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Inverse problem for a nonlinear Helmholtz equation. (English) Zbl 1062.35173
Summary: This paper is devoted to the uniqueness of the coefficients \(\theta, \varphi\in L^\infty(\mathbb{R}^3)\), and \(\psi\in L^\infty (\mathbb{R}^3,\mathbb{R}^3)\) for the nonlinear Helmholtz equation \[ -\Delta v(x)-k^2v(x)=\theta(x)v(x)F\biggl(\bigl | v(x)\bigr|\biggr) \] and \[ -\Delta v(x)-k^2v(x)=\bigl(\varphi(x)v(x)+ i\psi(x).\nabla v(x)\bigr)\bigl |\nabla v(x)\bigr|^r\bigl| v(x)\bigr |^s. \] For small values of \(\lambda\), a solution \(v\) is uniquely constructed by adding a small outgoing perturbation to the plane wave \(x\to\lambda e^{ikx. d}\), where \(|d|=1\) and \(\lambda\geq 0\). We can write \(v=v(x,\lambda,d)= \lambda e^{ikx.d}+u^s_infty (x/|x|,d,\lambda)e^{ik|x|}/|x|+ O(1/ |x|^2)\) for large \(|x|\). For a fixed \(k>0\), we would like to prove that \(\theta,\varphi\) and div \(\psi\) can be uniquely reconstructed from the behaviour of \(u^s_\infty(x/|x|,d, \lambda)\) as \(\lambda\to 0\). We prove the uniqueness in this paper.

MSC:
35R30 Inverse problems for PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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