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Journal ArticleDOI

Dynamics of lattice kinks

TL;DR: In this paper, the authors consider a class of Hamiltonian nonlinear wave equations governing a field defined on a spatially discrete one-dimensional lattice, with discreteness parameter, d = h −1, where h > 0 is the lattice spacing.
About: This article is published in Physica D: Nonlinear Phenomena.The article was published on 2000-08-01 and is currently open access. It has received 51 citations till now. The article focuses on the topics: Breather & Phase portrait.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, a review of recent developments in the study of the Discrete Nonlinear Schrodinger (DNLS) equation is presented in one and two spatial dimensions, concerning ground and excited states, their construction, stability and bifurcations.
Abstract: In this paper we review a number of recent developments in the study of the Discrete Nonlinear Schrodinger (DNLS) equation. Results concerning ground and excited states, their construction, stability and bifurcations are presented in one and two spatial dimensions. Combinations of such steady states lead to the study of coherent structure bound states. A special case of such structures are the so-called twisted modes and their two-dimensional discrete vortex generalization. The ideas on such multi-coherent structures and their interactions are also useful in treating finite system effects through the image method. The statistical mechanics of the system is also analyzed and the partition function calculated in one spatial dimension using the transfer integral method. Finally, a number of open problems and future directions in the field are proposed.

340 citations

Journal ArticleDOI
TL;DR: In this paper, a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states was studied and the authors showed that the general large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation.
Abstract: We prove for a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.

181 citations

Journal ArticleDOI
TL;DR: In this article, a class of nonlinear Schr\"odinger systems (NLS) having two nonlinear bound states was studied and the authors showed that the general large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation.
Abstract: We prove for a class of nonlinear Schr\"odinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as {\it ground state selection}. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear Master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree-Fock type.

83 citations

Journal ArticleDOI
TL;DR: In this article, a new class of discretizations of partial differential equations (PDEs) that preserve a (momentum-like) integral of the motion is presented, which results in an effective translational invariance for the dynamical problem and the absence of a Peierls-Nabarro barrier.

60 citations

Journal ArticleDOI
TL;DR: For the nonlinear Klein-Gordon type models, a general method of discretization was proposed in this article in which the static kink can be placed anywhere with respect to the lattice.
Abstract: For the nonlinear Klein–Gordon type models, we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice. These discrete models are, therefore, free of the Peierls–Nabarro potential. Previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method. A relevant physical consequence of our findings is the existence of a wide class of discrete Klein–Gordon models where slow kinks practically do not experience the action of the Peierls–Nabarro potential. Such kinks are not trapped by the lattice and they can be accelerated by even weak external fields.

49 citations

References
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Book
01 Jan 1961

20,079 citations

Book
01 Jan 1955
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
Abstract: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.

7,071 citations

Book
28 Dec 2011
TL;DR: In this paper, the authors present an approach for solving the panel flutter problem using a Second Order Equation (SOPE) and a Semigroup Theory. But their approach is limited to the case when the case is 1 < 0 and the case where 0 < 0.
Abstract: 0.- 4.5. The Case ?1 < 0.- 4.6. More Scaling.- 4.7. Completion of the Phase Portraits.- 4.8. Remarks and Exercises.- 4.9. Quadratic Nonlinearities.- 5. Application to a Panel Flutter Problem.- 5.1. Introduction.- 5.2. Reduction to a Second Order Equation.- 5.3. Calculation of Linear Terms.- 5.4. Calculation of the Nonlinear Terms.- 6. Infinite Dimensional Problems.- 6.1. Introduction.- 6.2. Semigroup Theory.- 6.3. Centre Manifolds.- 6.4. Examples.- References.

1,481 citations

Journal ArticleDOI
TL;DR: In this paper, the authors reported the observation of discrete spatial optical solitons in an array of 41 waveguides, where light was coupled to the central waveguide and when sufficient power was injected, the field was localized close to the input waveguiders and its distribution was successfully described by the discrete nonlinear Schrodinger equation.
Abstract: We report the observation of discrete spatial optical solitons in an array of 41 waveguides. Light was coupled to the central waveguide. At low power, the propagating field spreads as it couples to more waveguides. When sufficient power was injected, the field was localized close to the input waveguides and its distribution was successfully described by the discrete nonlinear Schr\"odinger equation.

1,097 citations

Book
01 Jan 2001
TL;DR: Topological approach: Finite dimensions Topological degree in Banach space Bifurcation theory Further topological methods Monotone operators and the min-max theorem Generalized implicit function theorems Bibliography as mentioned in this paper
Abstract: Topological approach: Finite dimensions Topological degree in Banach space Bifurcation theory Further topological methods Monotone operators and the min-max theorem Generalized implicit function theorems Bibliography.

905 citations