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Coupled mode equations and gap solitons for the 2d GrossPitaevskii equation with a nonseparable periodic potential
 Physica D, 238(910):860 – 879
, 2009
"... equation with a nonseparable periodic potential ..."
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Spectral Stability of Nonlinear Waves in Dynamical Systems
, 2007
"... Partial differential equations that conserve energy can often be written as infinitedimensional hamiltonian systems ofthe following general form: ~ ~ = J E' ( u(t)), u(t) E X where: J: X7 X is a symplectic matrix and E: X7 R is a C2 functional defined on some Hilbert space X. A critical p ..."
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Cited by 5 (0 self)
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Partial differential equations that conserve energy can often be written as infinitedimensional hamiltonian systems ofthe following general form: ~ ~ = J E' ( u(t)), u(t) E X where: J: X7 X is a symplectic matrix and E: X7 R is a C2 functional defined on some Hilbert space X. A critical point of this equation is a point </> E X such that E'(</ » = O. We investigate the spectral stability of solutions in a neighborhood of the critical point by using the linearized Hamiltonian system ~ ~ = J E " (</>)v. The main objective of this thesis is to develop analysis of the spectral properties of the nonselfadjoint operator J E " (</ » using the Pontryagin space decomposition. We adopt parallel computations on Sharcnet clusters to study eigenvalues and eigenvectors of J E " (</ » numerically. The structure of the thesis is as follows. The brief introduction to the spectral stability theory is given in Chapter 1. Count of spectrally unstable eigenvalues of the linearized Hamiltonian system using the indefinite metric approach is given in Chapter 2.
Broad band solitons in a periodic and nonlinear Maxwell system
 SIAM J. Appl. Dynam. Syst
"... We consider the onedimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite timescales, the coupled evolution o ..."
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We consider the onedimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite timescales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so called extended nonlinear coupled mode equations, xNLCME. Numerical studies support the existence of longlived spatially localized coherent structures, featuring a slowly varying envelope and a train of carrier shocks. Thus, it is natural to study the localized coherent structures of xNLCME. In this paper we explore, by analytical, asymptotic and numerical methods, the existence and properties of spatially localized structures of the xNLCME system, which arises for a refractive index profile consisting of periodic array of Dirac delta functions. We consider, in particular, the limit of small amplitude solutions with frequencies near a bandedge. In this case, stationary xNLCME is wellapproximated by an infinite system of coupled, stationary, nonlinear Schrödinger equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a oneparameter family of equations, xNLS, which interpolates between infinitely many decoupled NLS equations ( = 0) and xNLS ( = 1). Using bifurcation methods we show existence of solutions for a range of ∈ (−0, 0) and, by a numerical continuation method, establish the continuation of certain branches all the way to = 1. Finally, we perform timedependent simulations of truncated xNLCME and find the smallamplitude nearbandedge gap solitons to be robust to both numerical errors and the NLS approximation.
unknown title
, 2008
"... Justification of the coupledmode approximation for a nonlinear elliptic problem with a periodic potential ..."
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Justification of the coupledmode approximation for a nonlinear elliptic problem with a periodic potential
BLOCKDIAGONALIZATION OF THE LINEARIZED coupledmode systems
, 2005
"... We consider the Hamiltonian coupledmode system that occur in nonlinear optics, photonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the linearized coupledmode system, which is equivalent to a fourbyfour Dirac system with signindefinite metric. In the ..."
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We consider the Hamiltonian coupledmode system that occur in nonlinear optics, photonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the linearized coupledmode system, which is equivalent to a fourbyfour Dirac system with signindefinite metric. In the special class of symmetric nonlinear potentials, we construct a blockdiagonal representation of the linearized equations, when the spectral problem reduces to two coupled twobytwo Dirac systems. The blockdiagonalization is used in numerical computations of eigenvalues that determine stability of gap solitons.
unknown title
, 2008
"... Coupledmode equations and gap solitons in a twodimensional nonlinear elliptic problem with a separable periodic potential ..."
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Coupledmode equations and gap solitons in a twodimensional nonlinear elliptic problem with a separable periodic potential
unknown title
, 2008
"... Coupledmode equations and gap solitons in a twodimensional nonlinear elliptic problem with a separable periodic potential ..."
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Coupledmode equations and gap solitons in a twodimensional nonlinear elliptic problem with a separable periodic potential
Coupled Mode Equations and Gap Solitons for the 2D
, 2009
"... GrossPitaevsky equation with a nonseparable periodic potential ..."
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UNIVERSITÄT KARLSRUHE Coupled Mode Equations and Gap Solitons for the 2D Gross Pitaevskii equation with a nonseparable periodic potential
, 2008
"... GrossPitaevskii equation with a nonseparable periodic potential ..."
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"... are, therefore, linear and a linear PML may be used. In the presence of solitons propagating at different ..."
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are, therefore, linear and a linear PML may be used. In the presence of solitons propagating at different