Stationary Localized Solutions Research Articles

Stationary localized solutions in binary convection in slightly inclined rectangular cells.

We analyze numerically the effect of a slight inclination in the lowest part of the snaking branches of convectons that are present in negative separation ratio binary mixtures in two-dimensional elongated rectangular cells. The exploration reveals the existence of novel stationary localized solutions with striking spatial features different from those of convectons. The numerical continuation of these solutions with respect to the inclination of the cell unveils the existence of even further families of localized structures that can organize in closed branches. A variety of localized solutions coexist for the same heating and inclination, depicting a highly complex scenario for solutions in the lowest part of the snaking diagrams for moderate to high heating. The different localized solutions obtained in the horizontal cell are discussed in detail.

Electromagnetic solitons and their stability in relativistic degenerate dense plasmas with two electron species

The evolution of electromagnetic (EM) solitons due to nonlinear coupling of circularly polarized intense laser pulses with low-frequency electron-acoustic perturbations is studied in relativistic degenerate dense astrophysical plasmas with two groups of electrons: a sparse population of classical relativistic electrons and a dense population of relativistic degenerate electron gas. Different forms of localized stationary solutions are obtained and their properties are analyzed. Using the Vakhitov-Kolokolov stability criterion, the conditions for the existence and stability of a moving EM soliton are also studied. It is noted that the stable and unstable regions shift around the plane of soliton eigenfrequency and the soliton velocity due to the effects of relativistic degeneracy, the fraction of classical to degenerate electrons and the EM wave frequency. Furthermore, while the standing solitons exhibit stable profiles for a longer time, the moving solitons, however, can be stable or unstable depending on the degree of electron degeneracy, the soliton eigenfrequency and the soliton velocity. The latter with an enhanced value can eventually lead to a soliton collapse. The results should be useful for understanding the formation of solitons in the coupling of highly intense laser pulses with slow response of degenerate dense plasmas in the next generation laser-plasma interaction experiments as well as the novel features of x-ray and γ-ray pulses that originate from compact astrophysical objects.

Existence of localized radial patterns in a model for dryland vegetation

Abstract Localized radial patterns have been observed in the vegetation of semi-arid ecosystems, often as localized patches of vegetation or in the form of ‘fairy circles’. We consider stationary localized radial solutions to a reduced model for dryland vegetation on flat terrain. By considering certain prototypical pattern-forming systems, we prove the existence of three classes of localized radial patterns bifurcating from a Turing instability. We also present evidence for the existence of localized gap solutions close to a homogeneous instability. Additionally, we numerically solve the vegetation model and use continuation methods to study the bifurcation structure and radial stability of localized radial spots and gaps. We conclude by investigating the effect of varying certain parameter values on the existence and stability of these localized radial patterns.

Data-driven intrinsic localized mode detection and classification in one-dimensional crystal lattice model

In this work we propose Support Vector Machine classification algorithms to classify one-dimensional crystal lattice waves from locally sampled data. Different learning datasets of particle displacements, momenta and energy density values are considered. Efficiency of the classification algorithms is further improved by two dimensionality reduction techniques: Principal Component Analysis and Locally Linear Embedding. Robustness of classifiers is investigated and demonstrated. Developed algorithms are successfully applied to detect localized intrinsic modes in three numerical simulations considering a case of two localized stationary breather solutions, a single stationary breather solution in noisy background and two mobile breather collision.

Energy Efficiency Optimization for Millimeter Wave System With Resolution-Adaptive ADCs

This letter investigates the uplink of a multi-user millimeter wave (mmWave) system, where the base station (BS) is equipped with a massive multiple-input multiple-output (MIMO) array and resolution-adaptive analog-to-digital converters (RADCs). Although employing massive MIMO at the BS can significantly improve the spectral efficiency, it also leads to high hardware complexity and huge power consumption. To overcome these challenges, we seek to jointly optimize the beamspace hybrid combiner and the ADC quantization bits allocation to maximize the system energy efficiency (EE) under some practical constraints. The formulated problem is non-convex due to the non-linear fractional objective function and the non-convex feasible set which is generally intractable. In order to handle these difficulties, we first apply some fractional programming (FP) techniques and introduce auxiliary variables to recast this problem into an equivalent form amenable to optimization. Then, we propose an efficient double-loop iterative algorithm based on the penalty dual decomposition (PDD) and the majorization-minimization (MM) methods to find local stationary solutions. Simulation results reveal significant gain over the baselines.

