Milti-layer homocentric sphere and spherical symmetry for rogue wave light bullets in three-dimensional high-order nonlinear Schrödinger equation

We discover multi-layer homocentric spherical structures of rogue-wave light bullets in the spherically reduced (3+1)-dimensional higher-order nonlinear Schrödinger equation (HNLSE). These solutions belong to the ℓ = 0 invariant subspace of the full 3D operator and satisfy the complete radial equation exactly. These optical spheres are spherically reduced rogue-wave light bullets: they are localized in all three spatial directions and transient in time, but, unlike conventional traveling light bullets, they are stationary in the sense that their isosurfaces oscillate radially without translating -a localization arising from the rogue-wave nature extended to three dimensions, revealing single-, two-, and four-layer concentric spherical shells. We present also a linear-stability analysis of the optical sphere solutions. For the spherically reduced background, we derive the radial eigenvalue problem and identify the parameter regimes in which the background is modulationally unstable. A full nonlinear stability analysis remains an open problem. We uses the PINN framework to numerically simulate the rogue wave solutions in the HNLSE, and verifies the solution by comparing it with the exact analytical solution to confirm the solution accuracy. The multi-layer homocentric sphere with preserved spherical symmetry serves as the definitive fingerprint of the 3D rogue-wave light bullet, with implications for nonlinear optics, Bose–Einstein condensates, and ocean wave dynamics.