Rogue wave pattern of multi-component derivative nonlinear Schrödinger equations.

This paper studies the multi-component derivative nonlinear Schrödinger (n-DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n-DNLS equations. Specifically, we focus on the distinctive scenario where the (n+1)-order characteristic polynomial possesses an explicit (n+1)-multiple root. Additionally, we provide an in-depth analysis of the asymptotic dynamic behaviors and pattern classification inherent to the higher-order vector rogue wave solution of the n-DNLS equations, mainly when one of the internal arbitrary parameters is extremely large. These patterns are related to the root structures in the generalized Wronskian-Hermite polynomial hierarchies.

Phase characters of optical dark solitons with third-order dispersion and delayed nonlinear response.

Dark soliton is usually seen as one of the simplest topological solitons, due to phase shift across its intensity dip. We investigate phase characters of single-valley dark soliton (SVDS) and double-valley dark soliton (DVDS) in a single-mode optical fiber with third-order dispersion and delayed nonlinear response. Notably, two different phase shifts can produce an SVDS with the same velocity under some conditions, which is not admitted for a dark soliton with only the second-order dispersion and self-phase modulation, whose phase shift and velocity is a one-to-one match. This phase property of SVDS can be used to explain the generation of previously reported DVDS in Hirota equation and make DVDSs show two types of phase profiles. Moreover, the different topological vector potentials underlying the distinct phase profiles have been uncovered. We further explore the collision properties of the DVDSs by analyzing their topological phases. Strikingly, the inelastic collision can lead to the conversion between the two types of phase profiles for DVDS. The results reveal that inelastic or elastic collision can be judged by analyzing virtual topological magnetic fields.

Measuring the rogue wave pattern triggered from Gaussian perturbations by deep learning.

Weak Gaussian perturbations on a plane wave background could trigger lots of rogue waves (RWs), due to modulational instability. Numerical simulations showed that these RWs seemed to have similar unit structure. However, to the best of our knowledge, there are no relative results to prove that these RWs have the similar patterns for different perturbations, partly due to that it is hard to measure the RW pattern automatically. In this work, we address these problems from the perspective of computer vision via using deep neural networks. We propose a rogue wave detection network (RWD-Net) model to automatically and accurately detect RWs in the images, which directly indicates they have the similar computer vision patterns. For this purpose, we herein meanwhile have designed and release the corresponding dataset, termed as rogue wave dataset-10K (RWD-10K), which has 10191 RW images with bounding box annotations for each RW unit. In our detection experiments, we get 99.29% average precision on the test splits of the proposed dataset. Finally, we derive our metric, termed as the density of RW units, to characterize the evolution of Gaussian perturbations and obtain the statistical results on them.

Clustered piperidinium-functionalized poly(terphenylene) anion exchange membranes with well-developed conductive nanochannels.

Anion exchange membrane fuel cells (AEMFCs) attract considerable attention owing to their high-power density and potential utilization of cheap non-noble metal catalysts. However, anion exchange membranes (AEMs) still face the problems of low conductivity, poor dimensional and chemical stability. To address these issues, AEMs with clustered piperidinium groups and ether-bond-free poly(terphenylene) backbone (3QPAP-x, x = 0.3, 0.4, and 0.5) were designed. Transmission electron microscope results show that the clustered ionic groups are responsible for fabricating well-developed conductive nanochannels and restraining the swelling behavior of the membranes. 3QPAP-0.4 and 3QPAP-0.5 AEMs exhibit higher conductivity (117.5 mS cm-1, 80 °C) and lower swelling ratio than that of commercial FAA-3-50 (80.4 mS cm-1, 80 °C). The conductivity of 3QPAP-0.5 only decreased by 10.4% after treating with 1 M NaOH at 80 °C for 720 h. The Hofmann elimination degradation of the cationic groups is restrained by the long flexible alkyl chain between cations. Based on the high performance of 3QPAP-0.5, an H2-O2-type AEMFC reaches 291.2 mW cm-2 (60 °C), which demonstrates that the as-prepared AEMs are promising for application in fuel cells.

Multivalley dark solitons in multicomponent Bose-Einstein condensates with repulsive interactions.

