Inverse problem for a nonlinear Helmholtz equation

Abstract

This paper is devoted to the uniqueness of the coefficients θ,φ∈L∞(ℝ3), and ψ∈L∞(ℝ3,ℝ3) for the nonlinear Helmholtz equations −Δv(x)−k2v(x)=θ(x)v(x)F(|v(x)|) and −Δv(x)−k2v(x)=(φ(x)v(x)+iψ(x).∇v(x))|∇v(x)|r|v(x)|s. For small values of λ, a solution v is uniquely constructed by adding a small outgoing perturbation to the plane wave x→λeikx.d, where |d|=1 and λ⩾0. We can write v=v(x,λ,d)=λeikx.d+us∞(x/|x|,d,λ)eik|x|/|x|+O(1/|x|2) for large |x|. For a fixed k>0, we would like to prove that θ, φ and divψ can be uniquely reconstructed from the behaviour of us∞(x/|x|,d,λ) as λ→0. We prove the uniqueness in this paper.