Master projects for Mats Ehrnstrom

Equations as those mentioned above have a Hamiltonian energy of the form \[ \int ( u L u + N(u) ) dx, \] where \(N\) is an anti-derivative of \(n\), and \[ \int u^2 dx \] is a further conserved quantity for solutions. By minimising the Hamiltonian in an open ball in a Sobolev space, under the constraint that the second conserved quantity, which is the square of the \(L^2\)-norm, is fixed, one finds critical points that are in fact solutions to the equation. In variants to this, one fixes a different part of the energy, and instead maximises an energy. In both cases, the resulting waves have their main mass localised in space, so-called solitary waves.

Prerequisites: Basic knowledge of at least two of L^p-spaces, Sobolev spaces, functional analysis.

• Arnesen: Existence of solitary-wave solutions to nonlocal equations. Discrete Contin. Dyn. Syst. 36 (2016), 3483–3510.
• Hildrum: Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity. Nonlinearity 33 (2020), 1594–1624.
• Arnesen, Ehrnstrom, Stefanov: A maximisation technique for solitary waves: the case of the nonlocally dispersive Whitham equation. arXiv:2303.14036

Steady periodic waves in water-wave equations are most commonly constructed by perturbing the linear problem. This is local bifurcation: starting from a parameterised family of trivial solutions, such as running water with speed as the parameter, one can find certain values of the parameter where the implicit function theorem breaks down, and a different curve of solutions crosses the trivial one, yielding small-amplitude sinusoidal solutions. Continuing such a curve to larger solutions is called global bifurcation. In this project one analyzes small and larger periodic solution of water wave equations, using functional analysis and elliptic estimates.

Prerequisites: Fourier series; Functional analysis, alternatively god understanding of Linear methods; ODEs or PDEs.

• Ehrnstrom, Kalisch. Global bifurcation for the Whitham equation. Math. Model. Nat. Phenom. 8 (2013), 13–30.
• Aasen, Varholm. Traveling gravity water waves with critical layers. J. Math. Fluid Mech. 20 (2018), 161–187.
• Hildrum, Xue. Periodic Hölder waves in a class of negative-order dispersive equations. J. Differential Equations 343 (2023), 752–789.

The study of highest waves is about the existence, regularity and precise properties of solutions to water wave equations attaining a largest possible height. Not all equations possess such solutions: in water waves, it is primarily the gravitational force that enforces such a maximal height, and speed, of surface waves. The mathematical objective is mainly a regularity study, using ad hoc and very precise estimates to determine the singularity arising at the top of the wave, the possible angle or cusple, and further properties such as convexity between consequtive crests.

Prerequisites: Fourier series; Functional analysis, alternatively god understanding of Linear methods; handiness in analytic estimates. May be combined with, or reshaped as, a numerical project.

• Ehrnstrom, Wahlén: On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation. Ann. Inst. H. Poincaré C Anal. Non Linéaire 36 (2019), 1603–1637.
• Ehrnstrom, Mæhlen, Varholm: On the precise cusped behaviour of extreme solutions to Whitham-type equations. arXiv:2302.08856. In print in Ann. Inst. H. Poincaré C Anal. Non Linéaire.
• Ørke: Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations. arXiv:2201.13159. In print in Arkiv för MAtematik.

Dispersive PDEs model surfaces and media that are such that a wave’s velocity depends on its length. Water surfaces are one such medium, and there are numerous dispersive PDEs modelling water waves. A characteristic feature of many water wave models is that a wave can break (as water waves often do near the shore). Mathematically, we say that a function \(u\), depending on time \(t\) and space \(x\), breaks at time \(T\) if \(u_x\) becomes unbounded as t approaches T while u remains bounded. Weakly dispersive PDEs take the form \(\partial_t u + \partial_x u^2 = K*u\), where \(K\) is a convolution kernel in space, which is odd, \(K(x) = K(-x)\), and absolutely integrable. It is expected that all such equations feature wave breaking, but it has only been proved for the case where \(K\) is monotone for positive \(x\). This project generalizes the result to all \(K\) by employing new methods. It is also possible to extend the result to a periodic setting for more general dispersive PDEs. This project is in co-supervision with Ola Mæhlen, UiO. Prerequisites: The mathematics rely on the method of characteristic, fixed-point iteration, and operator splitting. * Mæhlen, Xue: One-sided Hölder regularity of global weak solutions of negative order dispersive equations. J. Differential Equations364(2023), 412–455.

Most water waves are symmetric. For travelling water waves, it has been proved a number of times, that under a large set of assumptions, all waves are ‘bell-shaped’, rising and falling once in each minimal period around a symmetric crest line. When surface tension is accounted for, there are however more complicated wave patterns, including waves with more than one crest in each minimal period. Recently, it was shown for the first time that periodic waves of that kind can be extended to asymmetric instances for the Whitham equation, a model equation for water waves. In this project, you walk through the quite intricate construction of such waves, but also try to find and extend them numerically along selected bifurcation curves or sheets. The project will combine theoretical and numerical aspects of mathematics, but allows the focus placed on either. For example, it is possible to extend this project to include interval arithmetic and computer-assisted proofs for higher wave numbers for a more numerical approach or to study the combinatorial construction for other equations for a more theoretical approach.

• On the bifurcation diagram of the capillary-gravity Whitham equation. Water Waves (2019) 1:275–313. Mats Ehrnström, Mathew A. Johnson, Ola I. H. Maehlen and Filippo Remonato.
• Asymmetric traveling wave solutions of the capillary-gravity Whitham equation. SIAM J. Math. Anal. (2024): 56:8096–8124. Ola Mæhlen and Douglas Svensson Seth.

A different problem than to solve a PDE, is how to construct one from data. This inverse problem is called data-driven PDEs, and includes both the study of a class of equations; how to ‘build’ them from potentially complex and noisy data; and how to check their validity and robustness. The latter can be done for example by solving the generated equations forward in time to compare with earlier unused parts of the data. The whole approach can be performed on general data, from data believed to describe a certain process, or — in the case of testing and verification – from solutions to a known system of differential equations.

Project 1: Nonlocal equations from periodic data In this project you use a recently developed Fourier-based method to build nonlocal PDEs from water-wave observations. Whereas in the earlier work, a local equation was studied by the means of localised waves, we here aim at finding a truly nonlocal equation by the means of periodic data. The work involves both analytical and numerical/coding skills, and lies on the verge of applied, theoretical and experimental fundemental science.

Project 2: Robustness for noise Noise is a common problem in all experimental sciences, and may have large implications on inverse problems in mathematics. Part of this challenge is inherent: it is not a priori clear what is noise, and what is a true signal. Another part is technical: exactly which procedure should be used to clear out the noise from the signal? We aim here to test the method developed in the below arXiv paper with respect to robustness towards noise. More generally, a goal is to explore more general approaches for noise reduction in the Fourier-based equation discovery method.

Source:

• arXiv: 2511.05486. Rediscovering shallow water equations from experimental data. Kjell S. Heinrich, Douglas S. Seth, Mats Ehrnstrom, Simen Å. Ellingsen

When water waves propagate over areas with varying currents or depth, the wave characteristics change. The aim of this project is to combine Fourier- and wavelet analysis with object tracking and feature extraction, and develop a method to estimate these changes from video observations of waves.

This project is in co-supervision with Adrian Kirkeby, Simula Research Laboratory.

Prerequisites: Knowledge of one or more of the following topics: partial differential equations, wave propagation, Fourier analysis, programming (Python, Julia, or Matlab). The project can be more mathematically or programming oriented, depending on the students preferences.

In co-supervision with Ingelin Steinsland.