Inverse scattering for the non‐linear Schrödinger equation: Reconstruction of the potential and the non‐linearity

Abstract

Abstract. We solve the inverse scattering problem for the nonlinear Schrödin-ger equation on R n , n ≥ 3:We prove that the small-amplitude limit of the scattering operator uniquely determines V j , j = 0, 1, · · · . Our proof gives a method for the reconstruction of the potentials V j , j = 0, 1, · · · . The results of this paper extend our previous results for the problem on the line.

Inverse $N$-body scattering with the time-dependent hartree-fock approximation

“…According to recent researches for inverse nonlinear scattering ( [31] and [32]), the method of the HVL also make it possible to recover nonlinearities. On the other hand, the small amplitude limit (SAL) of the scattering operator (Weder [35][36][37][38][39][40][41][42][43][44]) make it possible to recover both the potential and nonlinearities. This method of the SAL, however, fails to reconstruct the interaction potential in the HF equation (1).…”

Inverse scattering problems for the Hartree equation whose interaction potential decays rapidly

“…It was proved that if p and W satisfy suitable conditions, then the unknown W is uniquely reconstructed by 4) which is called the small amplitude limit, the notation "id " is the identity mapping. Later, Weder [14,15,16,17,18,19] proved that a more general class of nonlinearities is uniquely reconstructed, and moreover, a method is given for the unique reconstruction of the potential that acts as a linear operator. Unfortunately, the above methods to obtain the reconstruction formula are not applicable to the case of the Hartree term (V * |u| 2 )u (for details, see, e.g.…”

Inverse Scattering for the Nonlinear Schrödinger Equation with the Yukawa Potential

“…The above limit is called the small amplitude limit. Later, Weder [25,27,28,29,31] proved that a more general class of nonlinearities is uniquely reconstructed, and moreover, a method is given for the unique reconstruction of the potential that acts as a linear operator and that this problem was not considered in [19].…”

“…Using the method of [21,25,27,28,29], we see that S V 0 can be determined from the knowledge of S 1 . Theorem 1.2.…”