Project A14 • Nonlinear stability of periodic waves in dissipative-dispersive systems

Principal investigators

Project summary

Periodic traveling or standing waves are ubiquitous in nonlinear dynamical processes in many scientific disciplines and play a fundamental role in pattern formation. Therefore, the question of their dynamical, or nonlinear, stability is of great importance. In this project we focus on the nonlinear stability of periodic waves against localized perturbations in dispersive systems.

So far, nonlinear stability results for periodic waves in dispersive systems have been obtained against co-periodic or subharmonic perturbations using variational methods relying on Hamiltonian structure or other conserved quantities. Yet, from the perspective of pattern formation it is natural to consider localized perturbations on spatially unbounded domains, rendering the perturbed wave neither periodic nor localized and, thus, precluding existing variational arguments. Furthermore, many important nonlinear dynamical processes exhibit, besides dispersion, also dissipation, e.g. due to the presence of viscosity or diffusion. However, including such dissipative terms typically breaks Hamiltonian structure or other conservation laws.

This begs the question of whether there is an alternative, nonvariational approach to establish nonlinear stability of periodic waves in dispersive systems, which does work for localized perturbations and allows for dissipative terms. In this project we propose such an approach, which is based on spatio-temporal phase modulation and iterative estimates on the Duhamel formulation. Although these ideas have been successfully applied before to establish nonlinear stability of periodic waves in a number of dissipative systems such as reaction-diffusion models, the associated analyses make use of diffusive decay only. In order to exploit dispersive decay in the nonlinear iteration, we intend to employ the space-time resonances method leading to oscillatory integral operators, which can be controlled using (non-)stationary phase arguments and multi- linear analysis. We apply our approach to dissipative-dispersive systems, where the dispersive decay is essential to close the nonlinear stability argument. Natural candidates of such systems can be found in nonlinear optics, where systems of coupled Ginzburg–Landau-type equations arise. Our ultimate goal is to apply our techniques to purely dispersive systems of nonlinear Schrödinger-, Klein–Gordon-, or Korteweg—de Vries-type. Nonlinear stability of periodic waves in such systems against localized perturbations is still a largely unresolved problem.

Publications

1. E. Bukieda, L. Garénaux, and B. de Rijk. Orbital stability of harmonic waves in the Klein–Gordon equation against localized pertubations. J. Math. Phys., 67(10):Paper No. 101502, October 2026. URL https://doi.org/10.1063/5.0331730. [preprint] [files] [bibtex]

2. L. Garénaux and B. de Rijk. Long time behavior of small solutions in the viscoelastic Klein–Gordon equation. J. Nonlinear Waves, 2:e5, September 2026. URL https://doi.org/10.1017/jnw.2026.10032. [preprint] [bibtex]

3. J. Alexopoulos. Nonlinear dynamics of periodic Lugiato–Lefever waves against sums of co-periodic and localized perturbations. Phys. D, 488:135079, April 2026. URL https://doi.org/10.1016/j.physd.2025.135079. [preprint] [bibtex]

4. B. de Rijk and G. Schneider. Global existence and decay in multi-component reaction-diffusion-advection systems with different velocities: oscillations in time and frequency. Nonlinear Differ. Equ. Appl., 28(1):Paper No. 2, January 2021. URL https://doi.org/10.1007/s00030-020-00665-5. [preprint] [bibtex]

5. B. de Rijk and G. Schneider. Global existence and decay in nonlinearly coupled reaction-diffusion-advection equations with different velocities. J. Differential Equations, 268(7):3392–3448, March 2020. URL https://doi.org/10.1016/j.jde.2019.09.056. [preprint] [bibtex]

Preprints

6. J. Alexopoulos. $L^\infty$-regularity thresholds for a generalized Gross–Pitaevskii equation: dispersive blow-up vs. nonlinear stability. CRC 1173 Preprint 2026/53, Karlsruhe Institute of Technology, October 2026. [bibtex]

7. E. Bukieda and B. de Rijk. Nonlinear stability of periodic waves in the Korteweg-de Vries equation under localized perturbations. CRC 1173 Preprint 2026/28, Karlsruhe Institute of Technology, July 2026. [files] [bibtex]