Canonical ordinary differential equations for special functions of main, edge and corner wave catastrophes
by
Andrew S. Kryukovsky, Dmitry S. Lukin, Dmitry V. Rastyagaev
Dept. of Phys & Math Problems of Wave, Processes, MIPT, Institutsky per., 9, Dolgoprudny, Moscow Reg., 141700, Russia

Special functions of wave catastrophes (SWC) form the new class of special functions. There are leading terms of the uniform asymptotic solutions of partial (pseudo-) differential equations in the vicinity of singularities of the catastrophe types. Every SWC is a reference solution (in internal coordinates), which type depends on the type of the main, edge, corner or generalized edge catastrophe. Special functions of wave catastrophes are multiple integrals of fast oscillating exponential functions, which exponents are polynomials of the certain degree K (K-determined) named as universal unfoldings of singularities. Universal unfolding parameters are the SWC arguments. An integral multiplicity is defined by a singularity corank. In the case of the main SWC the integral multiplicity coincide with the catastrophe corank. In the case of the edge or corner catastrophe the multiplicity is one or two more than corank of the catastrophe in the maximum singularity restriction. Any SWC is a solution of a linear canonical system of differential equations of the second order. For the main SWC these equation systems are homogeneous, in the case of the edge SWC the SWC– restriction is contained on the right of one of the equations, for corner SWC these equations are two and so on. It is found out that canonical partial differential equation systems could be transformed in ordinary differential equation (ODE) systems in both the special function and its first derivatives. To receive these ODE systems the property of completeness of basic monomials defining perturbations of the universal unfoldings of singularities (catastrophes) have been used. Thus SWC calculation problem is reduced to standard methods of the ODE system calculation. The results of theoretical investigations are illustrated by numerical calculations of amplitude and phase structures of main, edge and corner SWC.

Date received: March 1, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caky-01.