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Discrete & Continuous Dynamical Systems - S

2012 , Volume 5 , Issue 5

Issue on recent progress on the long time behavior of coherent structures in discrete and continuous models

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Dmitry Pelinovsky, Milena Stanislavova and Atanas Stefanov
2012, 5(5): i-iii doi: 10.3934/dcdss.2012.5.5i +[Abstract](49) +[PDF](125.6KB)
Partial differential equations viewed as dynamical systems on an infinite-dimensional space describe many important physical phenomena. Lately, an unprecedented expansion of this field of mathematics has found applications in areas as diverse as fluid dynamics, nonlinear optics and network communications, combustion and flame propagation, to mention just a few. In addition, there have been many recent advances in the mathematical analysis of differential difference equations with applications to the physics of Bose-Einstein condensates, DNA modeling, and other physical contexts. Many of these models support coherent structures such as solitary waves (traveling or standing), as well as periodic wave solutions. These coherent structures are very important objects when modeling physical processes and their stability is essential in practical applications. Stable states of the system attract dynamics from all nearby configurations, while the ability to control coherent structures is of practical importance as well. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis of nonlinear equations of mathematical physics with a particular emphasis on existence and dynamics of localized modes. The unifying idea is to predict the long time behavior of these solutions. Three of the papers deal with continuous models, while the other three describe discrete lattice equations.

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Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations
Martina Chirilus-Bruckner, Christopher Chong, Oskar Prill and Guido Schneider
2012, 5(5): 879-901 doi: 10.3934/dcdss.2012.5.879 +[Abstract](134) +[PDF](521.6KB)
It is the purpose of this paper to prove error estimates for the approximate description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, like the Korteweg--de Vries (KdV) or the Nonlinear Schrödinger (NLS) equation. The proofs are based on a discrete Bloch wave transform of the underlying infinite-dimensional system of coupled ODEs. After this transform the existing proof for the associated approximation theorem for the NLS approximation used for the approximate description of oscillating wave packets in dispersive PDE systems transfers almost line for line. In contrast, the proof of the approximation theorem for the KdV approximation of long waves is less obvious. In a special situation we prove a first approximation result.
Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation
Hristo Genev and George Venkov
2012, 5(5): 903-923 doi: 10.3934/dcdss.2012.5.903 +[Abstract](103) +[PDF](486.8KB)
We study the Cauchy problem for the focusing time-dependent Schrödinger - Hartree equation $$i \partial_t \psi + \triangle \psi = -({|x|^{-(n-2)}}\ast |\psi|^{\alpha})|\psi|^{\alpha - 2} \psi, \quad \alpha\geq 2,$$ for space dimension $n \geq 3$. We prove the existence of solitary wave solutions and give conditions for formation of singularities in dependence of the values of $\alpha\geq 2$ and the initial data $\psi(0,x)=\psi_0(x)$.
The spectrum of travelling wave solutions to the Sine-Gordon equation
Christopher K. R. T. Jones and Robert Marangell
2012, 5(5): 925-937 doi: 10.3934/dcdss.2012.5.925 +[Abstract](60) +[PDF](445.9KB)
We investigate the spectrum of the linear operator coming from the sine-Gordon equation linearized about a travelling kink-wave solution. Using various geometric techniques as well as some elementary methods from ODE theory, we find that the point spectrum of such an operator is purely imaginary provided the wave speed $c$ of the travelling wave is not $\pm 1$. We then compute the essential spectrum of the same operator.
The Evans function and the Weyl-Titchmarsh function
Yuri Latushkin and Alim Sukhtayev
2012, 5(5): 939-970 doi: 10.3934/dcdss.2012.5.939 +[Abstract](86) +[PDF](605.0KB)
We describe relations between the Evans function, a modern tool in the study of stability of traveling waves and other patterns for PDEs, and the classical Weyl-Titchmarsh function for singular Sturm-Liouville differential expressions and for matrix Hamiltonian systems. Also, for the scalar Schrödinger equation, we discuss a related issue of approximating eigenvalue problems on the whole line by that on finite segments.
On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation
Tetsu Mizumachi and Dmitry Pelinovsky
2012, 5(5): 971-987 doi: 10.3934/dcdss.2012.5.971 +[Abstract](177) +[PDF](421.5KB)
Asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation was earlier established for septic and higher-order nonlinear terms by using Strichartz estimate. We use here pointwise dispersive decay estimates to push down the lower bound for the exponent of the nonlinear terms.
A lattice model for resonance in open periodic waveguides
Natalia Ptitsyna and Stephen P. Shipman
2012, 5(5): 989-1020 doi: 10.3934/dcdss.2012.5.989 +[Abstract](64) +[PDF](1695.6KB)
We present a discrete model of resonant scattering of waves by an open periodic waveguide. The model elucidates a phenomenon common in electromagnetics, in which the interaction of plane waves with embedded guided modes of the waveguide causes sharp transmission anomalies and field amplification. The ambient space is modeled by a planar lattice and the waveguide by a linear periodic lattice coupled to the planar one along a line. We show the existence of standing and traveling guided modes and analyze a tangent bifurcation, in which resonance is initiated at a critical coupling strength where a guided mode appears, beginning with a single standing wave and splitting into a pair of waves traveling in opposing directions. Complex perturbation analysis of the scattering problem in the complex frequency and wavenumber domain reveals the complex structure of the transmission coefficient at resonance.

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