Special Session 190: Amplitude equations for continuous, discrete, non-local, and stochastic nonlinear dispersive and dissipative dynamical systems

A dispersive relaxation mechanism: blow-up vs. stability in absence of any localization assumptions on initial data

Joannis Alexopoulos Karlsruhe Institute of Technology Germany Co-Author(s):

Abstract: The study of long-time behavior in dispersive equations typically relies on energy estimates that require localization assumptions on initial data. In this talk, we explore large-time dynamics without imposing any such localization conditions. We discuss a variant of the Gross-Pitaevskii equation on the real line which is used to model superfluidity properties of Bose-Einstein condensates. This model can be viewed as intermediate between the classical Gross-Pitaevskii equation, by keeping leading order dispersion, and the complex Ginzburg-Landau equation, by adding typical balance terms. We identify a relaxation mechanism close to the wave functions $re^{i\omega t + ikx}$, which arises through the interaction between diffusive- and dispersive effects. This relaxation process allows to show nonlinear stability of the wave functions in $C_b^k$ against perturbations from $C_b^{k+1+\sigma}$ for all $\sigma>0$. In contrast, we also establish finite time blow-up solutions in $C_b^k$ whenever we include initial $C_b^{k+1}$-perturbations: this shows that our stability result is sharp concerning global existence. To the best of our knowledge, this work provides a first analysis of long-time dynamics in dispersive models without any localization assumptions on initial data, and may motivate further research directions. Specifically, we expect that the blow-up vs. stability phenomenon also appears for the Lugiato-Lefever equation.

Validity of the Ginzburg-Landau Approximation for Pattern Forming Quasilinear Systems

Theo Belin Lund University France Co-Author(s):    Guido Schneider

Abstract: We present a method justifying that the Ginzburg-Landau equation is a valid modulation equation for a wide class of quasilinear dissipative systems. The modulation equation can be derived near the first instability of a Turing or Turing-Hopf bifurcation of spatially homogeneous steady states and correctly predicts the behavior of the solution on the long timescale. The proof of the approximation result relies on maximal regularity and a fixed point argument that can be expressed in a rather abstract way. This allows, for instance, to validate the modulation equation for initially nonlocalized perturbations. We show that the arguments can be applied to the quasilinear Klausmeier-Gray-Scott system as well as a quasilinear version of the B\`enard-Rayleigh convection model.

Heterogeneities in reaction-diffusion systems: A feature, not a perturbation

Martina Chirilus-Bruckner Leiden University Netherlands Co-Author(s):    Jolien Kamphuis, Lara van Vianen, Frits Veerman

Abstract: Spatial heterogeneities are often treated as an inconvenience: they break translation invariance, complicate spectral analysis, and blur bifurcation scenarios familiar from homogeneous media. In this talk, we argue that heterogeneity is not just a complication, but also a useful tool for generating, selecting, and stabilizing coherent structures. We develop an analytical framework for existence, stability, and bifurcations in reaction-diffusion systems with spatially dependent coefficients, focusing on front and wave train dynamics. The presentation is centered around two case studies. First, we examine heterogeneous front solutions in a FitzHugh-Nagumo equation, where spatial variability produces fronts that propagate at non-constant speeds through stationary heterogeneous background states. Second, we study wave trains governed by a Ginzburg-Landau amplitude equation arising as slow modulations in a Swift-Hohenberg model, showing how spatially non-uniform coefficients affect wave-number selection. A common thread is the use of perturbation techniques involving a small parameter. Importantly, this parameter does not measure the size of the spatial heterogeneity. Instead, it reflects scale separation or nearness to a critical regime, allowing order-one heterogeneities while preserving analytical tractability. The key novelty is extending classical perturbative approaches to include non-autonomous terms systematically. This yields solvability conditions and reduced modulation equations in spatially varying environments naturally.

