Schedule

Tuesday, August 20

10:00-10:20Coffee breakOutside of the lecture hall
10:20-10:30Welcome + info 
10:30-11:15Anna Geyer
On the transverse stability of smooth solitary waves in a two-dimensional Camassa–Holm equation
11:30-12:15Changzhen Sun
Transverse linear stability of line solitary gravity water waves
In this talk, we will talk about the transverse linear asymptotic stability of one-dimensional small-amplitude solitary waves of the gravity water-waves system. It is shown that the semigroup of the linearized operator about the solitary wave decays exponentially within a spectral subspace supplementary to the space generated by the spectral projection on continuous resonant modes. The key element of the proof is to establish suitable uniform resolvent estimates in the exponentially weighted energy space, for which we need to use different arguments depending on the size of transverse frequencies. As a corollary of our main result, we also get the spectral stability in the unweighted energy space.
12:15-14:00Lunch at Inspira 
14:15-15:00Massimiliano Berti
Long time dynamics of water waves
The dynamics of the water waves equations is governed by an infinite dimensional Hamiltonian and reversible system of quasi-linear equations. In the last years several important analytical results have been obtained concerning the long time dynamics of space periodic waves. In this talk I will present some long time existence results and the existence of time quasi-periodic traveling Stokes waves. These can be regarded as the nonlinear superposition of multiple periodic Stokes waves with rationally independent speeds, interacting each other and retaining forever a quasi-periodic structure.
15:00-15:30Coffee breakOutside of the lecture hall
15:30-16:15Dag Nilsson
A resonant Lyapunov centre theorem with an application to hydroelastic waves
We will present a generalized Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system with a nondegenerate 1:1 or 1:-1 semisimple resonance. In addition we allow for a non-constant symplectic structure and that the origin is in the spectrum of the linearized Hamiltonian vector field. As an application we will show how this theorem can be used to prove existence of doubly periodic hydroelastic waves by first formulating the hydrodynamic problem as a reversible Hamiltonian system, using Kirchgässner’s spatial dynamics approach. This talk is based on a joint work with Mark Groves and Rami Ahmad (Saarland University).
16:30-17:00Wei Lian
Transverse instability of line periodic waves to the KP-I equation
The passage from linear instability to nonlinear instability has been shown for 1D solitary waves under 2D perturbations. Although transverse instability of periodic waves to the KdV equation under the KP-I flow has been expected to be true from spectral instability for a long time, it has not been clear how to adapt the general instability theory for solitary waves to periodic waves until now. In this talk, we present how such an adaptation works with the aid of exponential trichotomies and multivariable Puiseux series. Joint work with E. Wahlén.
17:15-19:00Welcome receptionCoffee room, 4th floor

