# Description of the transient processes in waveguides with the multi-contour saddle point method

WHTW02 - WHT Follow on: the applications, generalisation and implementation of the Wiener-Hopf Method

A.V. Shanin, A.I. Korolkov, K.S. Kniazeva A problem of transient processes description in a waveguide is considered. The waveguide is supposed to be closed and homogeneous in the longitudinal direction x. The standard approach to this problem is to represent the field as a double Fourier integral over frequencies ω and wave numbers k and apply the residue theorem to obtain the field representation in the form of a sum of integrals. We propose to transform the letter representation into the integral over some contour on a complex manifold, which is embedded into the space of two complex variables ω and k. The manifold is the analytical continuation of the waveguide dispersion diagram. We find asymptotical estimation of the integral for large x and fixed x/t (t is time). For this purpose, we build a modification of the saddle point method. The modified saddle point method, or a multi-contour method, implies deformation of the integration contour on the manifold. In case the complex manifold is too complex, the contour deformation is difficult. Instead we propose to consider a set of the problems depending parametrically on real x/t, find a set of saddle points for all the problems, and classify the saddle points on contributing (active) and not contributing (not active) to the field. Following this plan, one obtains continuous branches of active saddle points on the analytically continued dispersion diagram, these branches form pulses, whose contribution to the field can be significant. The set of the saddle points for all real x/t is called a carcass of the dispersion diagram. Active carcass branches complement the usual dispersion diagram with real ω and k to form a set of points (ω, k) corresponding to the waves, which can propagate in the waveguide. We claim that waveguides can be classified by the type of their carcass. Here we show this technique for a number of model waveguides.

This talk is part of the Isaac Newton Institute Seminar Series series.