Forward looking session on the theory of water waves

On the afternoon of 6 August, the programme held a forward looking session, chaired by Tom Bridges, Mark Groves and Paul Milewski.  The session started with a backward looking session, reviewing and discussing the topics covered during the programme: water-wave stability (periodic waves in finite depth with surface tension, vorticity, and very shallow water; the Benjamin-Feir instability from many points of view; modulation instability for non-local operators; transverse instability of gen solitary waves; cascades induced by BF instability); large amplitude waves (experimentally generated, singularities of highest steady waves, splash singularities, role of vorticity, self-similarity); numerics (HOPS methods, high-performance numerics for standing waves, BEM & spectral methods, FEM & Galerkin methods, symplectic time integration, and CFD); 3D waves (lump solitary waves, 3D localized solitary waves, multi-periodic patterns); analysis and validity (Birkhoff normal form, IVP, DNO, small divisors, the unified transform, inverse problems, comparison of NLS breathers to Euler breathers).  Other topics of discussion included jet currents, statistical approaches, tsunamis, bottom topography, droplet dynamics, wave-energy harvesting, wind-wave interaction, new model equations, and particle paths in waves.  A lively discussion was held on the relative merits of the various results and approaches.  It was agreed that the programme, and the present state of the art, barely scratched the surface of the wide range of problems of interest in the theory of water waves.  Topics and problems flagged up for future work include: post-instability dynamics; the importance of well-posedness particularly for truncated models; the need for global well-posedness to validate nonlinear stability of solitary waves; what models and equations are appropriate for breaking, splashing, entrainment, and bubbles; a hierarchy of models and formulations; a encyclopedia of known results (analytical, numerical, rigorous) on all steady and time-periodic waves and their stability; new numerical methods and comparison between different methods; any theoretical development on (a) wind-wave interaction, (b) three dimensional waves of every shape and form, (c) extreme waves, and (d) any type of breaking waves.

Bokhove lectures on variational principles, water waves and discretization

Onno Bokhove began his talk at the summer school by showing a demonstration of wave breaking in a Hele-Shaw cell.  A snapshot of a similar demonstration is shown to the left.  A variational principle and Hamiltonian formulation for the Hele-Shaw model was derived including weak dissipation and forcing.  The non-autonomy was handled by lifting the the Hamiltonian formulation to a Kamiltonian formulation.  A discretization based on discontinuous Galerkin in space, with symplectic integration in time was introduced, including a new third order symplectic time-integration scheme.  Numerical error tests, comparisons with the experiment as well as comparisons with wave tank experiments at MARIN were presented. Background material for the talk includes Gagarina, Ambati, van der Vegt, & Bokhove (2014) and Thornton, van der Horn, Gagarina, Zweers, van der Meer, & Bokhove (2014).  The slides from the lectures are available for downloading here. Lecture notes will be forthcoming.

Bridges lectures on modulation of water waves

Tom Bridges lectured at the summer school on “modulation of water waves“.  The lectures gave an introduction to how modulation of relative equilibria can lead to well-known model equations like KdV, KP, Boussinesq, and Gardner equations.  The advantage of this approach is that the modulation equations emerge in a “universal form” with the coefficients determined by derivatives of conservation laws evaluated on the basic state.  This strategy leads to a range of new applications for such model equations.  Background material includes Bridges (2012), Bridges (2013), and Bridges (2014). The slides from Parts 1 and 2 of the lectures can be downloaded here and here.  Lecture notes will be forthcoming.

Nachbin lectures on conformal mapping and complex topography

Andre Nachbin lectured at the summer school on conformal mapping and complex topography.  The motivation is to understand waves over rapidly-varying topographies, inspired by theory for acoustic waves in heterogeneous random media.  In the theory, the principal tool in conformal mapping is the Schwarz-Christoffel transformation and the Schwarz-Christoffel Toolbox.  The talk started with a review of previous work of Hamilton (1977) and Nachbin (2003).  Then a derivation of the Boussinesq model in shallow water in the mapping coordinates was derived, based on three small parameters: nonlinearity, dispersion, and a topography parameter.  A key feature of the reduced system is the metric M which appears as a coefficient in the reduced Boussinesq equation.  It is an approximation to the Jacobian along the mapped surface. The second part of the talk discussed implementation details of the theory, implications for water waves, and extension to 3D.  Background reading includes Fokas & Nachbin (2012). Slides from the lectures can be downloaded here.  Lecture notes will be forthcoming.

