Inverse Problem for a Nonlinear Helmholtz Equation

This paper is devoted to the uniqueness of the coefficients θ, φ ∈ L∞(ℝ3), and ψ ∈ L∞ (ℝ3, ℝ3) for the nonlinear Helmholtz equations −△v(x) − k2v(x) = θ(x)v(x)F(|v(x) |) and −△v(x) − k2v(x) = (φ(x)v(x) + iψ(x).▽v(x)) ×|▽v(x) |r|v(x) |s. For small values of ⋋, a solution v is uniquely constructed by adding a small outgoing perturbation to the plane wave x → ⋋eikx.d, where |d| = 1 and ⋋ ≥ 0. We write the far-field expansion $$ v = v(x,\lambda, d) = \lambda {e^{{ikx.d}}} + u_{\infty }^s(\frac{x}{{\left| x \right|}},d,\lambda )\frac{{{e^{{ik\left| x \right|}}}}}{{\left| x \right|}} + O(\frac{1}{{{{\left| x \right|}^2}}}) $$ for large |x|, and we prove, for a fixed k, the uniqueness for the reconstruction of θ, φ and div ψ from the behaviour of $$ u_{\infty }^s(\frac{x}{{\left| x \right|}},d,\lambda ) $$ when ⋋ → 0.