Beals Richard, Sattinger David H, Szmigielski Jacek
Department of Mathematics, Yale University, New Haven, CT 06520-8283, USA.
Philos Trans A Math Phys Eng Sci. 2007 Sep 15;365(1858):2299-312. doi: 10.1098/rsta.2007.2010.
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 stronger assumptions, the Camassa-Holm spectral problem and the string density problem can be transformed to the Schrödinger spectral problem and its inverse problem. Recent results exploiting this transformation are reviewed briefly.
