Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications (15w5045)

Arriving in Banff, Alberta Sunday, May 17 and departing Friday May 22, 2015


(University of Colorado, Boulder)

(SUNY Buffalo)

Gennady El (Loughborough University)

(University of Colorado, Boulder)


The main objectives of the proposed meeting are:

  1. To bring together experts from a range of backgrounds: from pure mathematicians to experimental physicists -- to enable the exchange of ideas for recent and emerging problems of dispersive hydrodynamics.
  2. To facilitate breakthroughs in several directions of dispersive hydrodynamic research such as multidimensional DSWs, the mathematical description of DSWs in nonlocal media, integrable turbulence and the quantitative, experimental verification of existing DSW theories.

There are many fundamental mathematical problems emerging in the study of multiscale, dispersive hydrodynamic phenomena. The workshop will focus on the following topical directions of research:

  1. Analytical and numerical description of multidimensional DSWs in KP or other integrable and nonintegrable equations. An extension of the existing theory for one-dimensional DSWs to two spatial dimensions is a long-standing problem posing a number of mathematical challenges. Despite recent progress in the understanding of pre-breaking KP dynamics as well as careful numerical simulations of some multidimensional dispersive regularization problems, breakthroughs in this area still lie ahead. A closely related direction which has been under active development in recent years is the theory of integrable hydrodynamic type systems in higher dimensions.
  2. Modulation theory for hydrodynamic systems with non-strict hyperbolicity/linear degeneracy regularized by dispersive terms. The majority of existing DSW studies are related to systems which can be characterized as genuinely nonlinear, strictly hyperbolic conservation laws modified by small dispersion terms. The DSWs generated in such systems represent dispersive counterparts of classical, Lax shocks. At the same time, there exists a rich mathematical theory of nonclassical shock waves in hydrodynamic systems lacking genuine nonlinearity and/or strict hyperbolicity. The study of purely dispersive counterparts of nonclassical, dissipative shock resolutions is one of the outstanding problems in dispersive hydrodynamics. There have been several recent important works in this direction but a unified mathematical framework is yet to be developed.
  3. Rigorous analysis of DSWs and their interactions via the inverse scattering transform (IST). Along with providing rigorous justification of the results from modulation theory, the IST method yields the description of many subtle details not captured by the formal modulation analysis. A number of fundamental results have recently been obtained in the rigorous description of DSW generation and interaction problems for the KdV and defocusing NLS equations. The extension to other integrable systems (modified KdV, vector NLS, etc.) is needed.
  4. Incoherent wave structures in dispersive hydrodynamics. This is a new, rapidly developing direction involving the mathematical description of a range of macroscopically incoherent wave structures in dispersive hydrodynamics. The novel lines of investigation include integrable turbulence, soliton gases, incoherent DSWs, effectively viscous fluid dynamics in multidimensional dispersive hydrodynamics systems, and nonlinear wave counterparts of quantum mechanical effects in disordered media (e.g. Anderson localization). A closely related range of dispersive-hydrodynamic problems attracting recent interest concerns the formation of rogue waves in shallow and deep water as well as in nonlinear optical systems.
  5. Universality of wave breaking in dispersive hydrodynamics. The rigorous mathematical description of the emergence of a DSW in the space-time vicinity of a gradient catastrophe is of fundamental importance for the foundations of dispersive hydrodynamics. There has been notable progress in proving Dubrovin's Universality conjecture for a number of integrable systems, with many deep, fascinating results obtained in this direction. There are still many unanswered questions, especially related to Universality in non-integrable (including viscously modified) systems.
  6. DSWs in focusing media, in nonlocal media and in systems with higher order and nonlinear dispersion. DSW phenomena are not limited to hyperbolic systems modified by weak dispersive terms. The dispersive resolutions of gradient catastrophes can also occur in focusing media as well as in media characterized by nonlocal nonlinearities and higher order dispersive mechanisms. Relevant mathematical models include the focusing NLS equation and its modifications due to non-locality, as well as systems with higher order/nonlinear dispersion arising in nonlinear optics, shallow-water dynamics and other areas. The main mathematical approach in the analysis of the dispersive regularization of gradient catastrophe in the integrable focusing NLS equation is the powerful Riemann-Hilbert steepest descent method, which has enabled a number of recent significant advances in the analytical description of semi-classical "elliptic" dynamics. The outstanding problems in this area include the investigation of higher-order breaking dynamics for analytical data and the evolution of non-analytic data. Of special interest are the recently proposed possible connections between the generation of multiperiodic breather type structures in the resolution of focusing singularities and rogue wave formation. The description of DSWs in nonlocal media (e.g. in liquid crystals) and in systems with higher order/nonlinear dispersion (e.g. fully nonlinear shallow water, nonlinear optics with higher order dispersion) is of great importance for applications and has only recently begun to be explored, posing a number of challenging mathematical questions.
  7. Quantitative comparison of results of DSW experiments with existing theories. Despite the numerous existing theoretical results in the area of DSWs, the quantitative experimental verification of existing theory is presently lacking. There are only a handful of experimental results confirming certain features of DSW dynamics but no detailed comparison is available. Such a comparison, however, is vital for the further development of dispersive hydrodynamics as a fundamental discipline that stands on its own much like classical, viscous fluid dynamics.

A meeting of this scale that brings together an international collection of leading experts in the mathematical and physical description of dispersive hydrodynamic phenomena has not occurred before. The two recent, smaller, more focused meetings at SISSA, Trieste (2012) and CIRM, Marseille (2013) on the subject of DSWs have been highly successful. They have clearly demonstrated the vital necessity for the further exchange of new ideas, especially between representatives of different directions in the study of multiscale nonlinear coherent structures in dispersive media. It is a unique feature of this proposed workshop that it will bring together an exceptionally broad spectrum of top researchers with mutually complementing expertise, to make possible further breakthroughs in dispersive hydrodynamics.