Stokes phenomenon and matched asymptotic expansions

This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

Three routes to the exact asymptotics for the one-dimensional quantum walk

We demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path integral representation. This is significantly easier to use than the Darboux method. It also provides a single integral representation for the wavefunction that works over the full range of positions, $n,$ including throughout the transitional range where the behaviour changes from oscillatory to exponential. Previous analyses of this system have run into difficulties in the transitional range, because the approximations on which they were based break down here. The fact that there are two different kinds of approach to this problem (Path Integral vs. Schr\"{o}dinger wave mechanics) is ultimately a manifestation of the equivalence between the path-integral formulation of quantum mechanics and the original formulation developed in the 1920s. We discuss how and why our approach is related to the two methods that have already been used to analyse these systems.Comment: 25 pages, AMS preprint format, 4 figures as encapsulated postscript. Replaced because there were sign errors in equations (80) & (85) and Lemma 2 of the journal version (v3

Asymptotics and special functions

Asymptotics and Special Function

Hard wall - soft wall - vorticity scattering in shear flow

An analytically exact solution, for the problem of lowMach number incident vorticity scattering at a hard-soft wall transition, is obtained in the form of Fourier integrals by using theWiener-Hopf method. Harmonic vortical perturbations of inviscid linear shear flow are scattered at the wall transition. This results in a far field which is qualitatively different for low shear and high shear cases. In particular, for high shear the pressure (apparently driven by the mean flow) does not decay and its Fourier representation involves a diverging integral which is to be interpreted in generalised sense. Then the incompressible hydrodynamic (Wiener-Hopf) "inner" solution is matched asymptotically to an acoustic outer field in order to determine the sound associated to the scattering. The qualitative difference between low and high shear is also apparent here. The low shear case matches successfully. In the high shear case only a partial matching was possible