Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2002-09-23
International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 49, pp. 3123-3142
Nonlinear Sciences
Exactly Solvable and Integrable Systems
16 pages; keywords Darboux (Moutard) transformation, Classical acoustic spectral problem, Reflexionless potentials, Solitons
Scientific paper
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 Schr\"{o}dinger 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 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.
Rudnev Misha
Yurov Artyom V.
Yurova Alla A.
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