Henrik Kalisch

Filippo Remonato, Henrik Kalisch

The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth. In this work, the effect of incorporating capillarity into the Whitham equation is in focus. The capillary Whitham equation is a nonlocal equation similar to the usual Whitham equation, but containing an additional term with a coefficient depending on the Bond number T which measures the relative strength of capillary and gravity effects on the wave motion. A spectral collocation scheme for computing approximations to periodic traveling waves for the capillary Whitham equation is put forward. Numerical approximations of periodic traveling waves are computed using a bifurcation approach, and a number of bifurcation curves are found. Our analysis uncovers a rich structure of bifurcation patterns, including subharmonic bifurcations, as well as connecting and crossing branches. Indeed, for some values of the Bond number T, the bifurcation diagram features distinct branches of solutions which intersect at a secondary bifurcation point. The same branches may also cross without connecting, and some bifurcation curves feature self-crossings without self-connections.

Henrik Kalisch, Daulet Moldabayev, Olivier Verdier

In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling-wave solutions of nonlinear dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} + \mathcal{L} u_x = 0, \end{equation*} where $\mathcal{L}$ is a self-adjoint operator, and $f$ is a real-valued function with $f(0) = 0$. The SpectraVVave code uses a continuation method coupled with a spectral projection to compute approximations of steady symmetric solutions of this equation. The code is used in a number of situations to gain an understanding of traveling-wave solutions. The first case is the Whitham equation, where numerical evidence points to the conclusion that the main bifurcation branch features three distinct points of interest, namely a turning point, a point of stability inversion, and a terminal point which corresponds to a cusped wave. The second case is the so-called modified Benjamin-Ono equation where the interaction of two solitary waves is investigated. It is found that is possible for two solitary waves to interact in such a way that the smaller wave is annihilated. The third case concerns the Benjamin equation which features two competing dispersive operators. In this case, it is found that bifurcation curves of periodic traveling-wave solutions may cross and connect high up on the branch in the nonlinear regime.

Artur Matysiak, Volker Roeber, Henrik Kalisch et al.

Artificial neural networks (ANNs) have evolved from the 1940s primitive models of brain function to become tools for artificial intelligence. They comprise many units, artificial neurons, interlinked through weighted connections. ANNs are trained to perform tasks through learning rules that modify the connection weights. With these rules being in the focus of research, ANNs have become a branch of machine learning developing independently from neuroscience. Although likely required for the development of truly intelligent machines, the integration of neuroscience into ANNs has remained a neglected proposition. Here, we demonstrate that designing an ANN along biological principles results in drastically improved task performance. As a challenging real-world problem, we choose real-time ocean-wave prediction which is essential for various maritime operations. Motivated by the similarity of ocean waves measured at a single location to sound waves arriving at the eardrum, we redesign an echo state network to resemble the brain's auditory system. This yields a powerful predictive tool which is computationally lean, robust with respect to network parameters, and works efficiently across a wide range of sea states. Our results demonstrate the advantages of integrating neuroscience with machine learning and offer a tool for use in the production of green energy from ocean waves.