Abstract
Currently, in nonlinear optics, models associated with various types of the nonlinear Schrödinger equation (scalar (NLS), vector (VNLS), derivative (DNLS)), as well as with higher and mixed equations from the corresponding hierarchies are usually studied. Typical tools for solving the problem of propagation of optical nonlinear waves are the forward and inverse nonlinear Fourier transforms. One of the methods for reconstructing a periodic nonlinear signal is based on the use of spectral data in the form of spectral curves. In this paper, we study the properties of the spectral curves for all the derivatives NLS equations simultaneously. For all the main DNLS equations (DNLSI, DNLSII, DNLSIII), we have obtained unified Lax pairs, unified hierarchies of evolutionary and stationary equations, as well as unified equations of spectral curves of multiphase solutions. It is shown that stationary and evolutionary equations have symmetries, the presence of which leads to the existence of holomorphic involutions on spectral curves. Because of this symmetry, spectral curves of genus g are covers over other curves of genus M and N=g−M, where M is a number of phase of solutions. We also showed that the number of the genus g of the spectral curve is related to the number of phases M of the solution of one of the two formulas: g=2M or g=2M+1. The third section provides examples of the simplest solutions.
Highlights
The main tools for the study of nonlinear optical signals are the forward and inverse nonlinear Fourier transforms [1,2,3,4,5], and the main models of nonlinear optics are the scalar, vector, and derived nonlinear Schrödinger equations, as well as their higher forms from the corresponding hierarchies
A key feature of these equations is the fact that they are integrable nonlinear evolutionary differential equations
Integrable nonlinear equations can usually be obtained as conditions for the compatibility of two linear differential equations, called a Lax pair
Summary
The main tools for the study of nonlinear optical signals are the forward and inverse nonlinear Fourier transforms [1,2,3,4,5], and the main models of nonlinear optics are the scalar, vector, and derived nonlinear Schrödinger equations, as well as their higher forms from the corresponding hierarchies. Is simultaneously the solution of the Equation (1) (see [8]) In this case, the equation of the spectral curve associated with this matrix Q( x ) has the form det(νI − M) = R(ν, λ) = 0,. Symmetry 2021, 13, 1203 from the condition of compatibility of the Equations (1) and (7) we obtain an integrable evolutionary nonlinear equation from the corresponding hierarchy. That is, using the structure of the monodromy matrix, we can construct the corresponding hierarchy of integrable nonlinear equations. Let us note that there are gauge transformations that transform these equations into each other and preserve the magnitude of the solution (see, for example, [16,21,29,30,31]) Each of these nonlinear equations corresponds to its own matrix U.
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