8547327
Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations
Three new iteration methods, namely the squared-operator method, the modified squared-operator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrodinger equations with periodic potentials, and isolated solitons in GinzburgLandau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.
18517436
Why Newton's method is hard for travelling waves: Small denominators, KAM theory, Arnold's linear Fourier problem, non-uniqueness, constraints and erratic failure
Nonlinear travelling waves and standing waves can computed by discretizing the appropriate partial differential equations and then solving the resulting system of nonlinear algebraic equations. Here, we show that the ''small denominator'' problem of Kolmogorov-Arnold-Moser (KAM) theory is equally awkward for numerical algorithms. Furthermore, Newton's iteration combined with continuation in a parameter often exhibits ''erratic failure'' even in the absence of bifurcation. Wave resonances can interlock a countable infinity of branches in an extremely complex topology, as will be illustrated through the fifth-degree Korteweg-deVries equation. Continuation can easily jump, unsuspected, from one branch to another. Constraints, sometimes finite and sometimes infinite in number, are usually needed to specify a unique solution. This confluence of numerical difficulties can be overcome only by combining the latest numerical algorithms with a strong understanding of travelling wave physics.
18096951
Internal oscillations and instability characteristics of (2+1)-dimensional solitons in a saturable nonlinear medium.
Two related problems on scalar (2+1)-dimensional solitons in a saturable nonlinear medium are investigated. The first one is the internal oscillations of fundamental (single-hump) solitons. Internal modes which cause these oscillations, both with and without angular dependence, are discovered. The visual effect of angle-dependent internal modes on the soliton can be a rotation or spatially uneven breathing of the perturbed soliton. These internal oscillations are very robust and persist for a very long distance. The second problem is the instability of double-hump and radially symmetric solitons. Unstable eigenmodes of these solitons are presented. Contrary to intuition, the instability growth rates decrease to zero when the soliton power becomes high. Thus the instability is strongly suppressed at high powers. This phenomenon is corroborated by our direct numerical simulations.
123170
Localized excitations and their thresholds
We propose a numerical method for identifying localized excitations in discrete nonlinear Schrodinger type models. This methodology, based on the application of a nonlinear iterative version of the Rayleigh-Ritz variational principle yields breather excitations in a very fast and efficient way in one or higher spatial dimensions. The typical convergence properties of the method are found to be super-linear. The usefulness of this technique is illustrated by studying the properties of the recently developed theoretical criteria for the excitation power thresholds for nonlinear modes.
41891464
Convergence of Petviashvili's Iteration Method for Numerical Approximation of Stationary Solutions of Nonlinear Wave Equations
We analyze a heuristic numerical method suggested by V. I. Petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations. The method is used to construct numerically the solitary wave solutions, such as solitons, lumps, and vortices, in a space of one and higher dimensions. Assuming that the stationary solution exists, we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest. The theory is illustrated with examples of physical interest such as generalized Korteweg--de Vries, Benjamin--Ono, Zakharov--Kuznetsov, Kadomtsev--Petviashvili, and Klein--Gordon equations.
20459026
Spectral renormalization method for computing self-localized solutions to nonlinear systems.
A new numerical scheme for computing self-localized states--or solitons--of nonlinear waveguides is proposed. The idea behind the method is to transform the underlying equation governing the soliton, such as a nonlinear Schrödinger-type equation, into Fourier space and determine a nonlinear nonlocal integral equation coupled to an algebraic equation. The coupling prevents the numerical scheme from diverging. The nonlinear guided mode is then determined from a convergent fixed point iteration scheme. This spectral renormalization method can find wide applications in nonlinear optics and related fields such as Bose-Einstein condensation and fluid mechanics.
17743324
Optimizing Schrödinger Functionals Using Sobolev Gradients: Applications to Quantum Mechanics and Nonlinear Optics
In this paper we study the application of the Sobolev gradients technique to the problem of minimizing several Schrodinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics. We show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete Fourier basis and a fast Fourier transform. We show that the Sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators.
14293581
Computing the Ground State Solution of Bose-Einstein Condensates by a Normalized Gradient Flow
In this paper, we present a continuous normalized gradient flow (CNGF) and prove its energy diminishing property, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the CNGF. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD) method, the other is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for the linear case and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g., Crank--Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving the energy diminishing property of the CNGF. Numerical results in one, two, and three dimensions with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam, are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the CNGF and its BEFD discretization can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.
