Research / Partial differential equations

Dynamics of Partial Differential Equations

A main part of my research focuses on developing a functional analytic approach to study the dynamics of PDEs. Combining numerical analysis, topological methods, nonlinear analysis, scientific computing, continuation methods, functional analysis and fixed point theory, one of my goals is to develop a general computational theory to obtain computer-assisted proofs of existence of dynamical objects arising in infinite dimensional dynamical systems: steady states, periodic orbits, traveling waves, invariant manifolds, connecting orbits, bifurcations, etc.

Rigorous integration of parabolic PDEs

Following solutions of evolution equations in time with guaranteed accuracy is a central task of computer-assisted proofs. In [57] and [77], we solve initial value problems for semilinear parabolic PDEs using space-time expansions in Fourier and Chebyshev series, and in [71] we combine Chebyshev series with semigroup theory to treat higher-dimensional domains. These integrators give proofs of existence over long time intervals and, combined with trapping regions, global existence results: for instance, solutions of the Swift–Hohenberg equation converging to nontrivial equilibria, and a global solution of the 2D Navier–Stokes equations for a nontrivial initial condition.

Gap solitons in Bose–Einstein condensates

Gap solitons are localized solutions that play a central role in the physics of Bose–Einstein condensates. In [76], we turn numerical simulations of gap solitons in the one-dimensional Gross–Pitaevskii equation into rigorous proofs of existence. The key idea is to view each soliton as a homoclinic orbit of an associated dynamical system, and to derive verifiable conditions under which a numerically computed trajectory lies close to a true homoclinic orbit. This provides the first computer-assisted proofs of gap solitons in the Gross–Pitaevskii equation in non-perturbative parameter regimes.

PDEs on the disk via Zernike polynomials

Many physical problems are posed on geometries such as disks and cylinders, where standard Fourier methods no longer apply. In [74], we introduce a validated Matrix Multiplication Transform, a rigorous analogue of the discrete Fourier transform that allows nonlinear terms to be evaluated reliably in spectral methods. Combined with Zernike polynomials, a natural spectral basis on the disk, it gives an accurate and efficient way to multiply truncated series. We use this framework to prove the existence of solutions of two nonlinear elliptic PDEs on the disk.

Localized patterns on unbounded domains

Localized patterns on the whole space are challenging for computer-assisted proofs, since the problem is posed on an unbounded domain. In [69], we develop a fully spectral method, based on Fourier series, to prove the existence of solutions of semilinear PDEs on ℝ^m, and use it to prove the existence and stability of a soliton in the Kawahara equation. In [70], we prove the existence of stationary non-radial localized patterns in the planar Swift–Hohenberg equation, and show that each of them is the limit of an unbounded branch of spatially periodic solutions as the period tends to infinity.

Localized radial solutions of elliptic systems

Ground states of elliptic equations are fundamental spatial patterns, yet little is known about them for general systems of nonlinear elliptic equations. In [66], we present a general method to prove the existence of localized radially symmetric solutions of elliptic systems on ℝ^d, which reduce to non-autonomous ordinary differential equations in the radial variable. Combining dynamical systems theory with computer-assisted proofs, we establish localized radial solutions of the cubic Klein–Gordon equation in three dimensions, the Swift–Hohenberg equation in the plane and a three-component FitzHugh–Nagumo system.

Singularities and complex-time dynamics

Extending time to the complex plane reveals hidden singularities of evolution equations. In [61], we compute rigorous inclusions of solutions of nonlinear heat equations for complex time values, prove the existence of a branching singularity and establish global existence for an open set of solutions. In [59], we show that, for a class of nonlinear Schrödinger equations, an open set of solutions converges to zero in both time directions, which rules out any real-analytic conserved quantity. In [56] and [59], we prove heteroclinic orbits connecting nontrivial equilibria to zero, which in turn force the existence of unbounded solutions.

Rotation invariant and radially symmetric patterns

In [42], we introduce, through examples accessible to undergraduate students, a computer-assisted approach based on Taylor series to prove the existence of radially symmetric solutions of PDEs. In [60], we extend these ideas to prove the existence of rotation invariant patterns of a nonlinear Laplace–Beltrami equation on the sphere. The singularity at the pole is handled with a Taylor series expansion nearby, combined with a Chebyshev series expansion away from it, and the whole problem is solved at once with a Newton–Kantorovich argument.

Patterns in the phase-field-crystal model

The phase-field-crystal model describes the formation of crystalline structures at the atomic scale. In [58], we use rigorously validated numerics to study the model at this microscopic level. We prove the existence of critical points and local minimizers of the associated energy that correspond to classical candidate patterns, grain boundaries and localized patterns. We then explore how these patterns are related dynamically, both for fixed parameters and across parameter space, and formulate several conjectures about the connecting orbits between steady states.

