arXiv subjects

Björn de Rijk

Publications and source records attributed to Björn de Rijk.

At least 19 recordsLinked to original sources

arXiv2026

Dissipative bright and dark pulses under periodic forcing

We analyze the existence and stability of bright and dark pulses in a damped nonlinear Schrödinger (NLS) equation with periodic forcing. This Lugiato-Lefever equation is a canonical model in nonlinear optics describing pattern formation in a dissipative Kerr cavity driven by a bichromatic laser pump. Bifurcating from the bright and black NLS solitons, we rigorously construct bright and dark single-pulse solutions to the Lugiato-Lefever equation. Using a recently developed toolbox for concatenating and periodically extending pulse solutions in spatially periodic systems, we then obtain periodic multipulse solutions corresponding to experimentally observed broad-bandwidth frequency combs. While the existence and stability of the bright pulses follow from standard Lyapunov-Schmidt reduction and Krein index arguments, the construction of the dark pulses is substantially more delicate because the nonzero asymptotic states of the black NLS soliton yield neutral essential spectrum of the linearization. By carefully tracking the associated small spatial Floquet exponents along the bifurcation and employing exponential dichotomies, we establish the existence of dark pulse solutions. These consist of two domain walls connected by a long plateau with critical absolute spectrum. To our knowledge, this constitutes the first rigorous bifurcation result in the Lugiato-Lefever equation from the black NLS soliton.

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arXiv2026

Nonlinear stability of periodic waves in the Korteweg-de Vries equation under localized perturbations

We investigate the stability and asymptotic behavior of spatially periodic cnoidal waves in the Korteweg-de Vries equation subject to localized perturbations. Standard stability arguments in Hamiltonian systems break down in this setting, since localized perturbations preclude a characterization of stable periodic waves as strict minimizers of a suitable energy functional subject to finitely many constraints. As a result, the nonlinear stability of periodic waves under localized perturbations has remained a long-standing open problem in Hamiltonian systems, with previous results only addressing plane waves that can be reduced to constant states by passing to polar coordinates. In this paper, we develop a novel method that resolves this obstruction by combining variational arguments, Floquet-Bloch theory, and Duhamel-based estimates with spatiotemporal modulation. Our framework applies to general periodic waves in Hamiltonian systems with symmetry and reduces the nonlinear stability problem to verifying diffusive spectral stability conditions for the second variation of a suitable conserved energy. Applying our approach to cnoidal waves in the Korteweg-de Vries equation, we obtain the first nonlinear stability result for periodic waves in Hamiltonian systems under localized perturbations that cannot be reduced to constant states.

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arXiv2026

Selection of pushed pattern-forming fronts in the FitzHugh-Nagumo system

We establish nonlinear stability of fronts that describe the creation of a periodic pattern through the invasion of an unstable state. Our results concern pushed fronts, that is, fronts whose propagation is driven by a localized mode at the front interface. We prove that these pushed pattern-forming fronts attract initial data supported on a half-line, and therefore determine both propagation speeds and selected wave numbers for invasion from localized initial conditions. This provides to our knowledge the first proof of the marginal stability conjecture for pattern-forming fronts, thereby confirming universal wave number selection laws widely used in the physics literature. We present our analysis in the specific setting of the FitzHugh-Nagumo system, but our methods can be applied to general dissipative PDE models which exhibit pattern formation. The main technical challenge is to control the interaction between the localized mode driving the propagation and outgoing diffusive modes in the wake of the front. Through a subtle far-field/core decomposition of the linearized evolution, we resolve this interaction and describe the nonlinear response of the front to perturbations as a dynamically driven phase mixing problem for the pattern in the wake. The methods we develop are generally useful in any setting involving the interaction of localized modes and outward diffusive transport, such as in the nonlinear stability of undercompressive viscous shock waves or source defects.