We obtain multivalley dark soliton solutions with asymmetric or symmetric profiles in multicomponent repulsive Bose-Einstein condensates by developing the Darboux transformation method. We demonstrate that the width-dependent parameters of solitons significantly affect the velocity ranges and phase jump regions of multivalley dark solitons, in sharp contrast to scalar dark solitons. For double-valley dark solitons, we find that the phase jump is in the range [0,2π], which is quite different from that of the usual single-valley dark soliton. Based on our results, we argue that the phase jump of an n-valley dark soliton could be in the range [0,nπ], supported by our analysis extending up to five-component condensates. The interaction between a double-valley dark soliton and a single-valley dark soliton is further investigated, and we reveal a striking collision process in which the double-valley dark soliton is transformed into a breather after colliding with the single-valley dark soliton. Our analyses suggest that this breather transition exists widely in the collision processes involving multivalley dark solitons. The possibilities for observing these multivalley dark solitons in related Bose-Einstein condensates experiments are discussed.

Parity-time-symmetric rational vector rogue waves of the n-component nonlinear Schrödinger equation.

Extreme events are investigated in the integrable n-component nonlinear Schrödinger (NLS) equation with focusing nonlinearity. We report novel multi-parametric families of rational vector rogue wave (RW) solutions featuring the parity-time ( PT) symmetry, which are characterized by non-identical boundary conditions for the components that are consistent with the degeneracy of n branches of Benjamin-Feir instability. Explicit examples of PT-symmetric rational vector RWs are presented. Subject to the specific choice of the parameters, high-amplitude RWs are generated. The effect of a small non-integrable deformation of the 3-NLS equation on the excitation of vector RWs is discussed. The reported results can be useful for the design of experiments for observation of high-amplitude RWs in multi-component nonlinear physical systems.

Coupling hollow Fe3O4 nanoparticles with oxygen vacancy on mesoporous carbon as a high-efficiency ORR electrocatalyst for Zn-air battery.

Designing a low-cost, high-efficiency and robust doped-carbon-based oxygen reduction reaction electrocatalyst for large-scale implementations of fuel cells is highly desirable but challenging. In this work, we report a new type of hollow Fe3O4 with oxygen vacancy incorporating on mesoporous carbon prepared by pyrolyzing mesoporous carbon enriched with oxygen-containing functional groups, in combination with ferric acetylacetonate. The catalysts possess high specific surface area with predominantly mesoporous architecture and ultrahigh nitrogen content (up to 7.47 wt%). Benefiting from the integration of abundant active nitrogen and Fe-Nx species, and synergistic effect between Fe3O4 nanoparticles cooperated with oxygen vacancy and N-doped carbon, the half-wave potential of the preparing hybrid catalyst is 30 mV more positive than that of the commercial Pt/C catalyst in alkaline medium, and exhibits a high selectivity (4 e- process), and outstanding long-term stability. More importantly, the C-FePPDA-900 catalyst displays a high power density (106 mW cm-2) and specific capacity of 724 mAh gzn-1 when it is used as an air cathode catalyst in a specifically assembling Zn-air cell, superior to those of most reported catalysts.

Nondegenerate bound-state solitons in multicomponent Bose-Einstein condensates.

We investigate nondegenerate bound-state solitons systematically in multicomponent Bose-Einstein condensates, through developing the Darboux transformation method to derive exact soliton solutions analytically. In particular, we show that bright solitons with nodes correspond to the excited bound states in effective quantum wells, in sharp contrast to the bright solitons and dark solitons reported before (which usually correspond to ground state and free state, respectively). We further demonstrate that bound-state solitons with nodes are induced by incoherent superposition of solitons in different components. Moreover, we reveal that the interactions between these bound-state solitons are usually inelastic, caused by the incoherent interactions between solitons in different components and the coherent interactions between solitons in the same component. Additionally, the detailed spectral stability analysis demonstrates the stability of nondegenerate bound-state solitons. The bound-state solitons can be used to study many different physical problems, such as beating dynamics, spin-orbit coupling effects, quantum fluctuations, and even quantum entanglement states.

Mechanism of Kuznetsov-Ma breathers.

We discuss how to understand the dynamical process of Kuznetsov-Ma breather, based on some basic physical mechanisms. It is shown that the dynamical process of Kuznetsov-Ma breather involves at least two distinctive mechanisms: modulational instability and the interference effects between a bright soliton and a plane-wave background. Our analysis indicates that modulational instability plays dominant roles in the mechanism of Kuznetsov-Ma breather admitting weak perturbations, and the interference effect plays a dominant role for the Kuznetsov-Ma breather admitting strong perturbations. For intermediate cases, the two mechanisms are both greatly involved. These characters provide a possible way to understand the evolution of strong perturbations on a plane-wave background.

