arXiv subjects

Dominic Blanco

Publications and source records attributed to Dominic Blanco.

6 recordsLinked to original sources

arXiv2026

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

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 $k \geq 1$. 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.

math.AP↗

arXiv2026

Proving the existence of localized patterns, periodic solutions, and branches of periodic solutions in the 1D Thomas model

In this paper, we present a general framework for constructively proving the existence of stationary localized solutions, spatially periodic solutions, and branches of spatially periodic solutions in the 1D Thomas model. Specifically, we develop the necessary analysis to compute explicit upper bounds required in a Newton--Kantorovich approach. Given an approximate solution $\bar{\mathbf{u}}$, this approach relies on establishing that a well-chosen fixed point map is contracting on a neighborhood $\bar{\mathbf{u}}$. For this matter, we construct an approximate inverse of the linearization around $\bar{\mathbf{u}}$, and establish sufficient conditions under which the contraction is achieved. This provides a framework for which computer-assisted analysis can be applied to verify the existence and local uniqueness of solutions in a vicinity of $\bar{\mathbf{u}}$, and control the linearization around $\bar{\mathbf{u}}$. Furthermore, as the Thomas model has a non-polynomial nonlinearity, we will need to use different techniques to handle it during our analysis. Our contributions are to provide a partial answer to how one can approach rigorously verifying results in the Thomas model, to adapt and combine previously developed techniques to apply to the Thomas model, and to perform the computer-assisted analysis to obtain such results. The code to perform the rigorous proofs is available on Github.

math.AP↗

arXiv2026

Proving periodic solutions and branches in the 2D Swift Hohenberg PDE with hexagonal and triangular symmetry

In this article, we enforce space group symmetries in Fourier series to rigorously prove the existence of smooth, periodic solutions in partial differential equations (PDEs) with hexagonal and triangular symmetries. In particular, we provide the necessary analytical and numerical tools to construct Fourier series of functions on the hexagonal lattice. This allows one to build approximate solutions that are periodic. Moreover, to generate the periodic tiling, we can use one symmetric hexagon for $D_6$ symmetry and two symmetric triangles for $D_3$ symmetry. We derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, $\overline{u}$. More specifically, we verify a condition based on the computation of explicit bounds. The strategy for constructing $\overline{u}$, the approximate inverse, and the computation of these bounds will be presented. We demonstrate our approach on the 2D Swift-Hohenberg PDE by proving the existence of $D_3$ and $D_6$ periodic solutions. We then perform proofs of branches of solutions by using Chebyshev series. The algorithmic details to perform the proof can be found on Github.

math.NA↗

arXiv2025

Proving the existence of localized patterns and saddle node bifurcations in 1D activator-inhibitor type models

In this paper, we present a general framework for constructively proving the existence and stability of stationary localized 1D solutions and saddle-node bifurcations in activator--inhibitor systems using computer-assisted proofs. Specifically, we develop the necessary analysis to compute explicit upper bounds required in a Newton--Kantorovich approach. Given an approximate solution $\bar{\mathbf{u}}$, this approach relies on establishing that a well-chosen fixed point map is contracting on a neighborhood $\bar{\mathbf{u}}$. For this matter, we construct an approximate inverse of the linearization around $\bar{\mathbf{u}}$, and establish sufficient conditions under which the contraction is achieved. This provides a framework for which computer-assisted analysis can be applied to verify the existence and local uniqueness of solutions in a vicinity of $\bar{\mathbf{u}}$, and control the linearization around $\bar{\mathbf{u}}$. Furthermore, we extend the method to rigorously establish saddle-node bifurcations of localized solutions for the same type of models, by considering a well--chosen zero--finding problem. This depends on the rigorous control of the spectrum of the linearization around the bifurcation point. Finally, we demonstrate the effectiveness of the framework by proving the existence and stability of multiple steady-state patterns in various activator--inhibitor systems, as well as a saddle--node bifurcation in the Glycolysis model.

math.AP↗

arXiv2025

Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE

In this article, we extend the framework developed in \cite{unbounded_domain_cadiot} to allow for rigorous proofs of existence of smooth, localized solutions in semi-linear partial differential equations possessing both space and non-space group symmetries. We demonstrate our approach on the Swift-Hohenberg model. In particular, for a given symmetry group $\mathcal{G}$, we construct a natural Hilbert space $H^l_{\mathcal{G}}$ containing only functions with $\mathcal{G}$-symmetry. In this space, products and differential operators are well-defined allowing for the study of autonomous semi-linear PDEs. Depending on the properties of $\mathcal{G}$, we derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, $u_0$. More specifically, combining a meticulous analysis and computer-assisted techniques, the Newton-Kantorovich approach is validated thanks to the computation of some explicit bounds. The strategy for constructing $u_0$, the approximate inverse, and the computation of these bounds will depend on the properties of $\mathcal{G}$ and its maximal square lattice space subgroup, $\mathcal{H}$. More specifically, we consider three cases: $\mathcal{G}$ is a space group which can be represented on the square lattice, $\mathcal{G}$ is not a space group which can be represented on the square lattice and the symmetry of $\mathcal{H}$ isolates the solution, and where $\mathcal{G}$ is not a space group which can be represented on the square lattice and the symmetry of $\mathcal{H}$ does not isolate the solution. We demonstrate the methodology on the 2D Swift-Hohenberg PDE by proving the existence of various dihedral localized patterns. The algorithmic details to perform the computer-assisted proofs can be found on Github.

math.AP↗

arXiv2024

The 2D Gray-Scott system of equations: constructive proofs of existence of localized stationary patterns

In this article, we present a comprehensive framework for constructing smooth, localized solutions in systems of semi-linear partial differential equations, with a particular emphasis to the Gray-Scott model. Specifically, we construct a natural Hilbert space $\mathcal{H}$ for the study of systems of autonomous semi-linear PDEs, on which products and differential operators are well-defined. Then, given an approximate solution $\mathbf{u}_0$, we derive a Newton-Kantorovich approach based on the construction of an approximate inverse of the linearization around $\mathbf{u}_0$. In particular, we derive a condition under which we prove the existence of a unique solution in a neighborhood of $\mathbf{u}_0$. Such a condition can be verified thanks to the explicit computation of different upper bounds, for which analytical details are presented. Furthermore, we provide an extra condition under which localized patterns are proven to be the limit of an unbounded branch of (spatially) periodic solutions as the period tends to infinity. We then demonstrate our approach by proving (constructively) the existence of four different localized patterns in the 2D Gray-Scott model. In addition, these solutions are proven to satisfy the $D_4$-symmetry. That is, the symmetry of the square. The algorithmic details to perform the computer-assisted proofs are available on GitHub.

math.AP↗