• 16.03.2026 We have opened the registration. You can apply until May 31st, 2026.

This summer school is aimed at PostDocs, PhD and advanced master students with a background in analysis of partial differential equations. It focuses on the stability of nonlinear waves in evolution equations. These special solutions arise in many applied problems where they model, for instance, water waves, nerve impulses in axons, or pulses in optical fibers. Understanding the dynamic stability of these waves is therefore a fundamental question in both theory and applications.

The program of the summer school begins with a short recap on the stability theory for fixed points in ordinary differential equations, providing the foundation for extending these ideas to nonlinear PDEs. The course then develops two complementary approaches: a Duhamel-based framework for dissipative PDEs, and a variational framework tailored to Hamiltonian systems. Both nonlinear stability approaches rely heavily on the spectral analysis of differential operators, for which the school will provide the necessary tools and techniques.

Besides the lectures, the summer school will feature a number of group projects that allow participants to further explore the material. The results and observations from these projects will be presented at the end of the week.

Lecturers

• Mariana Haragus (FEMTO-ST Besançon, France)
  Title: Stability of nonlinear waves using spectral and semigroup methods
  Abstract:
  Starting from well known stability results for equilibria of ordinary differential equations, we present several tools used in the study of the stability of stationary solutions of partial differential equations (PDEs). One motivation for the choice of these tools is the question of stability of periodic waves for the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. The lectures plan is as follows.
  1. Introduction: brief review of the stability of equilibria for finite dimensional dynamical systems; stability notions for stationary solutions of PDEs; nonlinear waves of the Lugiato-Lefever equation.
  2. Spectral stability: spectra of closed linear operators; essential and point spectra of differential operators with asymptotically constant coefficients; spectra of differential operators with periodic coefficients; Floquet-Bloch decomposition.
  3. Linear stability: semigroups of closed linear operators; analytic and continuous semigroups; existence theorems; spectral mapping theorems, decomposition and bounds.
  4. Nonlinear stability: Duhamel's formula; asymptotic stability in the case of stable spectrum and isolated zero eigenvalue; asymptotic stability of periodic waves; co-periodic perturbations; localized perturbations; application to the Lugiato-Lefever equation.

• Todd Kapitula (Calvin Univ., Michigan, USA)
  Title: Eigenvalues and spectral/dynamical stability of nonlinear waves
  Abstract:
  The nonlinear stability of stationary waves is often determined by the spectrum. If the wave is localized it is relatively straightforward to determine the essential spectrum; however, the point spectrum is a different matter. Focusing on Hamiltonian systems we will develop tools to locate the point spectrum. The lecture plan is as follows:
  1. Applications: consider the spectral and orbital stability of nonlinear waves for Bose-Einstein condensates, photorefractive media with potential wells, KdV-like equations, and Boussinesq-like equations.
  2. Hamiltonian systems, II: develop the HKI theory for second-order in time systems, construct the Krein matrix, use the Krein matrix to reprove the classic Grillakis/Jones instability index.
  3. Hamiltonian systems, I: use the Evans function to locate edge bifurcations for near integrable systems, talk about the Krein signature of point eigenvalues, develop the Hamiltonian-Krein index (HKI) theory for first-order in time systems, show the HKI can be used to determine nonlinear orbital stability.
  4. Evans function: construct the Evans function, discuss its properties, show how domain of analyticity is determined by essential spectrum, discuss analytic continuation to track eigenvalues emerging from the essential spectrum, use the orientation index to locate positive real eigenvalues.

Registration

Please use this link https://indico.kit.edu/e/summerschool2026 to register. Registration deadline is on May 31st, 2026. Acceptance notifications will be sent out by mid of June.

Notice: Please apply early for a visa (if you need one) to enter Germany.

Travel and accommodation

The summer school is free of charge but we cannot provide financial support. Please take care of your own travel and accommodation arrangements. 

Tentative schedule

This school will start on Monday August 31st, 2026, and will end on Friday September 4th, 2026.

Venue

Karlsruhe Institute of Technology
Campus South, Building 20.30
Englerstr. 2
76131 Karlsruhe

Further directions.

Contact

Christian Knieling and Barbara Valnion via E-Mail: admin∂waves.kit.edu

Local organizers

Christian Knieling, Wolfgang Reichel, Björn de Rijk, Guido Schneider, and Barbara Valnion