1. 373th Wilhelm und Else Heraeus Seminar
Anomalous Hamilton-Jacobi equation, long-tailed waiting time distributions, and travelling waves
Sergei Fedotov
School of Mathematics
The University of Manchester UK
with Vicenc Mendez of Barcelona University
Madrid 2003, 24 April
373th Wilhelm und Else Heraeus Seminar
2019/8/5

2. 373th Wilhelm und Else Heraeus Seminar
Contents
373th Wilhelm und Else Heraeus Seminar
2019/8/5

3. Reaction-transport equations of Fisher-KPP type
The classical Fisher-KPP equation
Example:
the memory effect
the long-range interaction
Fractional Fisher-KPP equation,
(Del Castillo-Negrete, Yuste )
373th Wilhelm und Else Heraeus Seminar
2019/8/5

4. CTRW theory and governing equation for the mesoscopic density of particles
the joint pdf for jump lengths and waiting times
the reaction rate term
the survival probability
the waiting time pdf
2019/8/5

5. Associated integro-differential equation
the memory effect
the reaction rate term
the long-range interaction
What about a travelling wave solution to this equation?
373th Wilhelm und Else Heraeus Seminar
2019/8/5

6. Travelling wave solution for integro-differential equation
with the initial condition:
Travelling wave solution:
Geometric approach
Hyperbolic scaling:
373th Wilhelm und Else Heraeus Seminar
2019/8/5

7. Eikonal equation for the position of front
The Hamilton-Jacobi equation:
for
Equation for the position of a reaction front:
Large deviations theory (Mark Freidlin):
If H does not depend on x
then the propagation rate u:
If we know the Hamiltonian H(p)
we know the propagation rate !!!!
373th Wilhelm und Else Heraeus Seminar
2019/8/5

8. Simple example: the Fisher-KPP equation
Mark Freidlin
The Hamilton-Jacobi equation:
hyperbolic scaling
potential energy
reaction rate parameter
Action for the Fisher-KPP equation:
Position of the reaction front:
Propagation rate: If
then an acceleration of front, jumps, etc.
373th Wilhelm und Else Heraeus Seminar
2019/8/5

9. 373th Wilhelm und Else Heraeus Seminar
Reaction-transport equations and corresponding Hamilton-Jacobi equations
The Hamiltonian:
The Hamilton-Jacobi equation:
the Laplace transform
The Hamilton-Jacobi equation:
373th Wilhelm und Else Heraeus Seminar
2019/8/5

10. Continuous-time random walks and Hamilton-Jacobi equation
joint pdf for jump lengths and waiting times
Moment generating functions:
Very important equation for us !!!!:
373th Wilhelm und Else Heraeus Seminar
2019/8/5

11. Anomalous Hamilton-Jacobi equation
Example: long-tailed waiting time density:
waiting time density:
Anomalous Hamilton-Jacobi equation: If
The propagation rate:
373th Wilhelm und Else Heraeus Seminar
2019/8/5

12. 373th Wilhelm und Else Heraeus Seminar
Reaction-transport equations with memory, long-range interactions and transmutations
Two balance equations:
Phys. Rev. E, 2004
373th Wilhelm und Else Heraeus Seminar
2019/8/5

13. System of integro-differential equations
373th Wilhelm und Else Heraeus Seminar
2019/8/5

14. Conclusions
We developed a Hamilton-Jacobi technique for the problem of wave propagation involving memory effects and long-range interactions. In particular, we found an explicit expression for the speed of propagating fronts for anomalous transport.
Non-universal behaviour: the macroscopic dynamics of the front depend on the choice of the underlying statistics for mesoscopic transport processes; CLT does not work since it involves parabolic scaling; an appropriate tool for finding the tails is the large deviation theory
2019/8/5