# Theoretical and Computational Aspects of Nonlinear Surface Waves (16w5112)

Arriving in Banff, Alberta Sunday, October 30 and departing Friday November 4, 2016

## Organizers

Mark Groves (Saarland University)

Guyenne Philippe (University of Delaware)

Emilian Parau (University of East Anglia, UK)

Erik Wahlen (Lund University)

## Objectives

This workshop will be organised around five main themes related to important mathematical and numerical issues about nonlinear waves arising in a variety of free-surface problems. These issues are among the list of open problems that were highlighted during the concluding session of the 2014 `Theory of Water Waves' program at the Isaac Newton Institute. Because the problems under consideration (water waves, hydroelastic waves, hydromagnetic waves, etc.) are similar in nature, they use similar formulations and thus can be addressed by closely related methods. The participants will reflect the multidisciplinary nature of these problems, including mathematical analysts, applied mathematicians, numerical analysts as well as fluid dynamicists, oceanographers and engineers. The main objective is to promote the direct interaction between experts from different communities and the cross-fertilization of ideas among them with a focus on the five themes to be described next. We believe this will pave the way for further significant progress and breakthroughs in the theory, numerics and applications of these problems.

1. Initial-value problems

There is now a wealth of local well-posedness results for various versions of the water-wave problem, each relying upon a different formulation. Global well-posedness results are also available for three-dimensional gravity waves on water of infinite depth, as well as global or almost global well-posedness results for two-dimensional gravity waves on water of infinite depth (depending on the initial data). Very recently, theories of singularity formation due to wave breaking and overturning have been developed. At the same time, a number of accurate and efficient numerical methods have been developed to solve the full Euler equations for irrotational water waves on finite or infinite depth, based on boundary integral equations, conformal mappings and Taylor series expansions. The main task consists in evaluating the Dirichlet--Neumann operator associated with the fluid domain. Special effort has also been devoted to designing numerical schemes that preserve important properties of the mathematical system such as energy conservation.

Specific topics include:

2. Coherent structures

Recent advances in this area include the development -- from a base of almost zero knowledge -- of existence theories for travelling waves with vorticity, standing waves (waves which are periodic in space and time) and three-dimensional surface waves in the context of the water-wave problem. Various types of coherent structures in two and three dimensions have been computed in the last few years using boundary integral equations and spectral methods.

Specific topics include:

3. Stability of coherent structures

One crucial question facing all developments, both analytical and numerical, in the theory of surface waves is that of stability: Once one has found a solution of interest (e.g., a traveling or standing wave) will it be observable in the laboratory setting? Will it be found in the open ocean? There are now a range of rigorous mathematical linear instability results for two-dimensional periodic wavetrains and solitary waves on deep water (Benjamin-Feir, superharmonic instabilities) and two-dimensional waves under three-dimensional perturbations. Conditional stability results for solitary waves are also available (`conditional' since they hold only over the unknown, possibly small, existence times of solutions).

Specific topics include:

4. Model equations

An important direction of inquiry in the general field of nonlinear waves is the development and application of simplified model equations. In asymptotic regimes where small parameters can be defined, many such models can be derived at various orders of approximation. Their interest lies in the fact that they are usually more tractable analytically and numerically than the full equations. As a consequence, they have been a popular tool in the engineering community where they have been used with varying degrees of success. The well-posedness as well as rigorous justification of model equations are fundamental mathematical questions that have led to an activity surge recently. Such analysis is crucial in determining their precise range of validity. Their numerical simulation also requires sophisticated numerical methods that are suited to their mathematical structure. In addition, some models may exhibit blow-up which calls for careful analysis and computation.

Specific topics include:

5. Waves in domains with complex geometry

In view of real-world applications, it is crucial to consider wave problems in domains with complex geometry or complex boundary conditions, e.g. surface waves propagating over topography near the shore or generated by a wavemaker in a laboratory basin. The modeling of surface wave propagation under wind forcing and bottom friction also poses many theoretical challenges. Other fluid examples include hydroelastic waves in ice of variable thickness and hydromagnetic waves under the influence of a multidirectional magnetic field. These effects add considerably to the technical difficulties of the problem and also require special treatment in the numerical schemes.

Specific topics include:

1. Initial-value problems

There is now a wealth of local well-posedness results for various versions of the water-wave problem, each relying upon a different formulation. Global well-posedness results are also available for three-dimensional gravity waves on water of infinite depth, as well as global or almost global well-posedness results for two-dimensional gravity waves on water of infinite depth (depending on the initial data). Very recently, theories of singularity formation due to wave breaking and overturning have been developed. At the same time, a number of accurate and efficient numerical methods have been developed to solve the full Euler equations for irrotational water waves on finite or infinite depth, based on boundary integral equations, conformal mappings and Taylor series expansions. The main task consists in evaluating the Dirichlet--Neumann operator associated with the fluid domain. Special effort has also been devoted to designing numerical schemes that preserve important properties of the mathematical system such as energy conservation.