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arXiv2026

Stability of coherent pattern formation through invasion in the FitzHugh-Nagumo system

We establish sharp nonlinear stability results for fronts that describe the creation of a periodic pattern through the invasion of an unstable state. The fronts we consider are critical, in the sense that they are expected to mediate pattern selection from compactly supported or steep initial data. We focus on pulled fronts, that is, on fronts whose propagation speed is determined by the linearization about the unstable state in the leading edge, only. We present our analysis in the specific setting of the FitzHugh-Nagumo system, where pattern-forming uniformly translating fronts have recently been constructed rigorously, but our methods can be used to establish nonlinear stability of pulled pattern-forming fronts in general reaction-diffusion systems. This is the first stability result of critical pattern-selecting fronts and provides a rigorous foundation for heuristic, universal wave number selection laws in growth processes based on a marginal stability conjecture. The main technical challenge is to describe the interaction between two separate modes of marginal stability, one associated with the spreading process in the leading edge, and one associated with the pattern in the wake. We develop tools based on far-field/core decompositions to characterize, and eventually control, the interaction between these two different types of diffusive modes. Linear decay rates are insufficient to close a nonlinear stability argument and we therefore need a sharper description of the relaxation in the wake of the front using a phase modulation ansatz. We control regularity in the resulting quasilinear equation for the modulated perturbation using nonlinear damping estimates.

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arXiv2026

Orbital stability of plane waves in the Klein-Gordon equation against localized perturbations

We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its stability analysis outside the scope of the classical orbital stability theory for Hamiltonian systems developed by Grillakis, Shatah, and Strauss. Inspired by Zhidkov's work on the stability of time-periodic, spatially homogeneous states in the nonlinear Schrödinger equation, we develop an alternative method that relies on an amplitude-phase decomposition and leverages conserved quantities tailored to the perturbation equation. We establish an orbital stability result of plane waves that is locally uniform in space, accommodating $L^2$-localized perturbations as well as unbounded phase modulations. Our result is sharp in the sense that it holds up to the spectral stability boundary.

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arXiv2026

Stability and dynamics of planar fronts in reaction-diffusion systems under nonlocalized perturbations

We analyze the stability and dynamics of bistable planar fronts in multicomponent reaction-diffusion systems on $\mathbb{R}^{d}$. Under standard spectral stability assumptions, we establish Lyapunov stability of the front against fully nonlocalized perturbations. Such perturbations could previously be treated only for scalar equations via comparison principles. We also prove that the leading-order dynamics of the perturbed front are governed by a modulation that tracks the motion of the front interface and evolves according to a viscous Hamilton-Jacobi equation. This effective description reveals that asymptotic orbital stability does not hold in general. However, asymptotic stability can be recovered by imposing localization of perturbations in the transverse spatial directions. The treatment of nonlocalized perturbations on $\mathbb{R}^{d}$ poses significant challenges, both at the linear and nonlinear level. At the linear level, the neutral translational mode gives rise to continuous spectrum which touches the origin and cannot be projected out by conventional means, resulting in merely algebraic decay rates for the residual. Our linear estimates are necessarily $L^{\infty}$-based, yielding significantly weaker decay rates than those available for $L^p$-localized perturbations. At the nonlinear level, quadratic gradient terms decay at a critical rate and cannot be treated perturbatively. We overcome these challenges by carefully decomposing the linearized dynamics, blending semigroup methods with ideas from the stability analysis of viscous shock waves, and introducing a novel nonlinear tracking scheme that combines spatiotemporal modulation with forcing techniques and the Cole-Hopf transform.

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arXiv2026

Diffusive synchronization of phase waves in the FitzHugh-Nagumo system

We analyze synchronization of relaxation oscillations in multiple-timescale reaction-diffusion systems. Interpreting synchronization as convergence to frequency-synchronized wave-train solutions, we resolve for the first time the case of phase waves. These waves are nearly phase-synchronized relaxation oscillations, featuring quasistationary plateaus of length $\varepsilon^{-1}$ separated by fast transition layers, where $\varepsilon\ll1$ is the timescale separation parameter. Tracking the decay of modulations via a Bloch-wave eigenfunction analysis, we find a remarkably weak interaction strength of order $\varepsilon^{8/3}$. This weak layer interaction and many of the technical difficulties arise from repeated scattering of eigenfunctions through fold points at the ends of the quasistationary plateaus. We capture this by combining a novel geometric desingularization approach with Lin's method, exponential trichotomies, and the Riccati transform. While our spectral stability analysis yields diffusive synchronization of all phase waves in the FitzHugh-Nagumo system, it also identifies potential finite-wavelength instabilities, which we realize in a system variant.

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arXiv2025

Global existence and decay of small solutions in a viscous half Klein-Gordon equation

We establish global existence and decay of solutions of a viscous half Klein-Gordon equation with a quadratic nonlinearity considering initial data, whose Fourier transform is small in L1 cap Linfty. Our analysis relies on the observation that nonresonant dispersive effects yield a transformation of the quadratic nonlinearity into a subcritical nonlocal quartic one, which can be controlled by the linear diffusive dynamics through a standard L1 - Linfty argument. This transformation can be realized by applying the normal form method of Shatah or, equivalently, through integration by parts in time in the associated Duhamel formula.

