Higher-dimensional islands of stability

Establish the existence and characterize the mechanism of higher-dimensional islands of stability for resonance-induced nonlinear bound states of the d=3 focusing NLS/GP equation, including the ranges of optical power in which near-zero-energy scattering resonances stabilize the states.

Background

The paper identifies a multidimensional extension as a principal motivation for the one-dimensional analysis. Numerical experiments for the three-dimensional NLS/GP equation reportedly show that a scattering resonance near zero energy can create intervals of optical power, or mass, in which resonance-induced nonlinear bound states are stabilized.

The existence and mathematical characterization of this phenomenon are not established by the present one-dimensional theory. The problem concerns determining whether these numerically observed stability intervals persist rigorously in higher dimensions and understanding their spectral or variational origin.