Remarkable Effect of Sodium Alginate Aqueous Binder on Anatase TiO2 as High-Performance Anode in Sodium Ion Batteries.

Sodium alginate (SA) is investigated as the aqueous binder to fabricate high-performance, low-cost, environmentally friendly, and durable TiO2 anodes in sodium-ion batteries (SIBs) for the first time. Compared to the conventional polyvinylidene difluoride (PVDF) binder, electrodes using SA as the binder exhibit significant promotion of electrochemical performances. The initial Coulombic efficiency is as high as 62% at 0.1 C. A remarkable capacity of 180 mAh g-1 is achieved with no decay after 500 cycles at 1 C. Even at 10 C (3.4 A g-1), it remains 82 mAh g-1 after 3600 cycles with approximate 100% Coulombic efficiency. TiO2 electrodes with SA binder display less electrolyte decomposition, fewer side reactions, high electrochemistry reaction activity, effective suppression of polarization, and good electrode morphology, which is ascribed to the rich carboxylic groups, high Young's modulus, and good electrochemical stability of SA binder.

3D Electronic Channels Wrapped Large-Sized Na3 V2 (PO4 )3 as Flexible Electrode for Sodium-Ion Batteries.

The development of portable and wearable electronics has aroused the increasing demand for flexible energy-storage devices, especially for the characteristics of high energy density, excellent mechanical properties, simple synthesis process, and low cost. However, the development of flexible electrodes for sodium-ion batteries (SIBs) is still limited due to the intricate production methods and the relatively high-cost of current collectors such as graphene/graphene oxide and carbon nanotubes. Here, the hierarchical 3D electronic channels wrapped large-sized Na3 V2 (PO4 )3 is designed and fabricated by a simple electrospinning technique. As flexible electrode material, it exhibits outstanding electrolyte wettability, together with ultrafast electronic conductivity and high Na-ion diffusion coefficients for SIBs, leading to superior electrochemical performances. A high reversible specific capacity of 116 mA h g-1 (nearly 99% of the theoretical specific capacities) can be obtained at the current density of 0.1 C. Even after a 300-fold current density increased (30 C), the discharge specific capacity of the flexible electrode still remains 63 mA h g-1 . Such an effective concept of fabricating 3D electronic channels for large-sized particles is expected to accelerate the practical applications of flexible batteries at various systems.

Quick Activation of Nanoporous Anatase TiO2 as High-Rate and Durable Anode Materials for Sodium-Ion Batteries.

To understand the slow capacity activation behavior of anatase TiO2 as a sodium-ion battery anode during cycling, a nanoporous configuration was designed and prepared. On the basis of the comprehension of the Na-ion storage mechanism, the behavior is demonstrated to be related with the gradual formation of amorphous phase resulting from the phase transition during discharge. In addition, the level of phase transition is determined by the discharge rates and cycle numbers, which strongly affects the electrochemical performance of anatase TiO2. Via a quick formation process of the amorphous phase in the initial cycles, the capacity activation is accelerated, and high initial capacity is achieved with no fading after 500 cycles. Particularly, anatase TiO2 displays surprisingly unique properties in the fast charge (even at 20 C, 6.7 A g-1) mode, delivering a 179 mA h g-1 charge capacity. This study is significant for the comprehensive understanding of the controversial sodium storage mechanisms and unclear special behaviors occurring in anatase TiO2, thus greatly contributing to better guidance on the computational studies and experiment technologies for further performance promotion.

Reply to "Comment on 'Defocusing complex short-pulse equation and its multi-dark-soliton solution' ".

Our paper [Phys. Rev. E 93, 052227 (2016)PREHBM2470-004510.1103/PhysRevE.93.052227], proposing an integrable model for the propagation of ultrashort pulses, has recently received a Comment by Youssoufa et al. [Phys. Rev. E 96, 026201 (2017)10.1103/PhysRevE.96.026201] about a possible flaw in its derivation. We point out that their claim is incorrect since we have stated explicitly that a term is neglected to derive our model equation in our paper. Furthermore, the integrable model is validated by comparing with the normalized Maxwell equation and other known integrable models. Moreover, we show that a similar approximation has to be performed in deriving the same integrable equation as explained in the Comment.

