Two parameters wronskian representation of solutions of nonlinear Schrödinger equation, eighth Peregrine breather and multi-rogue waves

In this paper, we present a representation of solutions of the one dimensional focusing nonlinear Schrödinger equation as a quotient of two wronskians depending on two parameters. Here, we give the complete proof of this representation. We have already constructed Peregrine breathers and their two parameter deformations until order 7. With this method, the construction of the explicit analytical expressions of Peregrine breather of order 8 was made for the first time. When parameters a or b are equal to 0, we recover the Peregrine breather of order 8; we obtain multi-rogue waves by deformation of parameters a and b. These expressions enable us to understand the evolution of the solutions. In the case of order 8, it is shown for high values of parameters a or b, the appearance of Peregrine breather of order 6.