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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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##### References:
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