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Seminar on Partial Differential Equations
Fall 2022
All talks in ZOOM, Friday at 2:00 PM
Click here for previous seminars of Fall 2021
| September 2 |
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| September 9 |
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| September 16 |
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| September 23 |
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| September 30 |
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| October 7 |
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| October 14 |
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| October 21 |
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| October 28 |
Dr. Maria Teresa Chiri (Queen's University)
Title: Controlling the spread of invasive biological species
Abstract: We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes' formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. Then we introduce a family of optimization problems for a moving set and show how these can be derived from the original parabolic problems, by taking a sharp interface limit. In connection with moving sets, we show some results on controllability, existence of optimal strategies, and necessary conditions.
This is a joint work with Prof. Alberto Bressan (Penn State University) and Dr. Najmeh Salehi (Saint Mary's College).
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| November 4 |
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| November 11 |
Dr. Ross Parker (SMU)
Title: Bright and dark multi-solitons in Hamiltonian systems
Abstract: We consider the existence and spectral stability of multi-pulse solitary wave solutions in two Hamiltonian systems: a nonlinear Schrödinger equation which incorporates both fourth and second-order dispersion terms (NLS4) and the fifth-order Korteweg-De Vries equation (KdV5). For NLS4, we first show that a discrete family of bright multi-pulse solutions exists, which is characterized by the distances between consecutive copies of the the primary solitary wave. We then reduce the spectral stability problem to computing the determinant of a matrix which is, to leading order, block diagonal. Under additional assumptions, which can be verified numerically and are sufficient to prove orbital stability of the primary solitary wave, we show that all bright multi-solitons are spectrally unstable. We then look at a similar problem for KdV5 on a periodic domain, and show that brief instability bubbles form when eigenvalues collide on the imaginary axis. Finally, we return to NLS4, this time to the dark soliton regime. Since dark solitons decay to a nonzero background, we impose either Neumann or periodic boundary conditions. Numerical results suggest that, in contrast to bright multi-solitons, dark multi-solitons can be spectrally neutrally stable. In addition, eigenvalue collisions on the imaginary axis produce similar instability bubbles to those found in KdV5. Results of numerical timestepping experiments are shown for all systems, and these are interpreted using the spectral computations.
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| November 18 |
Dr. Bao-Feng Feng (The University of Texas Rio Grande Valley)
Title: Algebraic structure and soliton solutions of integrable PDEs
Abstract:
In this talk, I will show some common features of integrable PDEs. Starting from
a discrete Kadomtsev-Petviashvili (KP) equation or the Hirota-Miwa equation, I will show the KP, modified KP and two-dimensional Toda-lattice hierarchies can be generated via Miwa transformation. By further reductions, the KdV, modified KdV, sine-Gordon and the NLS equations , together with their various soliton solutions, can be obtained. As an example, I will show how the rogue wave solutions to the NLS equations are derived.
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| November 25 |
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| December 2 |
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Small changes/shifts in the dates may
be possible.
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