Admissible Discrete Linear Propagators for High-Order Time Splittings of Rotational Nonlinear Schr\"odinger Equations with Arbitrary Three-Dimensional Rotation

We study robust high-order time splittings for nonlinear Schrödinger equations whose linear part is defined by the Laplacian and an arbitrary three-dimensional rotation operator.

After Fourier pseudospectral discretization, a continuous exact factorization of the linear flow need not yield a method self-adjoint fixed-grid propagator.

For the original stage-wise explicit exact integrator, we identify a quadratic even term in the local logarithm and show that its visibility is state-dependent, so the observed temporal order of accuracy can depend on the initial data.

We then formulate fixed-grid admissibility for discrete linear propagators and construct two admissible propagators for arbitrary three-dimensional rotation: a symmetrized explicit exact integrator and a palindromic generalized shear propagator.

Both are unitary, first-order consistent, method self-adjoint, and have odd local logarithms.

Numerical experiments verify the predicted defect mechanism and demonstrate recovery of the designed second-, fourth-, and sixth-order behavior with the admissible propagators.