Johannes Giannoulis | University of Ioannina/Greece (original) (raw)

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Papers by Johannes Giannoulis

Research paper thumbnail of Transport and generation of macroscopically modulated waves in diatomic chains

Eprint Arxiv 1105 1552, May 8, 2011

We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of... more We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of a single plane wave in a nonlinear diatomic chain with stabilizing on-site potentials including the case where a wave generates another wave via self-interaction. More precisely, we show that in typical chains acoustical waves can generate optical but not acoustical waves, while optical waves are always closed with respect to self-interaction.

Research paper thumbnail of Interaction of modulated pulses in scalar multidimensional nonlinear lattices

Applicable Analysis, 2010

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude... more We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

Research paper thumbnail of Three-wave interaction in discrete lattices

Research paper thumbnail of Transport and generation of macroscopically modulated waves in diatomic chains

We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of... more We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of a single plane wave in a nonlinear diatomic chain with stabilizing on-site potentials including the case where a wave generates another wave via self-interaction. More precisely, we show that in typical chains acoustical waves can generate optical but not acoustical waves, while optical waves are always closed with respect to self-interaction.

Research paper thumbnail of Young-measure solutions to a generalized Benjamin-Bona-Mahony equation

Mathematical Methods in the Applied Sciences, 2005

Research paper thumbnail of Interaction of modulated pulses in scalar multidimensional nonlinear lattices

Applicable Analysis, 2010

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude... more We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

Research paper thumbnail of Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential

We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical s... more We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.

Research paper thumbnail of High-frequency averaging in semi-classical Hartree-type equations

We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with... more We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new

Research paper thumbnail of Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data

In this paper we study the semiclassical limit of the Schr\"odinger equation. Under mild reg... more In this paper we study the semiclassical limit of the Schr\"odinger equation. Under mild regularity assumptions on the potential UUU which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In

Research paper thumbnail of DISPERSIVE EVOLUTION OF PULSES IN OSCILLATOR CHAINS WITH GENERAL INTERACTION POTENTIALS

We study the dispersive evolution of modulated pulses in a non- linear oscillator chain embedded ... more We study the dispersive evolution of modulated pulses in a non- linear oscillator chain embedded in a background fleld. The atoms of the chain interact pairwise with an arbitrary but flnite number of neighbors. The pulses are modeled as macroscopic modulations of the exact spatiotemporally peri- odic solutions of the linearized model. The scaling of amplitude, space and time is

Research paper thumbnail of Macroscopic dynamics in microscopic nonlinear oscillator chains

Research paper thumbnail of Macroscopic pulse evolution for a nonlinear oscillator chain

PAMM, 2004

We are interested in the macroscopic evolution of the modulation of a pulse which captures disper... more We are interested in the macroscopic evolution of the modulation of a pulse which captures dispersive effects in a nonlinear discrete model. In the weakly nonlinear setting the associated modulation equation turns out to be the nonlinear Schrödinger equation. We focus on the mathematical justification of this macroscopic limit. *

Research paper thumbnail of Continuum Descriptions for the Dynamics in Discrete Lattices: Derivation and Justification

Analysis, Modeling and Simulation of Multiscale Problems, 2006

The passage from microscopic systems to macroscopic ones is studied by starting from spatially di... more The passage from microscopic systems to macroscopic ones is studied by starting from spatially discrete lattice systems and deriving several continuum limits. The lattice system is an infinite-dimensional Hamiltonian system displaying a variety of different dynamical ...

Research paper thumbnail of The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities

Nonlinearity, 2004

We consider the nonlinear model of an infinite oscillator chain embedded in a background field. W... more We consider the nonlinear model of an infinite oscillator chain embedded in a background field. We start from an appropriate modulation ansatz of the space- time periodic solutions to the linearized (microscopic) model and derive formally the associated (macroscopic) modulation equation, which turns out to be the nonlinear Schrodinger equation. Then we justify this necessary condition rigorously for the case of nonlinearities with cubic leading terms, that is, we show that solutions which have the form of the assumed ansatz for t = 0 preserve this form over time-intervals with a positive macroscopic length. Finally, we transfer this result to the analogous case of a finite, but large periodic chain and illustrate it by a numerical example.

Research paper thumbnail of Lagrangian and Hamiltonian two-scale reduction

Journal of Mathematical Physics, 2008

Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challen... more Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system.

Research paper thumbnail of Semiclassical limit of quantum dynamics with rough potentials and well-posedness of transport equations with measure initial data

Communications on Pure and Applied Mathematics, 2011

Research paper thumbnail of Passage from Quantum to Classical Molecular Dynamics in the Presence of Coulomb Interactions

Communications in Partial Differential Equations, 2010

We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dyna... more We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dynamics (QMD) that applies to the standard Hamiltonians of molecular physics with Coulomb interactions. The derivation is valid away from possible electronic eigenvalue crossings.

Research paper thumbnail of Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

Journal of Differential Equations, 2008

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical sc... more We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.

