Green's functions and the Cauchy problem of the Burgers hierarchy and forced Burgers equation

TL;DR

This paper derives explicit Green's functions for the Burgers hierarchy with time-dependent coefficients, explores their properties, and applies them to solve forced Burgers equations and related stochastic processes.

Contribution

It provides closed-form Green's functions for the Burgers hierarchy of arbitrary order and connects these to solutions of forced Burgers equations and stochastic processes.

Findings

Explicit Green's functions expressed as hypergeometric functions.

Time-dependent solutions for the forced Burgers equation.

Connections established between Burgers hierarchy, Schrödinger, and Fokker-Planck equations.

Abstract

We consider the Cauchy problem for the Burgers hierarchy with general time dependent coefficients. The closed form for the Green's function of the corresponding linear equation of arbitrary order $N$ is shown to be a sum of generalised hypergeometric functions. For suitably damped initial conditions we plot the time dependence of the Cauchy problem over a range of $N$ values. For $N=1$, we introduce a spatial forcing term. Using connections between the associated second order linear Schr\"{o}dinger and Fokker-Planck equations, we give closed form expressions for the corresponding Green's functions of the sinked Bessel process with constant drift. We then apply the Green's function to give time dependent profiles for the corresponding forced Burgers Cauchy problem.