On inverse scattering for the two-dimensional nonlinear Schrödinger equation

The inverse scattering problem for the two-dimensional nonlinear Schrödinger equation i u t + Δ u = N ( u ) is studied. We assume that the unknown nonlinearity N of the equation satisfies N ∈ C ∞ ( C ; C ) , ( ∂ k ∂ ‾ ℓ N ) ( z ) = O ( | z | max ⁡ { 3 − k − ℓ , 0 } ) ( z → 0 ) and ( ∂ k ∂ ‾ ℓ N ) ( z ) = O ( e c | z | 2 ) ( | z | → ∞ ) for any non-negative integers k and ℓ . Here, ∂ N and ∂ ‾ N are the Wirtinger derivatives of N , and c is a positive constant. We establish a reconstraction formula of ( ∂ k ∂ ‾ ℓ N ) ( 0 ) ( k + ℓ ≥ 3 ) by the knowledge of the scattering operator for the equation.

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