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International Journal of Mathematics and Mathematical Sciences
Volume 2003, Issue 49, Pages 3123-3142

Darboux transformation for classical acoustic spectral problem

1Department of Mathematics, Kaliningrad State Technical University, Sovetsky Prospect 1, Kaliningrad 236000, Russia
2Department of Theoretical Physics, Kaliningrad State University, 14 Aleksandra Nevskogo Street, Kaliningrad 236041, Russia
3School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK

Received 3 January 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows steps similar to those for the Schrödinger operator. However, there is no one-to-one correspondence between the two problems. The technique developed enables one to construct new families of integrable potentials for the acoustic problem, in addition to those already known. The acoustic problem produces a nonlinear Harry Dym PDE. Using the technique, we reproduce a pair of simple soliton solutions of this equation. These solutions are further used to construct a new positon solution for this PDE. Furthermore, using the dressing-chain approach, we build a modified Harry Dym equation together with its LA pair. As an application, we construct some singular and nonsingular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.