Rigorous Validation of Cusp Bifurcations of Stationary Periodic Patterns in Partial Differential Equations

Abstract

In this paper, we present a computer-assisted framework for the rigorous validation of cusp bifurcations of spatially symmetric stationary periodic patterns in parabolic semilinear partial differential equations. Our approach extends to an infinite-dimensional setting the cusp map formulation previously developed in finite dimensions. We formulate the cusp conditions as a zero-finding problem on a Hilbert space of Fourier coefficients, whose non-degenerate solutions correspond to cusp bifurcation points. A key technical ingredient is a careful treatment of the adjoint multiplication operator in the resulting sequence space, which is more involved than in the finite-dimensional case. Starting from a numerically computed approximation, we develop a constructive Newton-Kantorovich argument to prove the existence and local uniqueness of a nearby zero of the cusp map. The non-degeneracy of this solution directly yields the non-vanishing of the cubic normal form coefficient $c$. To complete the verification, we rigorously enclose the spectrum of the linearized operator, confirming that exactly one eigenvalue has zero real part via Gershgorin-type estimates adapted to the symmetry structure of the problem. As a further consequence, the number $k$ of eigenvalues with strictly positive real part at the cusp and the sign of $c$ (both rigorously certified by the framework) together determine the stability structure of the three coexisting solutions inside the cusp region: bistability arises when $k=0$ and $c<0$, monostability when $k=0$ and $c>0$, and no stable solution exists when $k\ge 1$. We apply the method to the Swift--Hohenberg equation and the Gray--Scott system in both one and two spatial dimensions, obtaining rigorous proofs of cusp bifurcations in all four settings.