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Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

4th July 2022 to 16th December 2022
Michael Shearer
Gennady El
Mark Hoefer
Antonio Moro
Barbara Prinari

Programme image acknowledgments 

LeftSolitary wave transmission through a dispersive shock wave in a viscous fluid conduit, Boulder, CO, USA © Mark Hoefer, 2016.
RightUndular bore on the Severn river near Gloucester, UK © Mark Humpage, 2007.


Programme Theme

Dispersive hydrodynamics has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media, encompassing both dynamic and stochastic aspects of wave propagation. Recent theoretical and experimental developments have opened up new areas for research, with intriguing open issues in both theory and applications. These include the understanding of fundamental regularisation mechanisms of hydrodynamic singularities via the generation of dispersive shock waves (DSWs) and related phenomena. Physical examples of dispersive hydrodynamic phenomena include undular bores on rivers, in the ocean and atmosphere, nonlinear diffraction patterns in optics and quantum fluids, turbulence in fibre lasers and superfluids.

The mathematical programme weaves together research topics on integrable and nonintegrable dispersive partial differential equations, hyperbolic conservation laws, convex and nonconvex dispersive hydrodynamic systems. Numerous physical applications of dispersive hydrodynamics in geophysics, nonlinear optics, superfluids and magnetic materials will be explored. The programme is designed to encourage interactions amongst mathematicians, physicists, and engineers specialising in the analysis of dispersive systems, experiments and numerical simulations. Special emphasis is given to involving young researchers and to diversity among the participants.

The programme is organised around four overlapping themes:

1. Modulation theory and dispersive shock waves

  • Whitham modulation theory and dispersive shock waves (DSWs) in integrable and non-integrable systems
  • DSWs in non-convex and multi-dimensional systems
  • Singular limits of nonlinear dispersive and diffusive-dispersive waves
  • Nonlinear theory of modulational instability
  • Numerical methods:  structure-preserving geometric numerical integration, high-order methods

2. Analysis of dispersive hydrodynamic systems

  • Inverse scattering transform (IST) and related techniques for rigorous analyses of integrable PDEs with non-decaying initial conditions
  • Riemann-Hilbert techniques for semi-classical integrable equations
  • Boundary value problems for integrable equations
  • Stability of nonlinear periodic travelling waves and rigorous developments of modulation theory
  • Numerical realisations of IST based Riemann-Hilbert problem analyses

3. Random phenomena and dispersive hydrodynamics

  • Integrable turbulence and soliton gas
  • Rogue waves and other extreme events
  • Connections with statistical mechanics and random matrix models
  • Application in oceanography, nonlinear optics, and quantum superfluid turbulence
  • Monte Carlo numerical methods

4. Physical applications

  • Fluid dynamics:  oceanic and atmospheric internal/surface undular bores, interfacial waves
  • Spatial and temporal nonlinear optical media
  • Quantum fluids:  Bose-Einstein condensates, ultracold gases, superfluid turbulence
  • Magnetic materials
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons