Partial differential equations on graphs
This project with
Annie Rak started in the summer 2016
as a
HCRP project.
The topic is ``differential equations on graphs".
We explored in the summer 2016 first various dynamical systems on networks.
The topic of discrete PDEs allows for expository, experimental as well as theoretical
investigations in an area of mathematics which links discrete
mathematics with dynamical system theory and discrete stochastic processes
(as PDEs related to the Laplacian are always closely related to diffusion
processes). Here are some topics we played with:
- Heat and wave equations u'=-Lu, u''=-Lu
- Linear and nonlinear ave equations like u''=-Lu+sin(u)
- Geometric evolution in the geometry of graphs like PDEs on forms
- Transport and advection consensus dynamics like u'=-L_{+}u on a directed graph.
- Burger and fluid like equations u'= T(u,u) with quadratic difference operator
- Integrability, like Sine-Gordon which generalizes Volterra-Toda like systems.
Annie got HCRP funding also for the Fall of 2016 and used the topic
for her senior thesis (which will be submitted in November 2016).
I will add a bit more about Annie's specific work later, once the thesis
is finished. It is more focused on advection dynamics and its
relation with discrete stochastic processes defined on the graph.
PDE on graphs: Some discussion notes from Summer 2016 [PDF].
The following pictures show are some trajectories of the Sine-Gordon equation
u'' = -L u + sin(u)
on the graph K
_{2}, where the energy surface is
three dimensional and appears to be foliated by two-dimensional
tori:
The system appears to be integrable. It is a Hamiltonian system in a
four dimensional phase space with a three
dimensional energy surface. The orbits appear to be located on two dimensional
tori which indicates integrability. Nobody seems have looked at this system
on graphs although. If this case is integrable, it strongly suggests that in general, the
Sine-Gordon equation
u''=-Lu+sin(u)
is integrable for a general finite simple graph with Kirchhoff
Laplacian L. We of course know it for the one point graph, where it
is the mathematical pendulum. We also know that for the circular graph,
equilibria belong to Frenkel-Kantorova configurations.
We eventually did not get into integrability questions in this project.