Schedule

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.

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.

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.

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.

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).

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

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.