Inverse Scattering for the Nonlinear Schrödinger Equation with the Yukawa Potential

Abstract

We study the inverse scattering problem for the three dimensional nonlinear Schrödinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.2000 Mathematics Subject Classification. 35R30, 35P25, 35Q40.

Transverse instability of concentric soliton waves

“…Murphy (2019) and references therein, even though NLS with potentials could be suitable to inverse scattering transform analysis and represents an active area of research, cf. Sasaki (2008);Fajun & Li (2019). where p = S R is the momentum, while the factors λ∞ χ∞ and λ ∞ χ∞ may assume all possible sign combinations as per (2.21).…”

The scattering map determines the nonlinearity

“…There is a large body of literature concerning the recovery of the nonlinearity (as well as external potentials) from the scattering map in the setting of nonlinear dispersive equations (see e.g. [1,3,11,12,[14][15][16][17][19][20][21][22][23][24][25][26][27][28]). In general, these works either consider analytic nonlinearities or make other strong structural assumptions on the nonlinearity.…”

Recovery of a spatially-dependent coefficient from the NLS scattering map

“…Our results fit in the broader context of the recovery of the nonlinear terms from scattering data for nonlinear dispersive equations. For some further results of this type (primarily in the NLS setting), we refer the reader to [1,3,[11][12][13][14][19][20][21][22][23]. We also mention the related works [2,9], which considered the recovery of spatiallydependent coefficients in the nonlinearity using particular solutions rather than the scattering map.…”