Nonlinear Schrödinger equations with exceptional potentials
The author considers an initial boundary value problem, associated to a cubic nonlinear Schrödinger equation in which appears a real-valued given function V, as an exceptional potential. The nonlinear Schrödinger equation is related to the Ginzburg-Landau equation of superconductivity. The results from this paper are extensions of the results from paper [\textit{I. P. Naumkin}, J. Math. Phys. 57, No. 5, 051501, 31 p. (2016; Zbl 1343.81206)], of the same author. In previous paper the author consider a cubic nonlinear Schrödinger equation with a generic potential and proved the existence of unique continuous solutions which decay for long times as solutions to a linear free Schrödinger equation. Also, it is obtained a large time asymptotic representation for these solutions, which have a modified character and it is proved that as in the case of the free NLS equation, the cubic nonlinearity, in the case of generic potentials is critical from the point of view of large time asymptotic behavior. In present paper the author obtained also existence of unique continuous solutions, but by imposing the parity condition on the exceptional potential and on the initial data in order to control these functions The difference between the generic and exceptional cases becomes relevant when the small-energy behavior of the scattering coefficients and of the wave functions is considered. It is important to note that the author's considerations are based on a spectral theorem for the perturbed linear Schrödinger operator and a factorization technique, that allows the control on the resonant nonlinear term.
scientific article