Deformations of third-order Peregrine breather solutions of the nonlinear Schrödinger equation with four parameters

We present a new representation of solutions of the one-dimensional nonlinear focusing Schrödinger equation (NLS) as a quotient of two determinants. This formulation gives in the case of the order 3, new solutions with four parameters. This gives a very efficient procedure to construct families of quasirational solutions of the NLS equation and to describe the apparition of multirogue waves. With this method, we construct analytical expressions of four-parameters solutions; when all these parameters are equal to 0, we recover the Peregrine breather of order 3. It makes possible with this four-parameters representation, to generate all the types of patterns for the solutions, like the triangular configurations or the ring structures.