Modulational instability, quantum breathers and two-breathers in a frustrated ferromagnetic spin lattice under an external magnetic field**Project supported by the National Natural Science Foundation of China (Grant No. 11604121), the Scientific Research Fund of Hunan Provincial Education Department, China (Grant Nos. 16B210 and 16A170), and the Natural Science Fund Project of Jishou University, China (Grant No. jdx17036).

The modulational instability, quantum breathers and two-breathers in a frustrated easy-axis ferromagnetic zig-zag chain under an external magnetic field are investigated within the Hartree approximation. By means of a linear stability analysis, we analytically study the discrete modulational instability and analyze the effect of the frustration strength on the discrete modulational instability region. Using the results from the discrete modulational instability analysis, the presence conditions of those stationary bright type localized solutions are presented. On the other hand, we obtain the analytical expressions for the stationary bright localized solutions and analyze the effect of the frustration on their emergence conditions. By taking advantage of these bright type single-magnon bound wave functions obtained, quantum breather states in the present frustrated ferromagnetic zig-zag lattice are constructed. What is more, the analytical forms for quantum two-breather states are also obtained. In particular, the energy level formulas of quantum breathers and two-breathers are derived.

Invasion waves and pinning in the Kirkpatrick-Barton model of evolutionary range dynamics.

The Kirkpatrick-Barton model, well known to invasion biologists, is a pair of reaction-diffusion equations for the joint evolution of population density and the mean of a quantitative trait as functions of space and time. Here we prove the existence of two classes of coherent structures, namely "bounded trait mean differential" traveling waves and localized stationary solutions, using geometric singular perturbation theory. We also give numerical examples of these (when they appear to be stable) and of "unbounded trait mean differential" solutions. Further, we provide numerical evidence of bistability and hysteresis for this system, modeling an initially confined population that colonizes new territory when some biotic or abiotic conditions change, and remains in its enlarged range even when conditions change back. Our analytical and numerical results indicate that the dynamics of this system are more complicated than previously recognized, and help make sense of evolutionary range dynamics predicted by other models that build upon it and sometimes challenge its predictions.

Modulational Instability and Quantum Discrete Breather States of Cold Bosonic Atoms in a Zig-Zag Optical Lattice

A theoretical study on modulational instability and quantum discrete breather states in a system of cold bosonic atoms in zig-zag optical lattices is presented in this work. The time-dependent Hartree approximation is employed to deal with the multiple body problem. By means of a linear stability analysis, we analytically study the modulational instability, and estimate existence conditions of the bright stationary localized solutions for different values of the second-neighbor hopping constant. On the other hand, we get analytical bright stationary localized solutions, and analyze the influence of the second-neighbor hopping on their existence conditions. The predictions of the modulational instability analysis are shown to be reliable. Using these stationary localized single-boson wave functions, the quantum breather states corresponding to the system with different types of nonlinearities are constructed.

Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving next-nearest neighbor interactions

The dynamics and modulation instability of an ultracold gas of bosonic atoms in an optical lattice can be portrayed by a Bose–Hubbard model and the system parameters are mastered by laser light. Based on the time dependent Hartree approximation combined with the semi-discrete multiple-scale method, the equation of motion for single-boson wave function is found analytically, the existence conditions of appearance of bright stationary localized solitons solutions of this quantum Bose–Hubbard model are discussed. We find that the introduction of the next nearest neighbor interactions (NNNI) may change the stability property of the plane waves and may predict the formation of modulational instability in the wave number k = kmax and k = keBZ in the system. With the help of stationary localized single-boson wave functions obtained, the quantized energy level and the quantum breather state are determined. The performance of the analytical results are checked by numerical calculations. Furthermore, we have shown that the presence of the NNNI affect significatively the shape of the region of modulational instability and it is responsible of the appearance of new region of modulational instability that occurs for the k = kmax carrier wave. The formation conditions of the modulational instability region predicted by the analytical analysis of stationary localized solutions in the wave number k = kmax and k = keBZ are in good agreement with the forecast respectively.

