Rational solutions of (1+1)-dimensional Burgers equation and their asymptotic

TL;DR

This paper derives new rational and radical solutions for the (1+1)-dimensional Burgers equation under specific initial conditions, analyzing their asymptotic behavior as viscosity approaches zero.

Contribution

It introduces explicit rational solutions for low viscosity and inviscid cases, with a novel focus on solutions valid for Reynolds numbers that are multiples of two.

Findings

Viscous solutions are rational functions for certain Reynolds numbers.

Inviscid solutions are expressed in radicals.

Asymptotic expansion matches inviscid case at high Reynolds number.

Abstract

A special initial condition for (1+1)-dimensional Burgers equation is considered. It allows to obtain new analytical solutions for an arbitrary low viscosity as well as for the inviscid case. The viscous solution is written as a rational function provided the Reynolds number (a dimensionless value inversely proportional to the viscosity) is a multiple of two. The inviscid solution is expressed in radicals. Asymptotic expansion of the viscous solution at infinite Reynolds number is compared against the inviscid case. All solutions are finite, tend to zero at infinity and therefore are physically viable.