Search

Under investigation in this letter is an (3+1)-dimensional Hirota–Satsuma–Ito-like equation, which provide strong support for studying the dynamic behavior of nonlinear waves. Based on a special Cole–Hopf transformation and Hirota bilinear method, the bilinear form of the equation is obtained and this form has never been given. High-order breather solutions, lump solutions and mixed solutions are obtained by using complex conjugate parameters and long-wave limit method. Then, the influence of the coefficient $$g_{t}(t)$$ of the bilinear equation on the interaction of these solutions is analyzed by means of images. It can be found that $$g_{t}(t)$$ changes the interaction of the solutions by influencing the positions and trajectories of higher-order breather solutions, lump solutions and mixed solutions. We find that different values of g(t) make the interaction of solutions different. Finally, the mixed solution of the equation including a breather wave and a line rogue wave is obtained by using the test function, and its dynamic properties are illustrated by means of images.

The present paper studied multi-breather, multi-lump and hybrid solutions of a novel KP-like equation by applying the Hirota bilinear method. The multi-breather solutions including one-, two- and three-breather and hybrid solutions between breathers and solitons were obtained through applying the complex conjugate method on the N-soliton solution. Via using the long wave limit method with respect to N-solution, the solution expression of multi-lump solutions was acquired; therefore, by choosing appropriate parameters on 2-, 3-, 4-, 5- and 6-soliton, one-, two-, three-lump and two types of hybrid solutions between lumps and solitons were derived. Additionally, the one-breather and one-lump solutions have been studied in-depth. In order to illustrate the wave trajectories, wave shapes and the fusion and fission process, several cases of plots with physical interpretation were given. Furthermore, the obtained solutions can be widely used to explain many interesting physical phenomena in the nature.

In this paper, based on the bilinear form, we give multi-solitary wave solutions of the $$(4+1)$$ -dimensional Fokas equation. From the obtained multi-solitary wave solutions, with special parameters, we derive resonant solutions of N-solitary waves. For complex conjugate parameters, we analyze interactions of two periodic waves, interactions of a solitary wave and a periodic wave by making use of their phase shifts. Particularly, the intermediate processes of elastic interactions are analyzed in detail, and interesting fusional and fissionable phenomena are found. The asymptotic interactional behaviors for these solutions are analyzed theoretically and illustrated graphically.

In the present paper, taking advantage of the bilinear method, we attained N-soliton solutions of the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (gCH-KP) equation. By combining the long wave limit method and the complex conjugate method to N-soliton solutions, M-lump wave solutions including one-lump wave, two-lump wave and three-lump wave and two types of hybrid solutions including hybrid solution between lump wave and solitons and between M-lump wave and soliton were obtained. In addition, several groups of images exhibited their dynamic structure and physical properties with physically interpreted via symbolic computation. Finally, resonant multi-soliton wave solutions were obtained by carrying out the linear superposition principle and the progresses of wave motions with physical interpretation represented graphically. Furthermore, the received results have immensely augmented the exact solutions of (3+1)-dimensional gCH-KP equation on the available literature and enabled us to understand the nonlinear dynamic system deeply. In addition, the proposed methods will be a strong boost to the calculation method of nonlinear equations.

In the classically unbroken phase, 3d $O(N)$ symmetric $\phi^4$ vector models admit two equivalent descriptions connected by a strong-weak duality closely related to the one found by Chang and Magruder long ago. We determine the exact analytic renormalization dependence of the critical couplings in the weak and strong branches as a function of the renormalization scheme (parametrized by $\kappa$) and for any $N$. It is shown that for $\kappa=\kappa_*$ the two fixed points merge and then, for $\kappa, Comment: 38 pages, 12 figures; v3: version to appear in JHEP

The phenomenon, which is classified as a complex resonance and takes place in electromagnetic structures described by nonself-adjoint boundary value problems, is considered. It is shown that the oscillation adjoint to a source is formed, when complex waves that are complex conjugate by wave numbers and amplitudes are excited in couples. This oscillation is described by the self-consistent boundary value problem for the adjoint Helmholtz equation, i.e., for the equation, whose right-hand side is the solution of the homogeneous boundary value problem. The distinctive feature of the complex resonance is in the fact that it exists in the whole range of complex waves, when the source is necessarily present. Theoretical and experimental results of the investigation of the considered phenomenon are presented.