Free vibrations of a MEMS membrane

Micro-electro-mechanical systems (MEMS) are tiny devices in which an elastic membrane deflects under an electrostatic force. In [46], we study a nonlinear wave equation modeling such a membrane, which has a unique branch of stable steady states up to a critical value of the voltage. We prove that infinitely many branches of periodic solutions, or free vibrations, bifurcate from this branch as the voltage varies. We also compute these branches numerically, and use the same functional setting to validate rigorously the steady state at the critical value.

A Kuramoto–Sivashinsky equation with absorption

Traveling wave patterns of a signed Kuramoto–Sivashinsky equation with absorption are described by a fourth-order piecewise linear ordinary differential equation. Galaktionov and Svirshchevskii conjectured that this equation has a unique, asymptotically stable periodic orbit. In [44], we give a partial proof of this conjecture. A computer-assisted proof first establishes the existence of the periodic orbit, and the rigorous bounds obtained in phase space are then combined with the theory of discontinuous dynamical systems to prove that the orbit is asymptotically stable.

The ill-posed Boussinesq equation

Ill-posed equations do not generate a dynamical system and have received little attention in computer-assisted proofs. In [37], we develop computer-assisted techniques to study periodic orbits of ill-posed PDEs. As a case study, we consider the Boussinesq equation, which plays an important role in the theory of shallow water waves. Using the symmetries of the solutions to isolate them, we prove the existence of several periodic orbits at different parameter values, in a space of space-time Fourier coefficients with geometric decay.

Traveling waves in the suspension bridge equation

The suspension bridge equation describes the deflection of the roadway of a bridge, and its traveling waves correspond to disturbances propagating along the deck. In [36], we prove the existence of symmetric homoclinic orbits of the equation u′′′′ + βu′′ + e^u − 1 = 0 for all parameter values β ∈ [0.5, 1.9]. This gives a large continuous family of solitary traveling waves and partially solves a conjecture of Chen and McKenna [Philos. Trans. Roy. Soc. London Ser. A 355 (1997)]. The proof combines a parameterization of the stable manifold with a rigorous continuation argument.

The Kuramoto–Sivashinsky equation

The Kuramoto–Sivashinsky equation is a popular model of weak turbulence and spatiotemporal chaos. In [32], we develop a general framework for computer-assisted proofs of invariant objects of semilinear PDEs, such as equilibria, traveling waves, periodic orbits and their invariant manifolds. In [31], we apply it to the Kuramoto–Sivashinsky equation: we compute a global continuous branch of periodic orbits, rigorously enclose their Floquet exponents, and prove that some of these periodic orbits are unstable.

Breakdown of analyticity

The regularity of a solution strongly affects how well it can be approximated numerically. In [30], we develop a functional analytic framework for computer-assisted Fourier analysis in sequence spaces of varying regularity, from finitely differentiable to analytic, which yields existence and regularity results at the same time. We use it to study steady states of a spatially inhomogeneous Fisher equation near the breakdown of analyticity, and show that spaces of algebraically decaying coefficients are well suited to this regime.

Coexistence of hexagons and rolls

Hexagons and rolls are the two most common patterns in two-dimensional pattern formation, and they can coexist in space. In [21], we prove the existence of standing fronts connecting a region of rolls to a region of hexagons in a pattern formation model, settling a conjecture of Doelman et al. [European J. Appl. Math. 14 (2003)]. These fronts are heteroclinic solutions of a system of two second-order differential equations, which we obtain by solving a boundary value problem with boundary conditions in the stable and unstable manifolds.

Steady states on higher-dimensional domains

Identifying the equilibria is the first step to understanding the dynamics of a parabolic PDE. In [1, 2], we introduce validated continuation, which verifies that each equilibrium computed numerically along a branch corresponds to a true, locally unique equilibrium of the PDE, at less than twice the cost of standard continuation. In [6, 7], we extend the method to two- and three-dimensional domains and prove the existence of global smooth branches of equilibria of the Cahn–Hilliard equation. In [11], we show that one-dimensional estimates make such proofs much cheaper than refining the numerical computation.

Pulses in the Gray–Scott model

The Gray–Scott model describes an autocatalytic chemical reaction and is known for its rich variety of pulses and patterns. In [9], we prove the existence of a family of even homoclinic orbits of the Gray–Scott system, which correspond to symmetric pulses. To do so, we introduce a rigorous numerical technique for symmetric connecting orbits of ordinary differential equations, which solves a projected boundary value problem using high-order parameterizations of the invariant manifolds and explicit bounds on their coefficients.