Wednesday, August 21

9:15-10:00Mats Ehrnström
Asymmetric capillary–gravity water waves in the steady periodic setting
We discuss ongoing work on asymmetric capillary–gravity surface waves in the Euler equations. It has been known for a long time that the setting of weak surface tension allows for higher-dimensional bifurcation from still water, giving rise to multimodal waves with more than one crest in a period. These waves have, however, all been symmetric, although numerical calculations indicate the presence of truly asymmetric waves in the steady periodic setting. Recently, Mæhlen and Svensson Seth extended earlier bifurcation results for the gravity–capillary Whitham equation, showing that asymmetric solutions exist as natural extensions of bimodal waves. In this work, which is joint with Douglas Svensson Seth (NTNU) and Boris Buffoni (EPFL), we investigate the existence of such asymmetric solutions in the Euler equations.
10:00-10:30Coffee breakOutside of the lecture hall
10:30-11:15Christian Klein
Numerical study of fractional nonlinear Schrödinger equations
A numerical study of solutions to fractional nonlinear Schrödinger (fNLS) equations is presented. We discuss efficient numerical algorithms to compute fractional derivatives. For the focusing fNLS equation, solitons are constructed numerically and their stability is explored. The possibility of a blow-up of solutions to fNLS for smooth initial data is discussed.
11:30-12:15Dmitry Pelinovsky
Traveling waves in Babenko’s equation for water waves
We introduce a new model equation for Stokes gravity waves based on conformal transformations of Euler’s equations. The local version of the model equation is relevant for dynamics of shallow water waves. It allows us to characterize the traveling periodic waves both in the case of smooth and peaked waves and to solve the existence problem exactly, albeit not in elementary functions. Spectral stability of smooth waves with respect to co-periodic perturbations is proven analytically based on the exact count of eigenvalues in a constrained spectral problem.
12:15-14:00Lunch at Inspira 
14:15-15:00Vladimir Kozlov
On the first bifurcation of solitary waves
Solitary water waves on the vorticity flow in a two-dimensional channel of finite depth are considered. The main object of study is a branch of solitary waves starting from a laminar flow and then approaching an extreme wave. It is proved that there always exists a bifurcation point on such branches. Moreover, the first bifurcation occurs at a simple eigenvalue. The structure of the set of bifurcating solutions is described.
15:00-15:30Coffee breakOutside of the lecture hall
15:30-16:15Jörg Weber
Axisymmetric capillary water waves with vorticity and swirl
While the research on axisymmetric capillary water waves has a long history, only surprisingly little has been done when one wants to allow for vorticity (local spinning) and swirl (angular momentum). After introducing the problem and presenting a well-known formulation in terms of the Stokes stream function, we will explain the main analytical ingredients in order to rigorously construct travelling wave solutions with a free boundary. Then, two solution curves are constructed: First, we start at perfectly cylindrical jets, where we derive the dispersion relation and investigate it in detail in certain cases. Also, we will point out inherent connections to the famous Rayleigh instability observed in everyday life, e.g. when a tap drips. Second, we start at static configurations with constant mean curvature surfaces, so-called unduloids. There, as an interesting interplay between water waves, geometry, and elliptic integrals, we show rigorously that to any such configuration there connects a curve of non-static configurations, which confirms previous numerical observations. The talk is based on joint work with André Erhardt (WIAS), Anna-Mariya Otsetova (Aalto), and Erik Wahlén (Lund).
18:30-22:00DinnerAt Hos Talevski

Thursday, August 22

9:15-10:00Alberto Maspero
Infinitely many isolas of modulational instability for Stokes waves
We prove the long-standing conjecture regarding the existence of infinitely many high-frequency modulational instability “isolas” for a Stokes wave in arbitrary depth, subject to longitudinal perturbations. We completely describe the spectral bands with non-zero real part of the spectrum, away from the origin, of the water waves system linearized at a Stokes waves of small amplitude. The unstable spectrum is the union of isolas of elliptical shape, centered along the imaginary axis, and shrinking exponentially fast away from the origin. This is a joint work with M. Berti, L. Corsi and P. Ventura
10:00-10:30Coffee breakOutside of the lecture hall
10:30-11:15Bastian Hilder
Patterns and fronts in thermocapillary thin films.
11:30-12:15Luigi Roberti
Existence, uniqueness and stability for a model of the Antarctic Circumpolar Current
The Antarctic Circumpolar Current is the strongest persistent oceanic current; it flows all the way around Antarctica and plays a crucial role in determining the Earth’s climate, as it drives the water exchange between the Atlantic, Pacific, and Indian basins. In this talk, we will discuss a model of this flow that has been studied extensively over the past few years. All previous results deal with the stationary case, mostly focusing on properties of particular zonal (i.e., exclusively latitude-dependent) flows; on the contrary, we will consider the full time-dependent model and obtain global-in-time existence and uniqueness of a classical solution to sufficiently regular initial data. Moreover, we will discuss the stability of a certain family of stationary zonal flows, which is obtained by exploiting appropriate conservation laws in order to construct a Lyapunov functional.
12:15-14:00Lunch at Inspira