Background: The knowledge of parameter estimation for interatomic potentials is useful in the computation of the vibrational structure of van der Waals molecules. Methods: On the estimation of the Generalized Morse and Classical Lennard-Jones potential energy functions, complex conjugates eigenvalues may be obtained. Different approaches can be used to solve this resulting problem. A method that uses the objective least squares function method to estimate parameters of the interatomic potentials is employed. Results: Numerical simulation of the systems using metal atoms yields complex conjugates eigenvalues at some initial point. Conclusion: Other approaches of solving the complex conjugates eigenvalues problem are discussed comprehensively.

In this paper, a generalized $$(2+1)$$-dimensional nonlinear wave equation is obtained by extending the generalized $$(2+1)$$-dimensional Hirota bilinear equation into a more generalized form. The obtained new equation is useful in describing nonlinear wave phenomena in nonlinear optics, shallow water and oceanography. Based on the bilinear method, the N-soliton solutions of the generalized $$(2+1)$$-dimensional nonlinear wave equation are obtained. M-lump solutions are investigated by applying the long wave limit to the N-soliton solutions. The propagation orbits and velocities of the M-lump wave are analyzed. The high-order breather waves are obtained by establishing the complex conjugate relations in the parameters of the N-solitons. Furthermore, the interaction hybrid solutions are constructed, which contain hybrid solutions composed of breathers, solitons and lumps. The dynamical behaviors of the hybrid solutions are systematically analyzed via numerical simulations. The obtained results will enrich the study of theory of the nonlinear localized waves.

Under investigation in this letter is an extended (3 + 1)-dimensional Jimbo–Miwa (eJM) equation, which can be used to describe many nonlinear phenomena in mathematical physics. With the aid of Hirota bilinear method and long-wave limit method, M-order lumps which describe multiple collisions of lumps are derived. The propagation orbit, velocity and extremum of the 1-order lump solutions on (x, y) plane are investigated in detail. Resorting to the extended homoclinic test technique, we obtain the breather–kink solutions, rational breather solutions and rogue wave solutions for the eJM equation. Meanwhile, through analysis and calculation, the amplitude and period of breather–kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived. T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions. Periods of the 1-order breather solutions on the (x, y) plane are determined by $$k_{12}$$ and $$k_{12}p_{11}+k_{11}p_{12}$$, while locations are determined by $$k_{11}$$ and $$k_{11}p_{11}-k_{12}p_{12}$$. Furthermore, hybrid solutions composed of the kink solitons, breathers and lumps for the eJM equation are worked out. Some figures are given to display the dynamical characteristics of these solutions.

A new analytic method to calculate the coupling coefficient including self-inductances and mutual inductance between two arbitrary-shaped coils is proposed. The proposed method is applicable for all common coil geometries such as rectangular, square, polygonal, circular coils etc. In the formula, the coil function, which is a completely a new type of function, is defined. The differences in shape and position of the coils are only on the choice of the coil function and its complex conjugate. The layouts of the two coils including separation change and lateral misalignment are discussed. An experimental setup has been built to validate the analytic calculations. Theoretical and experimental results are then compared to be in good agreement, which verify the effectiveness of the proposed method.

In the framework of the Stueckelberg-Wheeler-Feynman concept of a ``one-electron Universe'' we consider a worldline implicitly defined by a system of algebraic (precisely, polynomial) equations. Collection of pointlike ``particles'' of two kinds on the worldline (or its complex extension) is defined by the real (complex conjugate) roots of the polynomial system and detected then by an external inertial observer through the light cone connections. Then the observed collective dynamics of the particles' ensemble is, generally, subject to a number of Lorentz invariant conservation laws. Remarkably, this poperty follows from the Vieta's formulas for the roots of the generating polynomial system. At some discrete moments of the observer's proper time, mergings and subsequent transmutations of a pair of particles-roots take place simulating thus the processes of annihilation/creation of a particle/antiparticle pair, Comment: 7 pages, twoside, 1 figure