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arXiv2025

Nonlinear dynamics of reaction-diffusion wave trains under large and fully nonlocalized modulations

We study the dynamics of periodic wave trains in reaction-diffusion systems on the real line under large, fully nonlocalized modulations. We prove that solutions with nearby initial data converge, at an enhanced diffusive rate, to a modulated wave train whose leading-order phase and wavenumber dynamics are governed by an explicit solution to the viscous Hamilton-Jacobi equation. This constitutes a global stability result: such initial data are generally not close to the large-time modulated wave train. In contrast to previous modulational stability results, our analysis does not require that the initial data approach phase shifts of the wave train at spatial infinity. The central methodological advance is a nontrivial extension of the recently developed $L^\infty$-stability theory to accommodate large phase modulations. This framework, based entirely on $L^\infty$-estimates, removes all localization requirements as imposed in the previous literature, allowing us to treat the full range of bounded modulational initial data under minimal regularity assumptions. The main technical contributions include: the strategic use of interpolation inequalities to balance smallness and temporal decay, and a detailed analysis of the linear dynamics under fully nonlocalized modulational data.

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arXiv2025

Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation

We investigate the long-time behavior of solutions with small initial data to the viscoelastic Klein-Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. We identify a sign condition for the remaining critical resonant term to be of absorption type, leading to global-in-time existence and diffusive decay of solutions with small initial data. Even when this condition fails, our analysis shows existence and diffusive decay of small solutions on exponentially long time intervals.

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arXiv2025

Existence and stability of soliton-based frequency combs in the Lugiato-Lefever equation

Kerr frequency combs are optical signals consisting of a multitude of equally spaced excited modes in frequency space. They are generated by converting a continuous-wave pump laser within an optical microresonator. It has been observed that the interplay of Kerr nonlinearity and dispersion in the microresonator can lead to a stable optical signal consisting of a periodic sequence of highly localized ultra-short pulses, resulting in broad frequency spectrum. The discovery that stable broadband frequency combs can be generated in microresonators has unlocked a wide range of promising applications, particularly in optical communications, spectroscopy and frequency metrology. In its simplest form, the physics in the microresonator is modeled by the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing. In this paper we demonstrate that the Lugiato-Lefever equation indeed supports arbitrarily broad Kerr frequency combs by proving the existence and stability of periodic solutions consisting of any number of well-separated, strongly localized and highly nonlinear pulses on a single periodicity interval. We realize these periodic multiple pulse solutions as concatenations of individual bright cavity solitons by phrasing the problem as a reversible dynamical system and employing results from homoclinic bifurcation theory. The spatial dynamics formulation enables us to harness general results, based on Evans-function techniques and Lin's method, to rigorously establish diffusive spectral stability. This, in turn, yields nonlinear stability of the periodic multipulse solutions against localized and subharmonic perturbations.

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arXiv2025

Multiple front and pulse solutions in spatially periodic systems

In this paper, we develop a comprehensive mathematical toolbox for the construction and spectral stability analysis of stationary multiple front and pulse solutions to general semilinear evolution problems on the real line with spatially periodic coefficients. Starting from a collection of $N$ nondegenerate primary front solutions with matching periodic end states, we realize multifront solutions near a formal concatenation of these $N$ primary fronts, provided the distances between the front interfaces is sufficiently large. Moreover, we prove that nondegenerate primary pulses are accompanied by periodic pulse solutions of large spatial period. We show that spectral (in)stability properties of the underlying primary fronts or pulses are inherited by the bifurcating multifronts or periodic pulse solutions. The existence and spectral analyses rely on contraction-mapping arguments and Evans-function techniques, leveraging exponential dichotomies to characterize invertibility and Fredholm properties. To demonstrate the applicability of our methods, we analyze the existence and stability of multifronts and periodic pulse solutions in some benchmark models, such as the Gross-Pitaevskii equation with periodic potential and a Klausmeier reaction-diffusion-advection system, thereby identifying novel classes of (stable) solutions. In particular, our methods yield the first spectral and orbital stability result of periodic waves in the Gross-Pitaevskii equation with periodic potential, as well as new instability criteria for multipulse solutions to this equation.

