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.
Dynamics and exact solutions of the generalized Harry Dym equation
