Two exact solutions of the Tzitzeica-Bullough-Dodd equation

Abstract

The Tzitzeica-Bullough-Dodd equation (TBD) occurs in many different fields, ranging from geometry of surfaces to gas dynamics. Two simple and direct methods, namely, the Hirota bilinear technique and the separation of variables approach, are employed to calculate expressions for a 4-soliton and doubly periodic arrays of vortices respectively. In terms of applications, the dependent variable in TBD can assume the role of a stream function in two dimensional hydrodynamics. The flow patterns associated with the latter solution will accordingly be periodic in two spatial directions. The stability of these toroidal flows is tested numerically by a semi-Lagrangian code developed by one of the authors (DG). The patterns appear to be stable and relax to an equilibrium configuration where the vorticity-stream function relationship is different from the conventional hyperbolic sine. © Freund Publishing House Ltd.