Lattice, Delay and Functional Differential Equations

McGill University,

Montreal, Quebec

Tony Humphries,

CRM Applied Mathematics Laboratory

McGill University

This workshop which took place in the math department of McGill University at Burnside Hall was focused on recent developments in the analysis and computation of lattice, delay and functional differential equations, with particular emphasis on advanced-retarded differential delay equations which define travelling wave solutions to lattice differential equations, and related problems. See below for the programme and links to the presentations, where available.

9:30-10:30 | Dimitri Breda, Udine |

"Pseudospectral approximation of
eigenvalues of derivative operators with non-local boundary
conditions" | |

Abstract (pdf) | |

Bonus: Here is Dimitri's Applied math seminar from 18th April: "Linear time delay systems: from characteristic roots... ...to stability charts" | |

10:30-11:00 | Coffee |

11:00-12:00 | Dmitry Pelinovsky |

"Bifurcations of travelling wave solutions in the discrete NLS lattices" | |

We study discrete NLS equations, which include the cubic NLS lattice with on-site interactions and the integrable Ablowitz--Ladik lattice. Standing wave solutions are known to exist in the discrete NLS equations outside of the spectral band. We study travelling wave solutions which have nonlinear resonances with unbounded spectral bands. We apply methods of normal forms and center manifold reductions and show that a continuous NLS equation with the third-order derivative term is a canonical normal form for the discrete NLS equation. Bifurcations of travelling breathers are analyzed in the framework of the third-order derivative NLS equation. | |

12:00-2:00 | Lunch |

2:00-3:00 | Pietro-Luciano Buono, UOIT |

"Restrictions and unfolding of local bifurcations in delay-
differential equations modelling biological phenomena" | |

I am going to present some delay-differential equation models of biological phenomena from physiology, neural networks and drug release. I will discuss the occurrence of Hopf and double Hopf bifurcations in these models and the restrictions on the dynamics depending on the number of delays present in the model. I will also present some results on the linear unfolding of these systems at Hopf and double Hopf bifurcations. | |

3:00-3:30 | Brian Moore |

"Bistable Waves for Differential-Difference Equations
with Inhomogeneous Diffusion" | |

Traveling wave solutions for lattice differential equations are defined by boundary value problems with advances and delays on an unbounded domain. Using a specific bistable nonlinearity for a spatially discrete Nagumo equation with a diffusion coefficient that is varied in some interval on the lattice, one is able to obtain explicit solutions by way of transform techniques. Of particular interest is the issue of propagation failure of these traveling waves depending on the inhomogeneous media through which they propagate, and recent results on this problem are illustrated in this talk. | |

3:30-4:00 | Coffee |

4:00-5:00 | Tony Humphries |

"Model Problems and Truncation of Advanced-Retarded
FDEs arising in Lattice Travelling Wave Problems" | |

The analysis and computation of Advanced-Retarded Functional Differential Equations is complicated by the presence of advances as well as delays. As a first step before computing a numerical solution the problem is usually approximated on a bounded domain. This truncation can be done in several ways, but the process is not well studied or understood. We will discuss the issues that arise, including the need for and construction of good model test problems with known solutions. | |

5:00-5:30 | Roy Wilds |

"A new method for finding traveling wave solutions to lattice
differential equations" | |

I will present a new method for obtaining traveling wave solutions in 1-dimensional Lattice Differential Equaitons (LDEs). The new approach is based on embedding the original problem in a related one which can be solved. Making use of available theory, we find that there exists a correspondence of solutions between the new approach and the original problem for a certain class of LDEs. Using the discrete Nagumo problem as a test case, numerical investigations will be presented which demonstrate the utility of the new method. |

[Applied Mathematics] [Mathematics and Statistics] [McGill University]

[CRM Applied Mathematics Laboratory] [CRM]