Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential
TL;DR: In this paper, the authors studied the one-dimensional biharmonic nonlinear Schrodinger (NLS) equation in the presence of an external periodic potential and determined the band-gap structure using the Floquet-Bloch theory and the shape of dispersion curves as a function of the fourth-order dispersion coupling constant.
Abstract: Localization and dynamics of the one-dimensional biharmonic nonlinear Schr\"odinger (NLS) equation in the presence of an external periodic potential is studied. The band-gap structure is determined using the Floquet-Bloch theory and the shape of its dispersion curves as a function of the fourth-order dispersion coupling constant $\ensuremath{\beta}$ is discussed. Contrary to the classical NLS equation ($\ensuremath{\beta}=0$) with an external periodic potential for which a gap in the spectrum opens for any nonzero potential, here it is found that for certain negative $\ensuremath{\beta},$ there exists a nonzero threshold value of potential strength below which there is no gap. For increasing values of potential amplitudes, the shape of the dispersion curves change drastically leading to the formation of localized nonlinear modes that have no counterpart in the classical NLS limit. A higher-order two-band tight-binding model is introduced that captures and intuitively explains most of the numerical results related to the spectral bands. Lattice solitons corresponding to spectral eigenvalues lying in the semi-infinite and first band gaps are constructed. In the anomalous dispersion case, i.e., $\ensuremath{\beta}l0$ (where for the self-focusing nonlinearity no localized nonradiating solitons exist in the absence of an external potential), nonlinear finite-energy stationary modes with eigenvalues residing in the first band gap are found and their properties are discussed. The stability of various localized lattice modes is studied via linear stability analysis and direct numerical simulation.
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TL;DR: The approximate shape of the fundamental pure-quartic soliton is derived and it is found that is surprisingly Gaussian, exhibiting excellent agreement with the experimental observations.
Abstract: Temporal optical solitons have been the subject of intense research due to their intriguing physics and applications in ultrafast optics and supercontinuum generation. Conventional bright optical solitons result from the interaction of anomalous group-velocity dispersion and self-phase modulation. Here we experimentally demonstrate a class of bright soliton arising purely from the interaction of negative fourth-order dispersion and self-phase modulation, which can occur even for normal group-velocity dispersion. We provide experimental and numerical evidence of shape-preserving propagation and flat temporal phase for the fundamental pure-quartic soliton and periodically modulated propagation for the higher-order pure-quartic solitons. We derive the approximate shape of the fundamental pure-quartic soliton and discover that is surprisingly Gaussian, exhibiting excellent agreement with our experimental observations. Our discovery, enabled by precise dispersion engineering, could find applications in communications, frequency combs and ultrafast lasers.
140 citations
TL;DR: A generalized nonlinear Schrödinger equation is solved and a family of pure-quartic solitons (PQSs) are found, existing through a balance of positive Kerr nonlinearity and negative quartic dispersion, which has oscillatory tails.
Abstract: We numerically solve a generalized nonlinear Schrodinger equation and find a family of pure-quartic solitons (PQSs), existing through a balance of positive Kerr nonlinearity and negative quartic dispersion. These solitons have oscillatory tails, which can be understood analytically from the properties of linear waves with quartic dispersion. By computing the linear eigenspectrum of the solitons, we show that they are stable, but that they possess a nontrivial internal mode close to the radiation continuum. We also demonstrate evolution into a PQS from Gaussian initial conditions. The energy-width scaling of PQSs differs strongly from that for conventional solitons, opening up possibilities for PQS lasers.
53 citations
TL;DR: In this article, the existence of 2D matter-wave solitons in the free space was predicted using the 2D Gross-Pitaevskii Equation (GPE).
Abstract: We study two-dimensional (2D) matter-wave solitons in the mean-field models formed by electric quadrupole particles with long-range quadrupole–quadrupole interaction (QQI) in 2D free space. The existence of 2D matter-wave solitons in the free space was predicted using the 2D Gross–Pitaevskii Equation (GPE). We find that the QQI solitons have a higher mass (smaller size and higher intensity) and stronger anisotropy than the dipole–dipole interaction (DDI) solitons under the same environmental parameters. Anisotropic soliton–soliton interaction between two identical QQI solitons in 2D free space is studied. Moreover, stable anisotropic dipole solitons are observed, to our knowledge, for the first time in 2D free space under anisotropic nonlocal cubic nonlinearity.
37 citations
TL;DR: In this article, the stability of solitons in different nonlinear media with the second-order and fourth-order diffraction and PT -symmetric potentials is studied.
26 citations
TL;DR: It is found that the relative strength of the FOD coefficient can utterly change the direction of the power flow, which may be used to control the energy exchange among gain or loss regions.
Abstract: We investigate the existence and stability of solitons in parity-time ($\mathcal{PT}$)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution and fourth-order diffraction (FOD). For the linear case, we demonstrate numerically that the FOD parameter can alter the $\mathcal{PT}$-breaking points. For nonlinear cases, the exact analytical expressions of the localized modes are obtained both in one- and two-dimensional nonlinear Schr\"odinger equations with self-focusing and self-defocusing Kerr nonlinearity. The effect of FOD on the stability structure of these localized modes is discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Examples of stable and unstable solutions are given. The transverse power flow density associated with these localized modes is also discussed. It is found that the relative strength of the FOD coefficient can utterly change the direction of the power flow, which may be used to control the energy exchange among gain or loss regions.
