Hybrid Chebyshev function bases for sparse spectral methods in parity-mixed PDEs on an infinite domain

We present a numerical spectral method to solve systems of differential equations on an infinite interval $y\in (-\infty, \infty)$ in presence of linear differential operators of the form $Q(y) \left(\partial/\partial_y\right)^b$ (where $Q(y)$ is a rational fraction and $b$ a positive integer). Even when these operators are not parity-preserving, we demonstrate how a mixed expansion in interleaved Chebyshev rational functions $TB_n(y)$ and $SB_n(y)$ preserves the sparsity of their discretization. This paves the way for fast $O(N\ln N)$ and spectrally accurate mixed implicit-explicit time-marching of sets of linear and nonlinear equations in unbounded geometries.