207052769
Ground States and Dynamics of Multicomponent Bose-Einstein Condensates
We study numerically the time-independent vector Gross--Pitaevskii equations (VGPEs) for ground states and time-dependent VGPEs with (or without) an external driven field for dynamics describing a multicomponent Bose--Einstein condensate (BEC) at zero or a very low temperature. In preparation for the numerics, we scale the three-dimensional (3d) VGPEs, approximately reduce it to lower dimensions, present a continuous normalized gradient flow (CNGF) to compute ground states of multicomponent BEC, prove energy diminishing of the CNGF, which provides a mathematical justification, and discretize it by the backward Euler finite difference (BEFD), which is monotone in linear and nonlinear cases and preserves energy diminishing property in the linear case. Then we use a time-splitting sine-spectral (TSSP) method to discretize the time-dependent VGPEs with an external driven field for computing dynamics of multicomponent BEC. The merits of the TSSP method for VGPEs are that it is explicit, unconditionally stable,...
119402331
Rayleigh functional for nonlinear systems
We introduce Rayleigh functional for nonlinear systems. It is defined using the energy functional and the normalization properties of the variables of variation. The key property of the Rayleigh quotient for linear systems is preserved in our definition: the extremals of the Rayleigh functional coincide with the stationary solutions of the Euler-Lagrange equation. Moreover, the second variation of the Rayleigh functional defines stability of the solution. This gives rise to a powerful numerical optimization method in the search for the energy minimizers. It is shown that the well-known imaginary time relaxation is a special case of our method. To illustrate the method we find the stationary states of Bose-Einstein condensates in various geometries. Finally, we show that the Rayleigh functional also provides a simple way to derive analytical identities satisfied by the stationary solutions of the critical nonlinear equations. We also show that the functional used in Phys. Rev. A 66, 036612 (2002) is erroneous.
2050784
A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity
The Petviashvili's iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: -Mu+up=0, where M is a positive definite self-adjoint operator and p=const. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.
43188859
Matrix analysis
Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.
120384250
Quantum mechanics: Non-relativistic theory,
The basic concepts of quantum mechanics Energy and momentum Schrodinger's equation Angular momentum Perturbation theory Spin The identity of particles The atom The theory of symmetry Polyatomic molecules Motion in a magnetic field Nuclear structure Elastic collisions Mathematical appendices.
53493166
Some properties of nonlinear wave systems
Nonlinear wave phenomena remain poorly understood, and significant results are limited to either very special examples or situations for which perturbation methods are adequate. The former case is typified by exactly solvable equations and the important role played by solitons. For the latter class of problems, the evolution of weak interactions, resonances, and slowly varying wave trains are well-known examples. Despite the limitations of each approach, these studies have stimulated a large volume of mathematical research and considerable interest in the applied disciplines. One objective of the present paper is to report on some results for nonlinear wave systems that do not fall into either of the above categories. The appro'ach taken is to study a particular case in some detail and, on the basis of the results obtained, to speculate on some anticipated general characteristics of nonlinear
122651582
Defect solitons in photonic lattices
Nonlinear defect-modes (defect-solitons) in optically-induced one-dimensional photonic lattices with a single-site repulsive defect are predicted and observed. These defect solitons bifurcate from linear defect modes and are fully stable.
120155656
Buckling and vibration of a rotating spoke
SummaryThe buckling and vibration of the spoke of a rotating wheel is examined.Since the axial load is a function of position closed form solutions for the eigenmodes are proscribed and recourse is made to regular and singular perturbation expansions in terms of several dimensionless parameters appearing in the governing equations. Some numerical results are also included in the interest of completeness.
38255655
Observation of second-band vortex solitons in 2D photonic lattices.
We demonstrate second-band bright vortex-array solitons in photonic lattices. This constitutes the first experimental observation of higher-band solitons in any 2D periodic system. These solitons possess complex intensity and phase structures, yet they can be excited by a simple highly localized vortex-ring beam. Finally, we show that the linear diffraction of such beams exhibits preferential transport along the lattice axes.
19275786
Observation of vortex-ring "discrete" solitons in 2D photonic lattices
We present the experimental observation of both on-site and off-site vortex ring solitons of unity topological charge in a nonlinear photonic lattice, along with a theoretical study of their propagation dynamics and stability
13999756
Observation of discrete vortex solitons in optically induced photonic lattices.
We report on the first experimental observation of discrete vortex solitons in two-dimensional optically induced photonic lattices. We demonstrate strong stabilization of an optical vortex by the lattice in a self-focusing nonlinear medium and study the generation of the discrete vortices from a broad class of singular beams.
