Complex-valued Burgers and KdV-Burgers equations

TL;DR

This paper investigates complex-valued solutions of Burgers and KdV-Burgers equations, demonstrating blow-up phenomena and establishing conditions for global convergence and regularity of solutions.

Contribution

It provides explicit initial data leading to blow-up and proves global convergence and regularity for solutions under mild conditions.

Findings

Existence of initial data causing blow-up at large times

Global convergence of series solutions

Regularity results for solutions with mild initial conditions

Abstract

Spatially periodic complex-valued solutions of the Burgers and KdV-Burgers equations are studied in this paper. It is shown that for any sufficiently large time T, there exists an explicit initial data such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of series solutions is established for initial data satisfying mild conditions.