Modulation equations for scalar Fermi-Pasta-Ulam-Tsingou systems on 2D square lattices

Ioannis Giannoulis University of Ioannina Greece Co-Author(s):    Bernd Schmidt, Guido Schneider

Abstract: We present results concerning the justification of modulation equations for the dynamics of atoms in a two-dimensional square lattice that interact with their nearest neighbors nonlinearly with respect to the strain between their scalar displacements. When the interaction forces are cubic we show that small macroscopically modulated amplitudes of rapidly oscillating plane waves evolve approximately according to a nonlinear Schroedinger equation, while in the case of quadratic interaction forces the corresponding modulation equation is a Davey-Stewartson system. Due to the dispersive scaling of the small amplitudes, the justification in the latter case is significantly more involved than in the former and necessitates the employment of normal-form transformation techniques.

Amplitude Equations for Nonlocal Swift-Hohenberg Equation

Christian Kuehn TUM Germany Co-Author(s):

Abstract: In this talk I am going to present various results regarding the derivation of amplitude equations for nonlocal PDEs. In particular, we shall consider nonlocal reaction terms as well as nonlocal fractional diffusion in the context of the Swift-Hohenberg equation. Interestingly, the resulting Ginzburg-Landau-type amplitude equations become fully local with the non-local influence only entering to leading-order in the coefficients. This shows an interesting decoupling effect near instability. I shall explain, how this effect arises in the proofs as well. The work is based upon the following papers: (1) Validity of amplitude equations for nonlocal nonlinearities, C. Kuehn and S. Throm, Journal of Mathematical Physics, Vol. 59, 071510, 2018. (2) The amplitude equation for the space-fractional Swift-Hohenberg equation, C. Kuehn and S. Throm, Physica D, Vol. 472, 134531, 2025.

Validity of the stochastic Ginzburg-Landau approximation in higher space dimensions -- A Wiener algebra approach --

Anna Logioti University of Stuttgart Germany Co-Author(s):    Guido Schneider

Abstract: We consider an anisotropic $d$-dimensional Swift-Hohenberg model $ \mathcal{O}(\varepsilon^2) $-close to the first instability, where $ 0 < \varepsilon \ll 1 $ is a small perturbation parameter. This model for pattern formation is perturbed with additive noise in time and space. By a multiple scaling ansatz we derive a stochastic $ d $-dimensional Ginzburg-Landau equation for the approximate description of the bifurcating solutions. We prove the validity of the approximation by this amplitude equation on its natural time scale in case of $ d $-dimensional periodic domains of length $ \mathcal{O}(1/\varepsilon) $ for the Swift-Hohenberg model under suitable conditions on the additive noise. In detail, we prove the validity of this approximation for noise whose set of Fourier coefficients with respect to $ x $ is in $ \ell^1 $ for fixed $ t \geq 0 $. Moreover, we improve existing approximation results in the sense that the stable part of the noise can be larger.

Bifurcation of cylindrical solutions in the spontaneous curvature model

Alexander Meiners University of Oldenburg Germany Co-Author(s):

Abstract: We consider the $L^2$ gradient flow of the Helfrich model for lipid bilayers. The model incorporates constraints on membrane inextensibility (area constraint) and the absence of osmotic exchange (volume constraint). These constraints give rise to Lagrange multipliers, which appear as non-local terms. The lipid bilayer serves as a simplified model for the shape of red blood cells as well as other self-organizing cellular structures in biology. We studied bifurcation of closed vesicles using pde2path. We extend this to cylindrical topology. Well-known phenomena are the pearling instability of the cylindrical shape and transitions to coiled structures solution. We find them and other bifurcations using center manifold analysis and numerical continuation. Due to the presence of Lagrange multipliers, we take a non-standard approach to derive amplitude equations for each bifurcation scenario. Within this framework, we analyze the stability of the bifurcation branches. To provide a broader perspective on the shape transitions of cylindrical structures, we employ numerical continuation, similar to our approach for closed vesicles. However the analysis shows some discrepancies with experiments.