Specific topics include:

- Global well-posedness results for gravity waves on water of finite depth, and for gravity-capillary water waves (finite or infinite depth)

- Well-posedness results for interfacial, hydroelastic and hydromagnetic waves

- Global well-posedness results for solutions in a neighbourhood of solitary waves, to enable the development of a satisfactory stability theory

- Development and optimization of numerical schemes to efficiently solve the three-dimensional initial-value problem on such computing platforms as GPUs or high-performance supercomputers that enable high machine precision.

- Development of numerical schemes to accurately compute two- and three-dimensional solutions with multivalued profiles such as overturning waves, and solutions exhibiting singularities like cusps or contact points.

- Development of discontinuous Galerkin methods that preserve the symplectic structure of the Hamiltonian equations for nonlinear water waves.

2. Coherent structures

Recent advances in this area include the development -- from a base of almost zero knowledge -- of existence theories for travelling waves with vorticity, standing waves (waves which are periodic in space and time) and three-dimensional surface waves in the context of the water-wave problem. Various types of coherent structures in two and three dimensions have been computed in the last few years using boundary integral equations and spectral methods.

Specific topics include:

- Theory and numerics for large-amplitude travelling waves

- Three-dimensional standing waves (waves which are periodic in time and space)

- The effect of vorticity in three spatial dimensions

- Construction and computation of breathers

- Construction and computation of coherent structures for interfacial, hydroelastic and hydromagnetic waves.

3. Stability of coherent structures

One crucial question facing all developments, both analytical and numerical, in the theory of surface waves is that of stability: Once one has found a solution of interest (e.g., a traveling or standing wave) will it be observable in the laboratory setting? Will it be found in the open ocean? There are now a range of rigorous mathematical linear instability results for two-dimensional periodic wavetrains and solitary waves on deep water (Benjamin-Feir, superharmonic instabilities) and two-dimensional waves under three-dimensional perturbations. Conditional stability results for solitary waves are also available (`conditional' since they hold only over the unknown, possibly small, existence times of solutions).

Specific topics include:

- Development of nonlinear (in)stability theories

- Dynamic stability of geniunely three-dimensional patterns (experiments and oceanographic observation show that surface water waves with hexagonally-shaped surfaces exist)

- Development of numerical continuation methods based on transformed field expansions and the Ablowitz-Fokas-Musslimani formulation to investigate the spectral stability of two- and three-dimensional periodic capillary-gravity or hydroelastic waves.

4. Model equations

An important direction of inquiry in the general field of nonlinear waves is the development and application of simplified model equations. In asymptotic regimes where small parameters can be defined, many such models can be derived at various orders of approximation. Their interest lies in the fact that they are usually more tractable analytically and numerically than the full equations. As a consequence, they have been a popular tool in the engineering community where they have been used with varying degrees of success. The well-posedness as well as rigorous justification of model equations are fundamental mathematical questions that have led to an activity surge recently. Such analysis is crucial in determining their precise range of validity. Their numerical simulation also requires sophisticated numerical methods that are suited to their mathematical structure. In addition, some models may exhibit blow-up which calls for careful analysis and computation.

Specific topics include:

- Derivation and justification of two- and three-dimensional models with vorticity in the long-wave (e.g. Green-Naghdi equations) and modulational regimes (e.g. NLS equations).

- Variational methods for deriving model equations based on Luke's Lagrangian and Zakharov's Hamiltonian formulations.

- Development of efficient high-order finite-difference, finite-volume and finite-element methods to solve Boussinesq and Green-Naghdi equations for operational applications in coastal engineering.

- Justification of kinetic models from weak turbulence theory.

- Derivation and justification of physically relevant model equations for hydroelastic and hydromagnetic waves in the long-wave and modulational regimes.

5. Waves in domains with complex geometry

In view of real-world applications, it is crucial to consider wave problems in domains with complex geometry or complex boundary conditions, e.g. surface waves propagating over topography near the shore or generated by a wavemaker in a laboratory basin. The modeling of surface wave propagation under wind forcing and bottom friction also poses many theoretical challenges. Other fluid examples include hydroelastic waves in ice of variable thickness and hydromagnetic waves under the influence of a multidirectional magnetic field. These effects add considerably to the technical difficulties of the problem and also require special treatment in the numerical schemes.

Specific topics include:

- Mathematical formulation for water waves with vorticity over topography.

- New two- and three-dimensional models for wind forcing and wave breaking.

- Development of well-balanced and positivity-preserving numerical schemes to accurately solve shallow-water, Boussinesq and Green-Naghdi equations for surface water waves over topography.

- Accurate modeling and computation of ship waves for complex hull geometries.

- Inverse problems of reconstructing the bottom topography from surface measurements and reconstructing the water surface from bottom pressure measurements.