27 citations · 69 across the 16 of their papers we have counts for
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Dissipative bright and dark pulses under periodic forcing
Lukas Bengel, Björn de Rijk
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 canonica…
Nonlinear stability of periodic waves in the Korteweg-de Vries equation under localized perturbations
Emile Bukieda, Björn de Rijk
We investigate the stability and asymptotic behavior of spatially periodic cnoidal waves in the Korteweg-de Vries equation subject to localized perturbations. Standard stability ar…
Multiple front and pulse solutions in spatially periodic systems
Lukas Bengel, Björn de Rijk
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 semi…
Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation
Louis Garénaux, Björn de Rijk
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 sp…
Stability of coherent pattern formation through invasion in the FitzHugh-Nagumo system
Montie Avery, Paul Carter, Björn de Rijk +1
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 cri…
Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
Mariana Haragus, Mathew A. Johnson, Wesley R. Perkins +1
We study the nonlinear dynamics of perturbed, spectrally stable -periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation wit…
Nonlinear stability and asymptotic behavior of periodic wave trains in reaction-diffusion systems against -perturbations
Björn de Rijk
We present a nonlinear stability theory for periodic wave trains in reaction-diffusion systems, which relies on pure -estimates only. Our analysis shows that localization…
Nonlinear stability of periodic roll solutions in the real Ginzburg-Landau equation against -perturbations
Bastian Hilder, Björn de Rijk, Guido Schneider
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 pe…
Nonlinear Modulational Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
Mariana Haragus, Mathew A. Johnson, Wesley R. Perkins +1
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 n…
Validity of Whitham's modulation equations for dissipative systems with a conservation law -- Phase dynamics in a generalized Ginzburg-Landau system --
Tobias Haas, Björn de Rijk, Guido Schneider
It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity…
Modulation equations near the Eckhaus boundary: the KdV equation
Tobias Haas, Björn de Rijk, Guido Schneider
We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation \begin{align*} \partial_T Ψ= (1+ i α) \p…
Diffusive stability against nonlocalized perturbations of planar wave trains in reaction-diffusion systems
Björn de Rijk, Björn Sandstede
Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate th…
Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation
Björn de Rijk, Arjen Doelman, Jens Rademacher
In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In su…