22 citations
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Book•
01 Jan 1989TL;DR: The field of nonlinear fiber optics has advanced enough that a whole book was devoted to it as discussed by the authors, which has been translated into Chinese, Japanese, and Russian languages, attesting to the worldwide activity in the field.
Abstract: Nonlinear fiber optics concerns with the nonlinear optical phenomena occurring inside optical fibers. Although the field ofnonlinear optics traces its beginning to 1961, when a ruby laser was first used to generate the second-harmonic radiation inside a crystal [1], the use ofoptical fibers as a nonlinear medium became feasible only after 1970 when fiber losses were reduced to below 20 dB/km [2]. Stimulated Raman and Brillouin scatterings in single-mode fibers were studied as early as 1972 [3] and were soon followed by the study of other nonlinear effects such as self- and crossphase modulation and four-wave mixing [4]. By 1989, the field ofnonlinear fiber optics has advanced enough that a whole book was devoted to it [5]. This book or its second edition has been translated into Chinese, Japanese, and Russian languages, attesting to the worldwide activity in the field of nonlinear fiber optics.
15,770 citations
Book•
01 Jan 2000TL;DR: This paper presents a meta-analyses of Chebyshev differentiation matrices using the DFT and FFT as a guide to solving fourth-order grid problems.
Abstract: Preface 1 Differentiation matrices 2 Unbounded grids: the semidiscrete Fourier transform 3 Periodic grids: the DFT and FFT 4 Smoothness and spectral accuracy 5 Polynomial interpolation and clustered grids 6 Chebyshev differentiation matrices 7 Boundary value problems 8 Chebyshev series and the FFT 9 Eigenvalues and pseudospectra 10 Time-stepping and stability regions 11 Polar coordinates 12 Integrals and quadrature formulas 13 More about boundary conditions 14 Fourth-order problems Afterword Bibliography Index
3,696 citations
Book•
19 Nov 2010
TL;DR: This book presents cutting-edge developments in the theory and experiments of nonlinear waves, and contains a large number of simple and efficient MATLAB codes for various nonlinear wave computations, which readers can easily adapt to solve their own problems.
Abstract: This book presents cutting-edge developments in the theory and experiments of nonlinear waves. Its comprehensive coverage of analytical methods for nonintegrable systems is the first of its kind. It also covers in great depth analytical methods for integrable equations, and comprehensively describes efficient numerical methods for all major aspects of nonlinear wave computations. In addition, the book presents the latest experiments on nonlinear waves in optical systems and Bose- Einstein condensates, especially in periodic media. The book contains a large number of simple and efficient MATLAB codes for various nonlinear wave computations, which readers can easily adapt to solve their own problems. The codes can also be found on an associated Web page. Audience: This book is intended for researchers and graduate students working in applied mathematics and various physical subjects, such as nonlinear optics, Bose Einstein condensates, and fluid dynamics, where nonlinear wave phenomena arise. Contents: List of Figures; Preface; Chapter 1. Derivation of nonlinear wave equations; Chapter 2. Integrable theory for the nonlinear Schrdinger equation; Chapter 3. Theories for integrable equations with higher-order scattering operators; Chapter 4. Soliton perturbation theories and applications; Chapter 5. Theories for nonintegrable equations; Chapter 6. Nonlinear wave phenomena in periodic media; Chapter 7. Numerical methods for nonlinear wave equations; Bibliography; Index.
881 citations
"Band gaps and lattice solitons for ..." refers background in this paper
...(27) are to truncate the domain on some large grid and represent the derivatives by differentiation matrices [29,30] or instead to represent the functions in terms of their respective Fourier series [37]....
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01 Jan 2010
TL;DR: Nonlinear effects in reduced-dimensional structures: From Wave Guide Arrays to Slow Light- Nonlinear Effects in One-Dimensional Photonic Lattices, Nonlinear Optical Waves in Liquid CrystallineLattices and Nonlinear Optics and Solitons in Photonic Crystal Fibres as discussed by the authors.
Abstract: Nonlinear Effects in Reduced-Dimensional Structures: From Wave Guide Arrays to Slow Light- Nonlinear Effects in One-Dimensional Photonic Lattices- Nonlinear Optical Waves in Liquid Crystalline Lattices- Nonlinear Optics and Solitons in Photonic Crystal Fibres- Spatial Switching of Slow Light in Periodic Photonic Structures- Nonlinear Effects in Multidimensional Lattices: Solitons and Light Localization- to Solitons in Photonic Lattices- Complex Nonlinear Photonic Lattices: From Instabilities to Control- Light Localization by Defects in Optically Induced Photonic Structures- Polychromatic Light Localisation in Periodic Structures- Periodic Structures for Matter Waves: From Lattices to Ratchets- Bose-Einstein Condensates in 1D Optical Lattices: Nonlinearity and Wannier-Stark Spectra- Transporting Cold Atoms in Optical Lattices with Ratchets: Mechanisms and Symmetries- Atomic Bose-Einstein Condensates in Optical Lattices with Variable Spatial Symmetry- Symmetry and Transport in a Rocking Ratchet for Cold Atoms- Metamaterials: From Linear to Nonlinear Features- Optical Metamaterials: Invisibility in Visible and Nonlinearities in Reverse- Nonlinear Metamaterials- Circuit Model of Gain in Metamaterials- Discrete Breathers and Solitons in Metamaterials
50 citations