Generation mechanisms of fundamental rogue wave spatial-temporal structure.

We discuss the generation mechanism of fundamental rogue wave structures in N-component coupled systems, based on analytical solutions of the nonlinear Schrödinger equation and modulational instability analysis. Our analysis discloses that the pattern of a fundamental rogue wave is determined by the evolution energy and growth rate of the resonant perturbation that is responsible for forming the rogue wave. This finding allows one to predict the rogue wave pattern without the need to solve the N-component coupled nonlinear Schrödinger equation. Furthermore, our results show that N-component coupled nonlinear Schrödinger systems may possess N different fundamental rogue wave patterns at most. These results can be extended to evaluate the type and number of fundamental rogue wave structure in other coupled nonlinear systems.

Defocusing complex short-pulse equation and its multi-dark-soliton solution.

In this paper, we propose a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrödinger (NLS) equation in the ultrashort-pulse regime. Furthermore, we construct the multi-dark-soliton solution for the defocusing complex short-pulse equation through the Darboux transformation and reciprocal (hodograph) transformation. One- and two-dark-soliton solutions are given explicitly, whose properties and dynamics are analyzed and illustrated.

W-shaped solitons generated from a weak modulation in the Sasa-Satsuma equation.

We study rational solutions of continuous wave backgrounds with the critical frequencies of the Sasa-Satsuma equation, which can be used to describe the evolution of the optical field in a nonlinear fiber with some high-order effects. We find a striking dynamical process that two W-shaped solitons are generated from a weak modulation signal on the continuous wave backgrounds. This provides a possible way to obtain stable high-intensity pulses from a low-intensity continuous wave background. The process involves both modulational instability and modulational stability regimes, in contrast to the rogue waves and W-shaped solitons reported before which involve modulational instability and stability, respectively. Furthermore, we present a phase diagram on a modulational instability spectrum plane for the fundamental nonlinear localized waves obtained already in the Sasa-Satsuma equation. The interactions between different types of nonlinear localized waves are discussed based on the phase diagram.

Integrable pair-transition-coupled nonlinear Schrödinger equations.

We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.

Vector bright solitons associated with positive coherent coupling via Darboux transformation.

Describing coherently coupled and orthogonally polarized waveguide modes in the Kerr medium, vector bright solitons associated with positive coherent coupling are studied in this paper. Some conserved quantities and infinitely many conservation laws are computed, and the existence of Lax pair indicates the integrability of the two-coupled nonlinear Schrödinger system with positive coherent coupling. Performing the iterative algorithm of Darboux transformation, we present formulas of one-, two-, and even N-soliton solutions. With appropriate choices of the phase parameters, collision mechanisms of vector bright solitons (of single-hump, double-hump, or flat-top profiles) are displayed, which show the elastic collision under the combined influences of group velocity dispersion, self-phase modulation, cross-phase modulation, and positive coherent coupling.

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation.

We investigate the solution in rational form for the Sasa-Satsuma equation on a continuous background which describes a nonlinear fiber system with higher-order effects including the third-order dispersion, Kerr dispersion, and stimulated inelastic scattering. The W-shaped soliton in the system is obtained analytically. It is found that the height of hump for the soliton increases with decreasing the background frequency in certain parameter regime. The maximum height of the soliton can be three times the background's height and the corresponding profile is identical with the one for the well-known eye-shaped rogue wave with maximum peak. The numerical simulations indicate that the W-shaped soliton is stable with small perturbations. Particularly, we show that the W-shaped soliton corresponds to a stable supercontinuum pulse by performing exact spectrum analysis.

High-order rogue waves in vector nonlinear Schrödinger equations.

We study the dynamics of high-order rogue waves (RWs) in two-component coupled nonlinear Schrödinger equations. We find that four fundamental rogue waves can emerge from second-order vector RWs in the coupled system, in contrast to the high-order ones in single-component systems. The distribution shape can be quadrilateral, triangle, and line structures by varying the proper initial excitations given by the exact analytical solutions. The distribution pattern for vector RWs is more abundant than that for scalar rogue waves. Possibilities to observe these new patterns for rogue waves are discussed for a nonlinear fiber.