Research paper thumbnail of Transport and generation of macroscopically modulated waves in diatomic chains

Eprint Arxiv 1105 1552, May 8, 2011

We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of... more We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of a single plane wave in a nonlinear diatomic chain with stabilizing on-site potentials including the case where a wave generates another wave via self-interaction. More precisely, we show that in typical chains acoustical waves can generate optical but not acoustical waves, while optical waves are always closed with respect to self-interaction.

Research paper thumbnail of Interaction of modulated pulses in scalar multidimensional nonlinear lattices

Applicable Analysis, 2010

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude... more We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

Research paper thumbnail of Three-wave interaction in discrete lattices

Research paper thumbnail of Transport and generation of macroscopically modulated waves in diatomic chains

We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of... more We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of a single plane wave in a nonlinear diatomic chain with stabilizing on-site potentials including the case where a wave generates another wave via self-interaction. More precisely, we show that in typical chains acoustical waves can generate optical but not acoustical waves, while optical waves are always closed with respect to self-interaction.

Research paper thumbnail of Young-measure solutions to a generalized Benjamin-Bona-Mahony equation

Mathematical Methods in the Applied Sciences, 2005

Research paper thumbnail of Interaction of modulated pulses in scalar multidimensional nonlinear lattices

Applicable Analysis, 2010

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude... more We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise, arbitrary-range, nonlinear interaction potentials and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high order of the scaling parameter, thereby deducing the resonance and non-resonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

Research paper thumbnail of Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential

We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical s... more We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.

Research paper thumbnail of High-frequency averaging in semi-classical Hartree-type equations

We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with... more We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new

Research paper thumbnail of Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data

In this paper we study the semiclassical limit of the Schr\"odinger equation. Under mild reg... more In this paper we study the semiclassical limit of the Schr\"odinger equation. Under mild regularity assumptions on the potential UUU which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In

Research paper thumbnail of DISPERSIVE EVOLUTION OF PULSES IN OSCILLATOR CHAINS WITH GENERAL INTERACTION POTENTIALS

We study the dispersive evolution of modulated pulses in a non- linear oscillator chain embedded ... more We study the dispersive evolution of modulated pulses in a non- linear oscillator chain embedded in a background fleld. The atoms of the chain interact pairwise with an arbitrary but flnite number of neighbors. The pulses are modeled as macroscopic modulations of the exact spatiotemporally peri- odic solutions of the linearized model. The scaling of amplitude, space and time is

Research paper thumbnail of Macroscopic dynamics in microscopic nonlinear oscillator chains

Research paper thumbnail of Macroscopic pulse evolution for a nonlinear oscillator chain

PAMM, 2004

We are interested in the macroscopic evolution of the modulation of a pulse which captures disper... more We are interested in the macroscopic evolution of the modulation of a pulse which captures dispersive effects in a nonlinear discrete model. In the weakly nonlinear setting the associated modulation equation turns out to be the nonlinear Schrödinger equation. We focus on the mathematical justification of this macroscopic limit. *

Research paper thumbnail of Continuum Descriptions for the Dynamics in Discrete Lattices: Derivation and Justification

Analysis, Modeling and Simulation of Multiscale Problems, 2006

The passage from microscopic systems to macroscopic ones is studied by starting from spatially di... more The passage from microscopic systems to macroscopic ones is studied by starting from spatially discrete lattice systems and deriving several continuum limits. The lattice system is an infinite-dimensional Hamiltonian system displaying a variety of different dynamical ...

Research paper thumbnail of The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities

Nonlinearity, 2004

We consider the nonlinear model of an infinite oscillator chain embedded in a background field. W... more We consider the nonlinear model of an infinite oscillator chain embedded in a background field. We start from an appropriate modulation ansatz of the space- time periodic solutions to the linearized (microscopic) model and derive formally the associated (macroscopic) modulation equation, which turns out to be the nonlinear Schrodinger equation. Then we justify this necessary condition rigorously for the case of nonlinearities with cubic leading terms, that is, we show that solutions which have the form of the assumed ansatz for t = 0 preserve this form over time-intervals with a positive macroscopic length. Finally, we transfer this result to the analogous case of a finite, but large periodic chain and illustrate it by a numerical example.

Research paper thumbnail of Lagrangian and Hamiltonian two-scale reduction

Journal of Mathematical Physics, 2008

Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challen... more Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system.

Research paper thumbnail of Semiclassical limit of quantum dynamics with rough potentials and well-posedness of transport equations with measure initial data

Communications on Pure and Applied Mathematics, 2011

Research paper thumbnail of Passage from Quantum to Classical Molecular Dynamics in the Presence of Coulomb Interactions

Communications in Partial Differential Equations, 2010

We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dyna... more We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dynamics (QMD) that applies to the standard Hamiltonians of molecular physics with Coulomb interactions. The derivation is valid away from possible electronic eigenvalue crossings.

Research paper thumbnail of Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential

Journal of Differential Equations, 2008

We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical sc... more We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.