed point theorem, and the solution has the certain asymptotic behavior at infinity. We also show the inverse problem that the semilinear function a(x, z) is uniquely determined from the scattering amplitude. The idea is the linearization that by using sources with several parameters we differentiate the nonlinear equation with respect to these parameter in order to get the linear one.

中文翻译：

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半线性薛定ding方程的正向和反向散射问题

我们研究了半线性薛定谔方程的直接和逆散射问题\（\德尔塔U + A（X，U）+ K ^ 2U = 0 \）在\（\ mathbb {R} ^ d \） 。在基于Banach不动点定理的小解的直接问题中，我们证明了适定性，并且该解在无穷大处具有一定的渐近性。我们还显示了反问题，即半线性函数a（x，  z）由散射幅度唯一确定。这个想法是线性的，即通过使用带有多个参数的源，我们可以针对这些参数对非线性方程进行微分，以得到线性方程。