Scalar Reduction of a Neural Field Model with Spike Frequency Adaptation

We study a deterministic version of a one- and two-dimensional attractor neural network model of hippocampal activity first studied by Itskov et al. [J. Neurosci., 31 (2011), pp. 2828--2834]. We analyze the dynamics of the system on the ring and torus domains with an even periodized weight matrix, assuming weak and slow spike frequency adaptation and a weak stationary input current. On these domains, we find transitions from spatially localized stationary solutions (``bumps'') to (periodically modulated) solutions (``sloshers''), as well as constant and nonconstant velocity traveling bumps depending on the relative strength of external input current and adaptation. The weak and slow adaptation allows for a reduction of the system from a distributed partial integro-differential equation to a system of scalar Volterra integro-differential equations describing the movement of the centroid of the bump solution. Using this reduction, we show that on both domains, sloshing solutions arise through an Andronov--H...

Bifurcation structure of cavity soliton dynamics in a vertical-cavity surface-emitting laser with a saturable absorber and time-delayed feedback

We consider a wide-aperture surface-emitting laser with a saturable absorber section subjected to time-delayed feedback. We adopt the mean-field approach assuming a single longitudinal mode operation of the solitary VCSEL. We investigate cavity soliton dynamics under the effect of time- delayed feedback in a self-imaging configuration where diffraction in the external cavity is negligible. Using bifurcation analysis, direct numerical simulations and numerical path continuation methods, we identify the possible bifurcations and map them in a plane of feedback parameters. We show that for both the homogeneous and localized stationary lasing solutions in one spatial dimension the time-delayed feedback induces complex spatiotemporal dynamics, in particular a period doubling route to chaos, quasiperiodic oscillations and multistability of the stationary solutions.

An Efficient Global Algorithm for Single-Group Multicast Beamforming

Consider the single-group multicast beamforming problem, where multiple users receive the same data stream simultaneously from a single transmitter. The problem is NP-hard and all existing algorithms for the problem either find suboptimal approximate or local stationary solutions. In this paper, we propose an efficient branch-and-bound algorithm for the problem that is guaranteed to find its global solution. To the best of our knowledge, our proposed algorithm is the first tailored global algorithm for the single-group multicast beamforming problem. Simulation results show that our proposed algorithm is computationally efficient (albeit its theoretical worst-case iteration complexity is exponential with respect to the number of receivers) and it significantly outperforms a state-of-the-art general-purpose global optimization solver called Baron. Our proposed algorithm provides an important benchmark for performance evaluation of existing algorithms for the same problem. By using it as the benchmark, we show that two state-of-the-art algorithms, semidefinite relaxation algorithm and successive linear approximation algorithm, work well when the problem dimension (i.e., the number of antennas at the transmitter and the number of receivers) is small but their performance deteriorates quickly as the problem dimension increases.

Quantum soliton in 1D Heisenberg spin chains with Dzyaloshinsky-Moriya and next-nearest-neighbor interactions.

We report in this work, an analytical study of quantum soliton in 1D Heisenberg spin chains with Dzyaloshinsky-Moriya Interaction (DMI) and Next-Nearest-Neighbor Interactions (NNNI). By means of the time-dependent Hartree approximation and the semi-discrete multiple-scale method, the equation of motion for the single-boson wave function is reduced to the nonlinear Schrödinger equation. It comes from this present study that the spectrum of the frequencies increases, its periodicity changes, in the presence of NNNI. The antisymmetric feature of the DMI was probed from the dispersion curve while changing the sign of the parameter controlling it. Five regions were identified in the dispersion spectrum, when the NNNI are taken into account instead of three as in the opposite case. In each of these regions, the quantum model can exhibit quantum stationary localized and stable bright or dark soliton solutions. In each region, we could set up quantum localized n-boson Hartree states as well as the analytical expression of their energy level, respectively. The accuracy of the analytical studies is confirmed by the excellent agreement with the numerical calculations, and it certifies the stability of the stationary quantum localized solitons solutions exhibited in each region. In addition, we found that the intensity of the localization of quantum localized n-boson Hartree states increases when the NNNI are considered. We also realized that the intensity of Hartree n-boson states corresponding to quantum discrete soliton states depend on the wave vector.