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arXiv2024

Nonlinear stability of periodic wave trains in the FitzHugh-Nagumo system against fully nonlocalized perturbations

Recently, a nonlinear stability theory has been developed for wave trains in reaction-diffusion systems relying on pure $L^\infty$-estimates. In the absence of localization of perturbations, it exploits diffusive decay caused by smoothing together with spatio-temporal phase modulation. In this paper, we advance this theory beyond the parabolic setting and propose a scheme designed for general dissipative semilinear problems. We present our method in the context of the FitzHugh-Nagumo system. The lack of parabolicity and localization complicates mode filtration in $L^\infty$-spaces using the Floquet-Bloch transform. Instead, we employ the inverse Laplace representation of the semigroup generated by the linearization to uncover high-frequency damping, while leveraging a novel link to the Floquet-Bloch representation for the smoothing low-frequency part. Another challenge arises in controlling regularity in the quasilinear iteration scheme for the modulated perturbation. We address this by extending the method of nonlinear damping estimates to nonlocalized perturbations using uniformly local Sobolev norms.

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arXiv2024

Nonlinear stability and asymptotic behavior of periodic wave trains in reaction-diffusion systems against $C_{\mathrm{ub}}$-perturbations

We present a nonlinear stability theory for periodic wave trains in reaction-diffusion systems, which relies on pure $L^\infty$-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers' equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. Yet, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole-Hopf transform to eliminate the critical Burgers'-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against $C_{\mathrm{ub}}$-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton-Jacobi and Burgers' equation, respectively.

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arXiv2024

Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

We study the nonlinear dynamics of perturbed, spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is (orbitally) asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential decay rates of perturbations depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in $N$. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in $N$. This in turn yields an improved nonuniform subharmonic stability result providing an $N$-independent ball of initial perturbations which eventually exhibit exponential decay at an $N$-dependent rate. Finally, we argue that our results connect in the limit $N \to \infty$ to previously established stability results against localized perturbations, thereby unifying existing theories.

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arXiv2022

Moving modulating pulse and front solutions of permanent form in a FPU model with nearest and next-to-nearest neighbor interaction

We consider a nonlinear chain of coupled oscillators, which is a direct generalization of the classical FPU lattice and exhibits, besides the usual nearest neighbor interaction, also next-to-nearest neighbor interaction. For the case of nearest neighbor attraction and next-to-nearest neighbor repulsion we prove that such a lattice admits, in contrast to the classical FPU model, moving modulating front solutions of permanent form, which have small converging tails at infinity and can be approximated by solitary wave solutions of the Nonlinear Schrödinger equation. When the associated potentials are even, then the proof yields moving modulating pulse solutions of permanent form, whose profiles are spatially localized. Our analysis employs the spatial dynamics approach as developed by Iooss and Kirchgässner. The relevant solutions are constructed on a five-dimensional center manifold and their persistence is guaranteed by reversibility arguments.

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arXiv2022

Nonlinear Modulational Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such solutions has only been established against co-periodic perturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbations which are localized, i.e., integrable on the line. Such localized perturbations naturally yield the absence of a spectral gap, so we must rely on a substantially different method with origins in the stability analysis of periodic waves in reaction-diffusion systems. The relevant linear estimates have been obtained in recent work by the first three authors through a delicate decomposition of the associated linearized solution operator. Since its most critical part just decays diffusively, the nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation. However, the modulated perturbation satisfies a quasilinear equation yielding an apparent loss of regularity. To overcome this obstacle, we incorporate tame estimates on the unmodulated perturbation, which satisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to close an independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLE against localized perturbations with precisely the same decay rates as predicted by the linear theory.

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arXiv2022

Nonlinear stability of periodic roll solutions in the real Ginzburg-Landau equation against $C_{\mathrm{ub}}^m$-perturbations

The real Ginzburg-Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against localized perturbations. It is the purpose of this paper to prove their stability against bounded perturbations, which are not necessarily localized. Since all state-of-the-art techniques rely on localization or periodicity properties of perturbations, we develop a new method, which employs pure $L^\infty$-estimates only. By fully exploiting the smoothing properties of the semigroup generated by the linearization, we are able to close the nonlinear iteration despite the slower decay rates. To show the wider relevance of our method, we also apply it to the amplitude equation as it appears for pattern-forming systems with an additional conservation law.

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