Kick-off 2nd funding period (2019-23)

Photo of a "2nd funding period" cakeKIT

On the occasion of the successful renewal of our CRC we will organize a workshop on Thursday 4th of July, 2019 starting at 15:00 with the scientific program (including a coffee break). Afterwards at 18:30 the joint dinner takes place.

15:00 Welcome
15:15 Peaked periodic waves in the reduced Ostrovsky equation: instability and uniqueness (access only with valid credentials)
held by Anna Geyer (TU Delft, Netherlands)
The reduced Ostrovksy equation is a model for small-amplitude long waves in a rotating fluid. Peaked periodic waves of this equation are known to exist since the late 1970's. In this talk I will present recent results in which we answer the long standing open question whether these solutions are stable. We first prove linear instability of the peaked periodic waves using semi-group theory and energy estimates. Moreover, we show that the peaked wave is unique and that the equation does not admit Hölder continuous solutions, which implies that the reduced Ostrovsky equation does not admit cusps. Finally, we show that the peaked wave is also spectrally unstable and by so doing we discover a new instability phenomenon. This is joint work with Dmitry Pelinovsky.
16:00 Coffee break
16:30 Numerical simulation of Bose-Einstein condensates (access only with valid credentials)
held by Daniel Peterseim (Univ. Augsburg, Germany)
This talk reviews some methodological and mathematical aspects of numerical simulations of Bose-Einstein condensates modeled by nonlinear Schrödinger equations. The first part is devoted to the computation of stationary states by a novel gradient flow for the nonlinear eigenvalue problem. The flow is based on an energy inner product that depends on time through the density of the flow itself. Its time discretization yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. The practical relevance of the approach will be demonstrated by numerical experiments involving localized eigenstates in disorder potentials and vortex lattices in rotating traps. The second part addresses the numerical analysis of classical time-stepping schemes for dynamical simulations under very low regularity assumptions that are compatible with rough, oscillatory ans disordered potentials. We prove convergence with rates for the mass- and energy conserving variant of the Crank-Nicolson time discretization scheme due to Sanz-Serna. While for sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step and the spatial mesh size, the sharpness of the rates and the necessity of some coupling condition will be discussed in a sequence of numerical experiments.

[1] P. Henning and D. Peterseim. Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with disorder potentials. M3AS 27(11):2147–2184, 2017.
[2] P. Henning and D. Peterseim. Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency. Arxiv e-prints 1812.00835, 2018.
[3] R. Altmann, P. Henning, D. Peterseim Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials. Arxiv e-prints 1803.09950, 2018.
[4] R. Altmann, D. Peterseim. Localized computation of eigenstates of random Schrödinger operators. Arxiv e-prints 1902.00730, 2019.
17:15 On supercritical defocusing dispersive equations outside a ball
held by Piero D'Ancona (Sapienza Univ. Rome, Italy)
I consider the defocusing semilinear wave and Schrödinger equations, with a power nonlinearity, defined on the outside of the unit ball of $\mathbb{R}^{n}$, and with Dirichlet conditions at the boundary. The power is assumed to be sufficiently large, $p>O(n)$, and the space dimension is 3 or larger. Even in the radial case, the corresponding problem on $\mathbb{R}^{n}$ is completely open. Here I construct a family of large global solutions, whose data are small perturbations of radial initial data in suitable weighted Sobolev norms of higher order. These solutions are unique in the family of energy class solutions satisfying an energy inequality.
18:30 Dinner

The talks will take place in seminar room 1.067 and the dinner is located in the atrium of building 20.30 at KIT.


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