Darboux transformation for classical acoustic spectral problem

A Yurova, A Yurov, M Rudnev

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)


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 the steps, similar to those for the Schrodinger operator. However, there is no one-to-one correspondence between the tow 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 non-linear 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 non-singular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.
Translated title of the contributionDarboux transformation for classical acoustic spectral problem
Original languageEnglish
Pages (from-to)3123 - 3142
JournalInternational Journal of Mathematics and Mathematical Sciences
Publication statusPublished - 2003


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