On Exponential Asymptotics for Nonseparable Wave Equations III: Approximate Spectral Bands of Periodic Potentials on Strips

1992 ◽  
Vol 52 (3) ◽  
pp. 730-745 ◽  
Author(s):  
R. E. Meyer ◽  
M. C. Shen
Keyword(s):  
Wave Equations ◽  
Spectral Bands ◽  
Periodic Potentials ◽  
Exponential Asymptotics

scholarly journals A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

2019 ◽  
Author(s):  
Kenneth T-R McLaughlin ◽  
Patrik V Nabelek
Keyword(s):  
Vries Equation ◽  
Hilbert Problem ◽  
Kdv Equation ◽  
Unique Determination ◽  
Existence Of A Solution ◽  
Spectral Bands ◽  
Periodic Potentials ◽  
De Vries ◽  
Explicit Time Dependence ◽  
Riemann Hilbert Problem

Abstract We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials as a Riemann–Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann–Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potentials evolve according to the Korteweg–de Vries Equation (KdV) equation, we use integrability to derive an associated Riemann–Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann–Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann–Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation.

scholarly journals Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials

2013 ◽  
Vol 131 (2) ◽  
pp. 149-178 ◽  
Author(s):  
Sean D. Nixon ◽  
T. R. Akylas ◽  
Jianke Yang
Keyword(s):  
Two Dimensional ◽  
Periodic Potentials ◽  
Exponential Asymptotics

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Causal Probabilistic Network and Power Spectral Estimation Used in Sleep Stage Classification

1997 ◽  
Vol 36 (04/05) ◽  
pp. 41-46
Author(s):  
A. Kjaer ◽  
W. Jensen ◽  
T. Dyrby ◽  
L. Andreasen ◽  
J. Andersen ◽  
...  
Keyword(s):  
Sleep Stage ◽  
Spectral Estimation ◽  
Interrater Agreement ◽  
Probabilistic Networks ◽  
Probabilistic Network ◽  
Spectral Bands ◽  
Power Spectral ◽  
Stage Classification ◽  
Sleep Stage Classification ◽  
The Brain

Abstract.A new method for sleep-stage classification using a causal probabilistic network as automatic classifier has been implemented and validated. The system uses features from the primary sleep signals from the brain (EEG) and the eyes (AOG) as input. From the EEG, features are derived containing spectral information which is used to classify power in the classical spectral bands, sleep spindles and K-complexes. From AOG, information on rapid eye movements is derived. Features are extracted every 2 seconds. The CPN-based sleep classifier was implemented using the HUGIN system, an application tool to handle causal probabilistic networks. The results obtained using different training approaches show agreements ranging from 68.7 to 70.7% between the system and the two experts when a pooled agreement is computed over the six subjects. As a comparison, the interrater agreement between the two experts was found to be 71.4%, measured also over the six subjects.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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