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

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

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

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

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

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

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

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

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

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

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

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

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

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