We consider the Sturm–Liouville operator perturbed by a convolution integral on finite interval with Dirichlet boundary conditions and discontinuity in middle of interval. study inverse problem recovering term from spectrum. The is reduced to solving so-called main nonlinear equation singularity. To derive this equation, we do detailed analysis kernels transformation operators for integro-diffe...

The inverse spectral transform for the Zakharov-Shabat equation on the semi-line x > 0 is reconsidered as a Hilbert problem. The boundary data induce an essential singularity as k → ∞ to one of the basic solutions. Then solving the inverse problem means solving a Hilbert problem with particular prescribed behavior. It is demonstrated that the direct and inverse problems are solved in a consiste...

Recently, the string density problem, considered in the pioneering work of M. G. Kreĭn, has arisen naturally in connection with the Camassa-Holm equation for shallow water waves. In this paper we review the forward and inverse string density problems, with some numerical examples, and relate it to the Camassa-Holm equation, with special reference to multi-peakon/anti-peakon solutions. Under str...