Related publications

1. [77]
   Recent advances about the rigorous integration of parabolic PDEs via fully spectral Fourier-Chebyshev expansions

   M. Cadiot and J.-P. Lessard

   Numerische Mathematik, 158(4): 1237–1289, 2026

   arXiv:2502.20644 doi:10.1007/s00211-026-01546-2 Codes on GitHub
2. [76]
   Computer-Assisted Proofs of Solitons in Bose-Einstein Condensates

   M. Ayala, C. García-Azpeitia and J.-P. Lessard

   Journal of Nonlinear Science, 36(2), Paper No. 28, 2026

   arXiv:2503.04701 doi:10.1007/s00332-026-10242-2 Codes on GitHub
3. [75]
   Periodic localized traveling waves in the two-dimensional suspension bridge equation

   L. van der Aalst, J.B. van den Berg and J.-P. Lessard

   Nonlinearity, 38(7), Paper No. 075029, 2025

   arXiv:2405.19759 doi:10.1088/1361-6544/ade5e5 Codes on GitHub
4. [74]
   Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs

   M. Cadiot, J. Jaquette, J.-P. Lessard and A. Takayasu

   Communications in Nonlinear Science and Numerical Simulation, Volume 151, Paper No. 109063, 2025

   arXiv:2411.18361 doi:10.1016/j.cnsns.2025.109063 Codes on GitHub
5. [71]
   A rigorous integrator and global existence for higher-dimensional semilinear parabolic PDEs via semigroup theory

   G. Duchesne, J.-P. Lessard and A. Takayasu

   Journal of Scientific Computing, 102(3), Paper No. 62, 66 pages, 2025

   arXiv:2402.00406 doi:10.1007/s10915-024-02785-x Codes on GitHub
6. [70]
   Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: constructive proofs of existence

   M. Cadiot, J.-P. Lessard and J.-C. Nave

   Journal of Differential Equations, Volume 414, 555–608, 2025

   arXiv:2403.10450 doi:10.1016/j.jde.2024.09.015 Codes on GitHub
7. [69]
   Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods

   M. Cadiot, J.-P. Lessard and J.-C. Nave

   SIAM Journal on Applied Dynamical Systems, 23(3): 1966–2017, 2024

   arXiv:2302.12877 doi:10.1137/23m1607507 Codes on GitHub
8. [66]
   Constructive proofs for localized radial solutions of semilinear elliptic systems on ^d

   J.B. van den Berg, O. Hénot and J.-P. Lessard

   Nonlinearity, 36(12): 6476–6512, 2023

   arXiv:2211.16445 doi:10.1088/1361-6544/ad0278
9. [61]
   Rigorous numerics for nonlinear heat equations in the complex plane of time

   A. Takayasu, J.-P. Lessard, J. Jaquette and H. Okamoto

   Numerische Mathematik, 151(3), 693–750, 2022

   arXiv:1910.12472 doi:10.1007/s00211-022-01291-2 Code
10. [60]
    Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: a Taylor-Chebyshev series approach

    J.B. van den Berg, G. Duchesne and J.-P. Lessard

    Journal of Computational Dynamics, 9(2): 253–278, 2022

    PDF doi:10.3934/jcd.2022005 Code
11. [59]
    Global dynamics in nonconservative nonlinear Schrödinger equations

    J. Jaquette, J.-P. Lessard and A. Takayasu

    Advances in Mathematics, Volume 398, Paper 108234, 2022

    arXiv:2012.09734 doi:10.1016/j.aim.2022.108234 Codes on GitHub
12. [58]
    Microscopic patterns in the 2D phase-field-crystal model

    G. Martine La Boissonière, R. Choksi and J.-P. Lessard

    Nonlinearity, 35(3), 1500–1520, 2022

    arXiv:2102.02338 doi:10.1088/1361-6544/ac4d91
13. [57]
    Validated forward integration scheme for parabolic PDEs via Chebyshev series

    J. Cyranka and J.-P. Lessard

    Communications in Nonlinear Science and Numerical Simulation, Volume 109, Paper No. 106304, 2022

    arXiv:2101.00684 doi:10.1016/j.cnsns.2022.106304
14. [56]
    Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

    J. Jaquette, J.-P. Lessard and A. Takayasu

    Communications in Nonlinear Science and Numerical Simulation, Volume 107, Paper No. 106188, 2022

    arXiv:2109.00159 doi:10.1016/j.cnsns.2021.106188 Codes on GitHub
15. [50]
    Spontaneous periodic orbits in the Navier-Stokes flow

