**Supported by**

** Speakers Program Abstracts Venue Information **** **

The workshop is organized jointly by the Institute for Analysis and the CRC wave phenomena. It is held on the occasion of Prof. Plum's recent retirement and in honor of his rich contributions to the analysis of PDEs and the field of computer assisted proofs. The workshop brings together internationally renowned experts in the areas of nonlinear analysis, spectral theory, and PDEs as well as colleagues, friends and students of Prof. Plum.

Maxim Breden (École Polytechnique, Paris, France)

Tomas Dohnal (Martin Luther University Halle-Wittenberg, Germany)

Jason Mireles-James (Florida Atlantic University, USA)

Dmitry Pelinovsky (McMaster University, Canada)

Christiane Tretter (University of Bern, Switzerland)

Yoshitaka Watanabe (Kyushu University, Japan)

Ian Wood (University of Kent, UK)

Wednesday | ||

Feb. 21, 2024 | ||

13:00 - 14:00 | Welcome | |

14:00 - 14:45 | J. Mireles-James: Parameterization method for invariant manifolds | |

14:50 - 15:35 | Y. Watanabe: Some computer-assisted proofs for nonlinear differential equations involved with self-similar blowup in wave equations | |

15:35 - 16:15 | Coffee Break | |

16:15 - 17:00 | Ch. Tretter: Challenges for non-selfadjoint spectral problems | |

17:05 - 17:50 | D. Pelinovsky: Existence of generalized breathers (modulating pulses) in periodic systems via spatial dynamics | |

18:00 - 19:00 | Get together with drinks & snacks, Atrium Buildg. 20.30 | |

19:30 | Conference Dinner | |

Thursday | ||

Feb. 22, 2024 | ||

09:00 - 09:45 | M. Breden: Rigorous enclosures of moment Lyapunov exponents | |

09:50 - 10:35 | I. Wood: Spectrum of the Maxwell equations for a flat interface between homogeneous dispersive media | |

10:35 -11:15 | Coffee Break | |

11:15 - 12:00 | T. Dohnal: Standing waves and wave-packets at a material interface in nonlinear Maxwell equations | |

12:10 - 13:00 | Farewell | |

Maxim Breden (École Polytechnique, Paris, France)

*Rigorous enclosures of moment Lyapunov exponents*

When studying random dynamical systems described by stochastic differential equations, Lyapunov exponents are powerful tools to measure qualitative behavior. Moments Lyapunov exponents capture finer information, such as fluctuations of finite time Lyapunov exponents, and yield large deviations estimates. These moments Lyapunov exponents can be obtained as principal eigenvalues of elliptic PDEs associated to the original SDEs. I will describe a computer-assisted approach that can be used to obtain rigorous enclosures of such eigenvalues, which involves techniques developed by Michael Plum. This is joint work with Alex Blumenthal (Georgia Tech), Maximilian Engel (U. of Amsterdam) and Alexandra Blessing-Neamtu (U. of Konstanz).

Tomas Dohnal (Martin Luther University Halle-Wittenberg, Germany)

*Standing waves and wave-packets at a material interface in nonlinear Maxwell equations*

The talk presents some recent results on wave propagation in Maxwell equations at an interface of two generally dispersive, i.e. frequency dependent, materials. One of the main physical applications of this set-up is to surface plasmon polaritons (SPPs) at a metal/dielectric interface. First, we discuss bifurcations of nonlinear SPPs from linear ones given by simple isolated eigenvalues of the corresponding operator pencil (collaboration with G. Romani and R. He). We also provide an asymptotic expansion of the bifurcating solutions. In the second part of the talk we look into wave-packets propagating along the (straight) interface of non-linear materials and prove a slowly varying envelope approximation for such solutions. In the case of non-dispersive materials the envelope equation is the nonlinear Schrödinger equation while for dispersive ones it is the complex Gizburg-Landau equation. In the latter case the well-posedness of the (temporally nonlocal) nonlinear Maxwell equations is obtained in exponentially weighted L^2 spaces using an abstract evolutionary equations approach.
The work on wavepackets is in collaboration with R. Schnaubelt (Karlsruhe), D. Tietz (Okinawa), M. Tira (Halle), and M. Waurick (Freiberg).

