Hierarchy of solutions to the NLS equation and multi-rogue waves

The solutions to the one dimensional focusing nonlinear Schrodinger equation (NLS) are given in terms of determinants. The orders of these determinants are arbitrarily equal to 2N for any nonnegative integer N and generate a hierarchy of solutions which can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N(N + 1) in x and t. These solutions depend on 2N - 2 parameters and can be seen as deformations with 2N - 2 parameters of the Peregrine breather PN : when all these parameters are equal to 0, we recover the PN breather whose the maximum of the module is equal to 2N+1. Several conjectures about the structure of the solutions are given.