Fixed-Grid Reversibility of High-Order Splittings for Nonlinear Schrödinger Equations with Arbitrary-Axis 3D Rotation

Abstract

High-order splittings for rotational nonlinear Schrödinger equations may use an exact continuous factorization of the Laplace--rotation flow, but its fixed-grid Fourier realization need not retain the symmetry required by standard symmetric compositions. For the specified realization of the 3D arbitrary-axis explicit exact integrator (EEI), we prove that the complete map has a nonzero quadratic coefficient in its local matrix logarithm for every nonzero rotation on centered even grids. The defect persists under every real consistent composition of the same unrepaired nonlinear sandwich through a positive sum-of-squares factor. We construct two self-adjoint remedies: an adjoint-symmetrized EEI and a palindromic shear alternative. Fourier-tail estimates illustrate the representation mechanism, while complete-map diagnostics and a three-dimensional dipolar benchmark are consistent with the predicted order behavior. An implementation audit separates floating-point coefficient cancellation from the exact-arithmetic symmetry defect.

Replacement of the original manuscript "Admissible Discrete Linear Propagators for High-Order Time Splittings of Rotational Nonlinear Schrödinger Equations with Arbitrary Three-Dimensional Rotation"