Nonlocal integrable PDEs from hierarchies of symmetry laws: The example of Pohlmeyer–Lund–Regge equation and its reflectionless potential solutions

By following the ideas presented by Fukumoto and Miyajima in Fukumoto and Miyajima (1996) we derive a generalized method for constructing integrable nonlocal equations starting from any bi-Hamiltonian hierarchy supplied with a recursion operator. This construction provides the right framework for the application of the full machinery of the inverse scattering transform. We pay attention to the Pohlmeyer–Lund–Regge equation coming from the nonlinear Schrödinger hierarchy and construct the formula for the reflectionless potential solutions which are generalizations of multi-solitons. Some explicit examples are discussed.