Jason Mireles-James (Florida Atlantic University, USA)

*Parameterization method for invariant manifolds*

The parameterization method is a functional analytic framework for studying invariant sets and their attached invariant manifolds. The idea is to formulate a conjugacy equation solved by the desired object, so that a dynamical problem is replaced by a "more elliptic" equation. The new problem is amenable to all the tools of computational mathematics, including computer assisted proofs based on a-posteriori analysis. I'll illustrate the parameterization method by discussing several example problems, and also show how it can be used to study more global dynamical objects like transverse heteroclinic and homoclinic connections.

Dmitry Pelinovsky (McMaster University, Canada)

*Existence of generalized breathers (modulating pulses) in periodic systems via spatial dynamics*

I will overview solutions of two different problems with the methods of spatial dynamics. In the first problem, I will consider the traveling modulating pulse solutions of a nonlinear wave equation with spatially periodic coefficients. The modulating pulses consist of a small amplitude pulse-like
envelope moving with a constant speed and modulating a harmonic carrier wave. In the second problem, I will consider homoclinic solutions of the Fermi-Pasta-Ulam lattice with time-periodic coefficients. These solutions model generalized breathers with oscillating ripples. For both problems, the solutions can be approximated by solitons of an effective nonlinear Schroedinger equation arising as the envelope equation. We construct a rigorous existence proof of such solutions on large domains in time and space by using spatial dynamics, invariant manifolds, and near-identity transformations. My talk is based on the joint papers with C. Chong, T. Dohnal, and G. Schneider.

Christiane Tretter (University of Bern, Switzerland)

*Challenges for non-selfadjoint spectral problems*

The theory of linear operators is an important tool to investigate the stability of physical systems or their time evolution. In turn, applications from physics and engineering
have contributed considerably to the advances of operator theory. The most prominent example of this fruitful interaction is the interplay between quantum mechanics and the theory of selfadjoint operators in Hilbert spaces.
In this talk more challenging applications are addressed which pose serious problems in analysis and computations due to their lack of symmetry; these include, in particular, problems in magnetohydrodynamics and hydrodynamics. Here numerical calculations may fail to produce reliable results and, thus, rigorous analytical information is highly desirable -- an aspect which has also played an important role in Michael Plum's research.

Yoshitaka Watanabe (Kyushu University, Japan)

*Some computer-assisted proofs for nonlinear differential equations involved with self-similar blowup in wave equations*

Some existing proofs of non-trivial solutions for nonlinear differential equations involved with self-similar blowup in three-dimensional wave equations are presented.
The proofs are computer-assisted based on a fixed-point and Newton-type formulation, and the results consider the effects of rounding errors of floating-point arithmetic on a computer. It is convinced that the existence of proofs of the non-trivial solutions will contribute to the solutions' stability and instability analysis.
This study is a joint work with Kaori Nagato-Plum and Michael Plum (KIT), Birgit Schörkhuber (University of Innsbruck), and Mitsuhiro T. Nakao (Waseda University).

Ian Wood (University of Kent, UK)

*Spectrum of the Maxwell equations for a flat interface between homogeneous dispersive media*

We determine and classify the spectrum of a non-selfadjoint operator pencil generated by the time-harmonic Maxwell problem with a nonlinear dependence on the frequency. More specifically, we consider one- and two-dimensional reductions for the case of two homogeneous materials joined at a planar interface. The dependence on the spectral parameter, i.e. the frequency, is in the dielectric function and we make no assumptions on its form. In order to allow also for non-conservative media, the dielectric function is allowed to be complex, yielding a non-selfadjoint problem. This is joint work with Malcolm Brown (Cardiff), Tomas Dohnal (Halle) and Michael Plum (Karlsruhe).

Mathematics Building 20.30, Englerstrasse 2, 76131 Karlsruhe.

Scientific Talks: Room 1.067

Coffee Breaks, Welcome, Farewell: Room 1.058

Please contact Wolfgang Reichel (*wolfgang.reichel@kit.edu*) and Xian Liao (*xian.liao@kit.edu*).

For administrative questions please contact Marion Ewald (*marion.ewald@kit.edu*).