Dave Smith

Assistant Professor of Science (Mathematics), Yale-NUS College

Workshop: Dispersive quantisation via the unified transform method

Overview

Dispersive quantisation was first observed by Talbot, a physical scientist and pioneer of photography [H. F. Talbot, Facts related to optical science. No. IV, Philos. Mag. 9 (1836), 401–407]. It describes a curious pattern formation / interference effect which appears in wave equations, including the celebrated time-dependent linear Schroedinger equation (LS), as first observed by Talbot, and the linearised Korteweg-de Vries equation (LKdV), as first observed by Olver [Dispersive Quantization, Amer. Math. Monthly 117 7 (2010), 599–610]. Specifically, at rational times the solution appears as a certain linear combination of spatial shifts of the initial datum, while at irrational times the solution assumes a fractal profile.

It is known that dispersive quantisation occurs for LS under a broad class of boundary conditions. For LKdV, only the simplest (periodic) boundary conditions have yet been analysed, and dispersive quantisation is observed in that case, but LKdV has not been studied with other types of boundary conditions. All the existing results rely upon the application of the Fourier series method but, in order to analyse more complicated boundary conditions for LKdV, a more advanced tool is required. The unified transform method has been shown to solve LKdV with all boundary conditions that make sense [D. A. Smith, Well-posed two-point initial-boundary value problems with arbitrary boundary conditions, Math. Proc. Cambridge Philos. Soc. 152 3 (2012), 473-496]. Therefore, it is expected that this method may be used to decide the appearance or nonappearance of dispersive quantisation in LKdV with general boundary conditions, and even to give a description of the mathematical mechanism for the effect.

Workshop details

Please contact the local organiser, Dave Smith, if you are interested in participating.

Confirmed Participants

  • Peter Olver, University of Minneapolis.
  • Beatrice Pelloni, Heriot-Watt University.
  • Dave Smith, Yale-NUS College.

Dates

Mon 6 to Fri 10 January 2020.

Venue

Kewalram Chanrai Room, Elm College, Yale-NUS College, Singapore.

The college is located in the UTown part of the main NUS campus, close to the junction of Clementi Road and Dover Road. The college is well served by local buses to "University Town", "New Town Sec Sch", "Aft Clementi Ave 1" and "Yale-NUS College".

There are a wide range of restaurants and canteens in UTown, within 10 minutes' walk of the conference venue.

Campus map. Google map.

Hotel

No rooms have been reserved at any hotels, but a good option is the Park Avenue Hotel, Rochester Drive. From the hotel, you can walk 200m north, cross the road and to bus stop "Buona Vista Stn Exit D", then catch bus 196 to stop "University Town"

Support

The organisers thank Yale-NUS College and the Yale-NUS College Workshop B grant for organisational and financial support of this workshop.

It is possible that there may be some support available for graduate students to attend the workshop. Please contact the organiser for more details.