14622682
Dipole and Quadrupole Solitons in Optically Induced Two-Dimensional Photonic Lattices: Theory and Experiment
Dipole and quadrupole solitons in a two-dimensional photorefractive optical lattice are investigated both theoretically and experimentally. It is shown theoretically that out-of-phase dipole solitons and quadrupole solitons exist and are linearly stable in the intermediate-intensity regime. In-phase dipole and quadrupole solitons, however, are always linearly unstable, but their instabilities are rather weak in the low-intensity regime. Experimentally, both types of dipole solitons are observed, and the experimental results agree qualitatively with the theoretical predictions. In addition, we have observed the anisotropic effect of the photorefractive crystal in the dipole-soliton formation.
122617329
Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications
We present an overview of nonlinear phenomena related to optical quadratic solitons—intrinsically multicomponent localized states of light, which can exist in media without inversion symmetry at the molecular level. Starting with presentation of a few derivation schemes of basic equations describing three-wave parametric wave mixing in di7ractive and=or dispersive quadratic media, we discuss their continuous wave solutions and modulational instability phenomena, and then move to the classi8cation and stability analysis of the parametric solitary waves. Not limiting ourselves to the simplest spatial and temporal quadratic solitons we also overview results related to the spatio-temporal solitons (light bullets), higher order quadratic solitons, solitons due to competing nonlinearities, dark solitons, gap solitons, cavity solitons and vortices. Special attention is paid to a comprehensive discussion of the recent experimental demonstrations of the parametric solitons including their interactions and switching. We also discuss connections of quadratic solitons with other types of solitons in optics and their interdisciplinary signi8cance. c
11927752
Two-dimensional optical lattice solitons.
We study various families of two-dimensional discrete or lattice solitons, and show that they are possible only when their power level exceeds a critical threshold. In addition, we show that gap-lattice solitons exist only when the lattice possesses a complete 2D band gap. Our results suggest that these conditions are universally valid, irrespective of the nature of the nonlinearity or the specific structure of the index lattice. The analysis explains fundamental aspects of behavior of two-dimensional discrete solitons that have been very recently observed in photosensitive optical crystals.
166916
Fundamental and vortex solitons in a two-dimensional optical lattice.
Fundamental and vortex solitons in a two-dimensional optically induced waveguide array are reported. In the strong localization regime the fundamental soliton is largely confined to one lattice site, whereas the vortex state comprises four fundamental modes superimposed in a square configuration with a phase structure that is topologically equivalent to the conventional vortex. However, in the weak localization regime, both the fundamental and the vortex solitons spread over many lattice sites. We further show that fundamental and the vortex solitons are stable against small perturbations in the strong localization regime.
10424505
Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices.
We overview our recent theoretical studies on nonlinear atom optics of the Bose-Einstein condensates (BECs) loaded into optical lattices. In particular, we describe the band-gap spectrum and nonlinear localization of BECs in one- and two-dimensional optical lattices. We discuss the structure and stability properties of spatially localized states (matter-wave solitons) in 1D lattices, as well as trivial and vortex-like bound states of 2D gap solitons. To highlight similarities between the behavior of coherent light and matter waves in periodic potentials, we draw useful parallels with the physics of coherent light waves in nonlinear photonic crystals and optically-induced photonic lattices.
21248490
Efficient Numerical Continuation and Stability Analysis of Spatiotemporal Quadratic Optical Solitons
A numerical method is set out which efficiently computes stationary (z-independent) two- and three-dimensional spatiotemporal solitons in second-harmonic-generating media. The method relies on a Chebyshev decomposition with an infinite mapping, bunching the collocation points near the soliton core. Known results for the type-I interaction are extended and a stability boundary is found by two-parameter continuation as defined by the Vakhitov--Kolokolov criteria. The validity of this criterion is demonstrated in (2+1) dimensions by simulation and direct calculation of the linear spectrum. The method has wider applicability for general soliton-bearing equations in (2+1) and (3+1) dimensions.
10990559
Stable vortex and dipole vector solitons in a saturable nonlinear medium.
We study both analytically and numerically the existence, uniqueness, and stability of vortex and dipole vector solitons in a saturable nonlinear medium in (2+1) dimensions. We construct perturbation series expansions for the vortex and dipole vector solitons near the bifurcation point, where the vortex and dipole components are small. We show that both solutions uniquely bifurcate from the same bifurcation point. We also prove that both vortex and dipole vector solitons are linearly stable in the neighborhood of the bifurcation point. Far from the bifurcation point, the family of vortex solitons becomes linearly unstable via oscillatory instabilities, while the family of dipole solitons remains stable in the entire domain of existence. In addition, we show that an unstable vortex soliton breaks up either into a rotating dipole soliton or into two rotating fundamental solitons.