Wu lectures on well posedness and singularities of water waves

The second set of lectures at the summer school were given by Sijue Wu on well-posedness of the initial value problem.  The first part of the lectures was an introduction to local and global existence for 2D waves.  The moving domain was mapped to the lower half plane using a Riemann mapping.  Tools included the theory of holomorphic functions, the Hilbert transform on the real line, commutator estimates, product formulae for singular integrals, and harmonic analysis.  A key step was finding the right formulation to which analysis in Sobolev spaces could be applied directly.  The extension to 3D involved generalizing complex analysis to Clifford analysis, introducing an analogue of the Hilbert transform, and key estimates for singular integrals.  Key background work includes Wu (1997), Wu (1999), Wu (2009), and Wu (2011). Slides from the lectures can be downloaded here.  Lecture notes will be forthcoming.

Ambrose lectures on the IVP and vortex sheets: analysis and computation

The first lecture of the summer school was by David Ambrose.  He gave a wide ranging talk on aspects of the initial value problem, for irrotational waves in 2D and 3D using a geometric formulation of the water wave problem.  The first half of the lectures was on 2D waves.  Although useful for water waves, the setting was the more general case of two fluids of differing densities.  The talk started with a review of the paper of Hou, Lowengrub & Shelley (1994).  The aim of HLS was a numerical scheme, but the lectures showed how it is also a starting point for a rigorous analysis of the IVP.  The key is to evolve geometric properties of the surface (curvature, length) combined with careful analysis of the Birkhoff-Rott integral.  The second part of the lectures showed how to generalize to 3D.  Background work includes Ambrose & Masmoudi (2005) and Ambrose & Masmoudi (2007). The slides from the lectures are available for downloading here. Lecture notes will be forthcoming.

Summer School on the Theory of Water Waves

From Wednesday 6 August to Friday 8 August, the Institute is hosting a Summer School on the Theory of Water Waves.  The school includes 15 hours of lectures and a forward looking session, given by David Ambrose, Sijue Wu, Andre Nachbin, Onno Bokhove and Tom Bridges.  A group photo from the school is shown on the left (click on the photo for full size).

Transverse instability of generalized solitary waves

On Tuesday afternoon 5 August, Erik Wahlen talked about the transverse instability of generalized solitary waves (gSWs).  gSWs are periodic with finite-amplitude tails at infinity, and localized variation in the middle.  The principal model was a 6th-order KP equation which reduces to 5th-order KdV (Kawahara) in one space dimension.  The parameter range of interest was Froude number near unity, and Bond number positive and near 1/3.  He first recalled the result of Haragus & Scheel (2006) which proved spectral stability of periodic solutions of the Kawahara equation to one-dimensional perturbations.  Then showed that they are unstable to transverse perturbations.  In the linearization about gSWs this instability shows up as unstable essential spectrum.  Hence the principal result is the gSWs are unstable as solutions of the 6th-order KP equation.  In work in progress, a proof is being developed for instability of gSWs for the full water-wave problem using spatial dynamics in the transverse direction.  This project is joint work with Mariana Haragus.  A video of the talk is here.

Bouncing droplets on a free surface and the quantum analogy

Robert Brady visited the programme from the University of Cambridge on Tuesday 5 August, to speak on “Why bouncing droplets are a pretty good model for quantum mechanics”.  This talk follows up on the talk of Paul Milewski on Friday 1 August.  He reviewed the experiments of Yves Couder, and then presented an analysis based more on quantum principles rather than fluid dynamic modelling.  The principal equation was a wave equation in the circular dish with a amplitude dependent speed c, emphasizing the role of Lorentz invariance.  He pointed out an interesting analogy between the Bjerknes force on bubbles and the analogue in quantum mechanics, including a Bjerknes analogue of Planck’s constant.  Other quantum phenomena have analogues in this hydrodynamic experiment, such as double-slit diffraction and tunneling.  A preprint on this topic, Brady & Anderson (2014), can be found on the arXiv.  A related paper, finding quantum phenomena, such as the Aharonov-Bohm effect, in water waves is the paper Berry, Chambers, Large, Upstill & Walmsley (1980).  A video of the talk is here.

Comparison of 5 methods for computing the DNO for water waves

A question that came up in the forward looking session was the comparison of different numerical methods for computing in the context of water waves.  One example along these lines is the recent study of Wilkening & Vasan (2014) which compares 5 methods of computing the Dirichlet-Neumann operator for the water wave problem.  The five methods were Craig-Sulem, Ablowitz-Fokas-Musslimani (AFM), dual AFM, boundary integral collocation and transformed field expansion.  Two key test cases were large-amplitude standing waves in deep water, and a pair of interacting travelling waves.  Details of the comparison can be found in the arXiv paper linked above.