POLE DYNAMICS AND OSCILLATIONS FOR THE COMPLEX BURGERS-EQUATION IN THE SMALL-DISPERSION LIMIT

A meromorphic solution to the Burgers equation with complex viscosity is analysed. The equation is linearized via the Cole-Hopf transform wh ich allows for a careful study of the behaviour of the singularities o f the solution. The asymptotic behaviour of the solution as the disper sion coefficient tends to zero is derived. For small dispersion, the t ime evolution of the poles is found by numerically solving a truncated infinite-dimensional Calogero-type dynamical system. The initial data are provided by high-order asymptotic approximations of the poles at the critical time t(s) for the dispersionless solution via the method of steepest descents. The solution is reconstructed using the pole exp ansion and the location of the poles. The oscillations observed via th e singularities are compared to those obtained by a classical stationa ry phase analysis of the solution as the dispersion parameter epsilon --> 0(+). A uniform asymptotic expansion as epsilon --> 0(+) of the di spersive solution is derived in terms of the Pearcey integral in a nei ghbourhood of the caustic. A continuum limit of the pole expansion and the Calogero system is obtained, yielding a new integral representati on of the solution to the inviscid Burgers equation.