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The One Dimensional Focusing Nonlinear Schrödinger Equation and Their Regular Solutions

Year: 2026
Type: article
Abstract: 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. Then we present various results obtained by the author in recent years. First, we construct representations of the solutions in terms of Wronskians. Then we degenerate solutions given in the frame of an algebro-geometric approach, using Riemann theta functions to get solutions in terms of Fredholm determinants of order 2N depending on 2N − 2 real parameters. We deduce representations of the solutions in terms of Wronskians of order 2N with elementary trigonometric functions depending on 2N − 2 real parameters. We also construct quasi-rational solutions expressed as a quotient of 2N-order determinants. We call these solutions N-order solutions, and they depend on 2N − 2 real parameters. When all these parameters are equal to zero, we recover the famous Peregrine breathers of order N. These solutions therefore appear as deformations of Peregrine breathers of order N. We determine the structure of the quasi-rational solutions to the (NLS) equation at order N and prove that they can be written as a quotient of two polynomials of degree N(N + 1) in x and t multiplied by a time-dependent exponential. It is proven that the maximum of the modulus of the Peregrine breather of order N is equal to 2N + 1. We construct the explicit solutions to the (NLS) equation up to order N = 13 depending on 2N − 2 real parameters and study the configurations of the modulus of these solutions in the plane of coordinates (x; t) in function of the different parameters.
Source: Mathematics
DOI: 10.3390/math14163024
Language: English
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7203896232
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Cites: 26
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