    J.B. van den Berg, M. Breden, J.-P. Lessard and L. van Veen

    Journal of Nonlinear Science, 31(2): paper No. 41 (61 pages), 2021

    arXiv:1902.00384 doi:10.1007/s00332-021-09695-4
16. [46]
    Free vibrations in a wave equation modeling MEMS

    C. García-Azpeitia and J.-P. Lessard

    SIAM Journal on Applied Dynamical Systems, 19 (4): 2749–2782, 2020

    arXiv:2003.01365 doi:10.1137/20m1332530
17. [44]
    Traveling wave oscillatory patterns in a signed Kuramoto-Sivashinsky equation with absorption

    Y. Alama and J.-P. Lessard

    Journal of Computational and Applied Mathematics, 372 (112608), 2020

    PDF arXiv:1812.01166 doi:10.1016/j.cam.2019.112608 Code
18. [42]
    Computer-assisted proofs for radially symmetric solutions of PDEs

    I. Balazs, J.B. van den Berg, J. Courtois, J. Dudas, J.-P. Lessard, A. Vörös- Kiss, JF Williams and X.Y. Yin

    Journal of Computational Dynamics, 5(1-2): 61–80, 2018

    PDF doi:10.3934/jcd.2018003
19. [37]
    Rigorous numerics for ill-posed PDEs: periodic orbits in the Boussinesq equation

    R. Castelli, M. Gameiro and J.-P. Lessard

    Archive for Rational Mechanics and Analysis, 228(1):129–157, 2018

    PDF arXiv:1509.08648 doi:10.1007/s00205-017-1186-0 Code
20. [36]
    Continuation of homoclinic orbits in the suspension bridge equation: a computer-assisted proof

    J.B. van den Berg, M. Breden, J.-P. Lessard and M. Murray

    Journal of Differential Equations, 264(5): 3086–3130, 2018

    PDF arXiv:1702.07412 doi:10.1016/j.jde.2017.11.011
21. [32]
    A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations

    R. de la Llave, J.-L. Figueras, M. Gameiro and J.-P. Lessard

    SIAM Journal on Applied Dynamical Systems 16(2): 1070–1088, 2017

    PDF arXiv:1605.01086 doi:10.1137/16m1073777
22. [31]
    A posteriori verification of invariant objects of evolution equations: periodic orbits in the Kuramoto-Sivashinsky PDE

    M. Gameiro and J.-P. Lessard

    SIAM Journal on Applied Dynamical Systems, 16(1): 687–728, 2017

    PDF doi:10.1137/16m1073789 Code
23. [30]
    Computer assisted Fourier analysis in sequence spaces of varying regularity

    J.-P. Lessard and J.D. Mireles James

    SIAM Journal on Mathematical Analysis, 49(1), 530–561, 2017

    PDF doi:10.1137/16m1056006 Code
24. [21]
    Stationary coexistence of hexagons and rolls via rigorous computations

    J.B. van den Berg, A. Deschênes, J.-P. Lessard and J.D. Mireles James

    SIAM Journal on Applied Dynamical Systems, 14 (2), 942–979, 2015

    PDF doi:10.1137/140984506 Code
25. [11]
    Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates

    M. Gameiro and J.-P. Lessard

    SIAM Journal on Numerical Analysis, 51(4), 2063–2087, 2013

    PDF doi:10.1137/110836651
26. [9]
    Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation

    J.B. van den Berg, J.-P. Lessard, J.D. Mireles James and K. Mischaikow

    SIAM Journal on Mathematical Analysis, 43 (4): 1557–1594, 2011

    PDF doi:10.1137/100812008
27. [7]
    Rigorous computation of smooth branches of equilibria for the three-dimensional Cahn-Hilliard equation

    M. Gameiro and J.-P. Lessard

    Numerische Mathematik, 117 (4): 753–778, 2011

    PDF doi:10.1007/s00211-010-0350-3
28. [6]
    Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs

    M. Gameiro and J.-P. Lessard

    Journal of Differential Equations, 249 (9): 2237–2268, 2010

    PDF doi:10.1016/j.jde.2010.07.002
29. [2]
    Validated continuation over large parameter ranges for equilibria of PDEs

    M. Gameiro, J.-P. Lessard and K. Mischaikow

    Mathematics and Computers in Simulation, 79 (4), 1368–1382, 2008

    PDF doi:10.1016/j.matcom.2008.03.014
30. [1]
    Validated continuation for equilibria of PDEs

    S. Day, J.-P. Lessard and K. Mischaikow

    SIAM Journal on Numerical Analysis, 45 (4), 1398–1424, 2007

    PDF doi:10.1137/050645968