Forced Integrable Systems - The Case of Sine-Gordon Equation by Vasudeva Murthy

Explore forced integrable systems through a 40-minute lecture focusing on the sine-Gordon equation. Delve into the inhomogeneously driven sine-Gordon equation and its applications to tectonic stress transfer and one-dimensional crack dynamics. Examine solutions for kinks and breathers under various boundary conditions, and investigate finite difference methods for heat equations with nonlocal boundary conditions. Learn about the generalized Dirichlet-to-Neumann map for nonlinear evolution PDEs. Gain insights into future research directions and participate in a Q&A session to deepen understanding of these complex mathematical concepts.

Forced integrable systems: the case of sine-Gordon equation
Revisiting the inhomogeneously driven sine-Gordon equation
Dynamics of a one-dimensional crack with variable friction
Linear stress-drop model
Sine-Gordon equation and its application to tectonic stress transfer
Dynamics of a one-dimensional crack
Figure 2: Solution of inhomogeneous sG equation with constant and Heaviside forcing for different times
Kink with No Flux Boundary Conditions:
Kink with Non-reflecting Boundary Conditions:
Acritical vs C Variation over different domain size
Breather with Non-Reflecting Boundary Conditions:
Breather with No Flux Boundary Conditions:
Variation of Acritical with C:
Conclusions
Future?
Finite Difference Methods for the heat Equation with a Nonlocal Boundary Condition
Nonlinear problems?
Theory
Keller & Hagstrom 1986
The technique
Need to do the same for the sine-Gordon equation
The Generalized Dirichlet-to-Neumann Map for Certain Nonlinear Evolution PDFs
Q&A