We consider the scattering transform for Schrödinger operators with energy dependent potentials. We prove unique invertibility of the transform when there are no bound states and find a simplified recovery formula. We construct as a special case a one-soliton "breather". As an application we prove a global existence theorem for a class of non-linear partial differential equations.
David H. Sattinger. Jacek Szmigielski. "Energy dependent scattering theory." Differential Integral Equations 8 (5) 945 - 959, 1995. https://doi.org/10.57262/die/1369056038