The One Dimensional Focusing Nonlinear Schrödinger Equation and Their Regular Solutions

We consider the classical one-dimensional focusing nonlinear Schrödinger equation (NLS).We give a brief overview of the techniques used to solve it since its formulation. Thenwe present various results obtained by the author in recent years. First, we constructrepresentations of the solutions in terms of Wronskians. Then we degenerate solutionsgiven in the frame of an algebro-geometric approach, using Riemann theta functions toget solutions in terms of Fredholm determinants of order 2N depending on 2N − 2 realparameters. We deduce representations of the solutions in terms of Wronskians of order 2Nwith elementary trigonometric functions depending on 2N − 2 real parameters. We alsoconstruct quasi-rational solutions expressed as a quotient of 2N-order determinants. Wecall these solutions N-order solutions, and they depend on 2N − 2 real parameters. Whenall these parameters are equal to zero, we recover the famous Peregrine breathers of orderN. These solutions therefore appear as deformations of Peregrine breathers of order N. Wedetermine the structure of the quasi-rational solutions to the (NLS) equation at order Nand prove that they can be written as a quotient of two polynomials of degree N(N + 1) inx and t multiplied by a time-dependent exponential. It is proven that the maximum of themodulus of the Peregrine breather of order N is equal to 2N + 1. We construct the explicitsolutions to the (NLS) equation up to order N = 13 depending on 2N − 2 real parametersand study the configurations of the modulus of these solutions in the plane of coordinates(x; t) in function of the different parameters.