Fully localized solitary waves for the forced Kadomtsev–Petviashvili equation

The Kadomtsev–Petviashvili (KP) equation is a generalized form of the Korteweg–de Vries equation for a three-dimensional channel flow. Three-dimensional fully localized solitary wave solutions can be obtained when the effect of surface tension is significant; we refer to this as the KP-I equation. An exact solution of a three-dimensional fully localized stationary solution for the KP-I equation has been found in the absence of any obstacle on the rigid channel bottom. However, three-dimensional fully localized stationary solutions for the KP-I equation have not been found in the presence of a forcing neither analytically nor numerically. This forcing term comes from some obstacles in the rigid channel bottom. In this work, we focus on the fully localized solitary wave solutions when a positive bump or a negative hole is given as the rigid bottom configuration. The forced KP-I equation is defined in an infinite domain and it is reduced to a finite computational domain by introducing artificial boundary conditions. Interestingly, there are at least two distinct stationary lump-type solutions for a positive bump, while there are at least three distinct stationary lump-type solutions for a negative hole. Furthermore, we investigate their numerical stability by solving the forced time-dependent KP-I equation using them as initial conditions. Our numerical results confirm that there exists a stable stationary lump-type solitary wave solution for both a positive bump and a negative hole when they evolve in time.

Stability of Dissipative Optical Solitons in the 2D Complex Swift-Hohenberg Equation

This article deals with stationary localized solutions of the (2D) two-dimensional complex Swift-Hohenberg equation (CSHE). Our approach is based on the semi-analytical method of collective coordinate approach. According to the parameters of the equation and a suitable choice of ansatz, the stationary dissipative solitons of the 2D CSHE equation are mapped. This approach allows to describe the influence of the parameters of the equation on the various physical parameters of the pulse and their dynamics. Finally, the major impact of spectral filtering terms on the dynamic of the solitons is demonstrated.

Stationary discrete solitons in a driven dissipative Bose-Hubbard chain

We demonstrate that stationary localized solutions (discrete solitons) exist in a one dimensional Bose-Hubbard lattices with gain and loss in the semiclassical regime. Stationary solutions, by defi- nition, are robust and do not demand for state preparation. Losses, unavoidable in experiments, are not a drawback, but a necessary ingredient for these modes to exist. The semiclassical calculations are complemented with their classical limit and dynamics based on a Gutzwiller Ansatz. We argue that circuit QED architectures are ideal platforms for realizing the physics developed here. Finally, within the input-output formalism, we explain how to experimentally access the different phases, including the solitons, of the chain.

Periodic and localized solutions in chains of oscillators with softening or hardening cubic nonlinearity

Spatially periodic and stationary localized solutions arising from the dynamics of chains of linearly coupled mechanical oscillators characterized by on site cubic nonlinearity are addressed aiming to explore their relationship with the underlying nonlinear wave propagation regions. Softening and hardening nonlinearities are considered, and regions of occurrence of discrete breathers and multibreathers associated with homoclinic or heteroclinic connections are identified.

Iterative Schemes for Bump Solutions in a Neural Field Model

We develop two iteration schemes for construction of localized stationary solutions (bumps) of a one-population Wilson–Cowan model with a smooth firing rate function. The first scheme is based on the fixed point formulation of the stationary Wilson–Cowan model. The second one is formulated in terms of the excitation width of a bump. Using the theory of monotone operators in ordered Banach spaces we justify convergence of both iteration schemes.

The short envelope soliton dynamics in inhomogeneous dispersive media with allowance for stimulated scattering by damped low-frequency waves

We consider the soliton dynamics in terms of the extended nonlinear Schrodinger equation taking into account the inhomogeneous linear second-order dispersion (SOD) and stimulated scattering by damped low-frequency waves (SSDW). It is shown that the wave number downshift due to SSDW is compensated by an upshift due to the SOD decrease on the spatial coordinate. A new class of stationary nonlinear localized solutions (solitons) arising as an equilibrium of SSDW and decreasing spatial SOD is found analytically within the framework of the extended inhomogeneous nonlinear Schrodinger equation. A regime of the dynamic equilibrium of SSDW and inhomogeneous dispersive medium with the soliton parameters periodically varied in time is found. Analytical and numerical results are in good agreement for this regime.

Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane

In this paper we present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincaré disc \(\mathbb{D }\). Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of \(\mathbb{D }\) also called H-planforms in reference with the “planforms” introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of \(\mathbb{D }\) and periodic in the “transverse” direction. We highlight our theoretical results with a selection of numerical simulations.