scholarly journals Spectral Curves for the Derivative Nonlinear Schrödinger Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1203
Author(s):  
Aleksandr O. Smirnov
Keyword(s):  
Nonlinear Optics ◽  
Spectral Data ◽  
Nonlinear Waves ◽  
Fourier Transforms ◽  
Spectral Curve ◽  
Evolutionary Equations ◽  
Nonlinear Schrödinger ◽  
Spectral Curves ◽  
The Third ◽  
Nonlinear Signal

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.

scholarly journals Spectral curves are transcendental

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
H. W. Braden
Keyword(s):  
Spectral Curve ◽  
Spectral Curves ◽  
Arithmetic Properties ◽  
A Charge

AbstractSome arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge $$n\ge 2$$ n ≥ 2 Euclidean BPS monopole is not defined over $$\overline{\mathbb {Q}}$$ Q ¯ if smooth.

scholarly journals Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Yun Wu ◽  
Zhengrong Liu
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Periodic Waves ◽  
The Third ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
Fourth Kind

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equationut+au2+bu4ux+γuxxx+δuxyy=0. We reveal four kinds of interesting bifurcation phenomena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.

Detection of the Third Heart Sound Based on Nonlinear Signal Decomposition and Time–Frequency Localization

2016 ◽  
Vol 63 (8) ◽  
pp. 1718-1727 ◽  
Author(s):  
Shovan Barma ◽  
Bo-Wei Chen ◽  
Wen Ji ◽  
Seungmin Rho ◽  
Chih-Hung Chou ◽  
...  
Keyword(s):  
Heart Sound ◽  
Signal Decomposition ◽  
Time Frequency ◽  
The Third ◽  
Third Heart Sound ◽  
Time Frequency Localization ◽  
Nonlinear Signal

Compactness of Minimizing Sequences in Nonlinear Schrödinger Systems Under Multiconstraint Conditions

2014 ◽  
Vol 14 (1) ◽  
Author(s):  
Norihisa Ikoma
Keyword(s):  
Nonlinear Optics ◽  
Orbital Stability ◽  
Minimizing Sequences ◽  
Minimizing Sequence ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Schrödinger System

AbstractIn this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrödinger system which appears in the field of nonlinear optics. As a consequence of the compactness of each minimizing sequence, the orbital stability of the set of all minimizers is obtained.

scholarly journals Generalized solutions describing singularity interaction

2002 ◽  
Vol 29 (8) ◽  
pp. 481-494 ◽  
Author(s):  
V. G. Danilov
Keyword(s):  
Nonlinear Waves ◽  
New Method ◽  
Generalized Solutions ◽  
Evolutionary Equations

We present a new method for constructing solutions to nonlinear evolutionary equations describing the propagation and interaction of nonlinear waves.

scholarly journals Novel Approaches to Smoothing and Comparing SELDI TOF Spectra

2005 ◽  
Vol 1 ◽  
pp. 117693510500100 ◽  
Author(s):  
Sreelatha Meleth ◽  
Isam-Eldin Eltoum ◽  
Liu Zhu ◽  
Denise Oelschlager ◽  
Chandrika Piyathilake ◽  
...  
Keyword(s):  
Spectral Analysis ◽  
Fourier Transforms ◽  
Area Under The Curve ◽  
Maximum Intensity ◽  
Data Sets ◽  
Intensity Level ◽  
Prominent Feature ◽  
Data Set ◽  
The Third ◽  
Novel Approaches

Background Most published literature using SELDI-TOF has used traditional techniques in Spectral Analysis such as Fourier transforms and wavelets for denoising. Most of these publications also compare spectra using their most prominent feature, ie, peaks or local maximums. Methods The maximum intensity value within each window of differentiable m/z values was used to represent the intensity level in that window. We also calculated the ‘Area under the Curve’ (AUC) spanned by each window. Results Keeping everything else constant, such as pre-processing of the data and the classifier used, the AUC performed much better as a metric of comparison than the peaks in two out of three data sets. In the third data set both metrics performed equivalently. Conclusions This study shows that the feature used to compare spectra can have an impact on the results of a study attempting to identify biomarkers using SELDI TOF data.

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