Existence and spectral instability of steady rolls in the Nikolaevskiy equation: a computer-assisted study

The Nikolaevskiy equation is a model of short-wave pattern formation coupled to a neutral long-wave mode, and it is believed that all of its spatially periodic steady states are unstable. Existing results are formal, numerical, or restricted to a small neighbourhood of onset. We give computer-assisted proofs of three results. First, for every wavenumber q in [0.9, 1.1] and every control parameter r between the onset value (1-q^2)^2 and 4, the equation has a nontrivial, real-analytic, odd steady roll; these rolls form a single continuous two-parameter family that bifurcates from the trivial state at onset, with an explicit amplitude bound proportional to the square root of the distance to onset. Second, every roll of this family with r in [0.25, 4] is spectrally unstable on the real line: its linearisation has a Bloch eigenvalue with real part at least 6.7e-4, and at least 0.13 for r >= 0.575. Third, at nine sample points we enclose, at a small Bloch number, the eigenvalue numerically identified with the long-wave (Matthews-Cox) branch; its real part is positive at seven points and negative at two, while strongly unstable eigenvalues exist at all nine. The proofs combine a parametrised Newton-Kantorovich theorem, a desingularisation near onset, Taylor-model bounds on parameter boxes, and a Weyl-sequence argument for the Bloch spectrum; all estimates are evaluated in rigorous interval arithmetic.