On the closeness of the dynamics for some nonlinear lattices

Simone Paleari Universit\\`{a} di Milano Italy Co-Author(s):    Tiziano Penati, Marco Calabrese

Abstract: We study the closeness between the Ablowitz-Ladik, Salerno, and discrete nonlinear Schroedinger lattices in regimes of small coupling $\epsilon$ and small energy norm $\rho$. Two approaches are compared: direct estimates in physical variables and a transformation to canonical symplectic coordinates via Darboux variables.

Time-Periodic Dynamics in Driven-Damped $\phi^4$ Models: From Reduced Chaos to PDE-Level Existence Theory

VASSILIOS M ROTHOS Aristostle University of Thessaloniki Greece Co-Author(s):    Vassilis Rothos

Abstract: We study time-dependent dynamics in driven-damped $\phi^4$ models with spatial and spatiotemporal modulations, focusing on the connection between reduced dynamical descriptions and rigorous PDE-level analysis. Collective-coordinate reductions reveal complex behavior, including separatrix splitting and chaotic kink motion under periodic forcing. However, the existence of time-periodic solutions at the level of the full PDE remains largely open. We address this by proving the existence of time-periodic solutions for a driven-damped $\phi^4$ equation on a bounded domain with periodic boundary conditions. The analysis is based on a Lyapunov-Schmidt reduction in a time-periodic Hilbert space. A key feature is that damping ensures uniform invertibility of the linearized operator, eliminating the need for non-resonance conditions typical of conservative systems. Our results provide a rigorous PDE-level counterpart to time-periodic responses observed in reduced models and numerical studies.

The NLS approximation for the Peregrine soliton and fractional dispersion

Nils Thorin Universit\"at Stuttgart Germany Co-Author(s):    Guido Schneider, Anna Logioti, Theo Belin

Abstract: In this talk I am going cover various topics regarding the validation of the NLS approximation in dispersive systems. In the first half of the talk we will study a family of solutions to the NLS called the Peregrine Soliton family, which are not covered by previously known approximation theory, as they do not approach zero for $|x|\to\infty$. Either the envelope spatially decays to zero over a temporally oscillating background, or it is fully spatially periodic, and special care is necessary when it comes to choice of function spaces to work in. The second half of the talk will instead focus on the NLS approximation in the presence of a fractional Laplacian. Still, a fully local NLS can be derived from the original, non-local system. The fact that the approximation is strongly localized around the wave number $k_0$ is used to, in lowest order, be able to use similar methods as in the regular case. Something of note is that the character of the derived NLS (focusing/defocusing) may change depending on choice of basic wave number, which is a stark difference to the classic case. This talk is based on joint work with Guido Schneider, Anna Logioti and Theo Belin.

Long-wave KdV hierarchy approximation of the NLS hierarchy with nonzero boundary condition

Robert Wegner Karlsruhe Institute of Technology Germany Co-Author(s):

Abstract: The nonlinear Schr\odinger and Korteweg--De Vries equations are both universal amplitude equations for one-dimensional nonlinear dispersive waves. As a consequence of their complete integrability, they both have an associated infinite hierarchy of nonlinear PDEs: the NLS and KdV hierarchies. We present an approximation of certain renormalized conserved quantities for the NLS hierarchy with nonzero boundary condition, in the long-wave regime, by the energies of the KdV hierarchy. We use this to prove that long-wave solutions to the NLS hierarchy with nonzero boundary condition are approximated by two solutions to KdV hierarchies on a certain timescale.

A variational reduction for the full-dispersion Davey--Stewartson equation

Espen Xylander Institute for analysis, dynamics and modeling, University of Stuttgart Germany Co-Author(s):

Abstract: In this talk, we discuss a method to derive the classical Davey--Stewartson equation from its full-dispersion counterpart. After a brief tour of the relevant parameter regimes, we prove an approximation result for the energy functionals associated with the equations, based on a Lyapunov--Schmidt-type reduction. We then touch upon some existence results.