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

We propose and investigate several complex versions of extensions and restrictions of the Skyrme model with a well-defined Bogomolny-Prasad-Sommerfield (BPS) limit. The models studied possess complex kink, anti-kink, semi-kink, massless and purely imaginary compacton BPS solutions that all have real energies. The reality of the energies for a particular solution is guaranteed when a modified antilinear CPT-symmetry maps the Hamiltonian functional to its parity time-reversed complex conjugate and the solution field to itself or a new field with degenerate energy. In addition to the known BPS Skyrmion configurations we find new types that we refer to as step, cusp, shell, and purely imaginary compacton solutions., 21 pages, 7 figures

The interpretation of optical spectra requires thorough comprehension of quantum mechanics, especially understanding the concept of angular momentum operators. Suppose now that a transformation from laboratory-fixed to molecule-attached coordinates, by invoking the correspondence principle, induces reversed angular momentum operator identities. However, the foundations of quantum mechanics and the mathematical implementation of specific symmetries assert that reversal of motion or time reversal includes complex conjugation as part of anti-unitary operation. Quantum theory contraindicates sign changes of the fundamental angular momentum algebra. Reversed angular momentum sign changes are of a heuristic nature and are actually not needed in analysis of diatomic spectra. This review addresses sustenance of usual angular momentum theory, including presentation of straightforward proofs leading to falsification of the occurrence of reversed angular momentum identities. This review also summarizes aspects of a consistent implementation of quantum mechanics for spectroscopy with selected diatomic molecules of interest in astrophysics and in engineering applications.

The effects of gain and loss on the band structures of a bulk topological dielectric photonic crystal (PC) with $C_{6v}$ symmetry and the PC-air-PC interface are studied based on first-principle calculation. To illustrate the importance of parity-time (PT) symmetry, three systems are considered, namely the PT-symmetric, PT-asymmetric, and lossy systems. We find that the system with gain and loss distributed in a PT symmetric manner exhibits a phase transition from a PT exact phase to a PT broken phase as the strength of the gain and loss increases, while for the PT-asymmetric and lossy systems, no such phase transition occurs. Furthermore, based on the Wilson loop calculation, the topology of the PT-symmetric system in the PT exact phase is demonstrated to keep unchanged as the Hermitian system. At last, different kinds of edge states in Hermitian systems under the influences of gain and loss are studied and we find that while the eigenfrequencies of nontrivial edge states become complex conjugate pairs, they keep real for the trivial defect states., 6 pages, 6 figures

In this paper, we investigate two extended higher-order KdV models (i.e., the extended Sawada–Kotera equation and the extended Lax equation), which can successfully describe propagation of dimly nonlinear long waves in fluids and ion-acoustic waves in harmonic sparklers. First, we present a general formula of multisoliton solutions of the two models. We then build the interaction solutions in terms of hyperbolic and sinusoidal functions by using multisoliton solutions with appropriate complex conjugate parameters controlling the phase shifts, propagation direction and energies of the waves. In particular, we present their collision solutions in the identical plane with different parametric constraints, which degenerate to the line rogue waves, x-shaped rogue waves, cnoidal periodic waves, interactions of rogue and bell waves, line breather and double breather waves. The dynamical characteristics of the wave solutions are shown graphically by choosing some special parameter values.

We study many-body quantum dynamics using Floquet quantum circuits in one space dimension as simple examples of systems with local interactions that support ergodic phases. Physical properties can be expressed in terms of multiple sums over Feynman histories, which for these models are paths or many-body orbits in Fock space. A natural simplification of such sums is the diagonal approximation, where the only terms that are retained are ones in which each path is paired with a partner that carries the complex conjugate weight. We identify the regime in which the diagonal approximation holds, and the nature of the leading corrections to it. We focus on the behaviour of the spectral form factor (SFF) and of matrix elements of local operators, averaged over an ensemble of random circuits, making comparisons with the predictions of random matrix theory (RMT) and the eigenstate thermalisation hypothesis (ETH). We show that properties are dominated at long times by contributions to orbit sums in which each orbit is paired locally with a conjugate, as in the diagonal approximation, but that in large systems these contributions consist of many spatial domains, with distinct local pairings in neighbouring domains. The existence of these domains is reflected in deviations of the SFF from RMT predictions, and of matrix element correlations from ETH predictions; deviations of both kinds diverge with system size. We demonstrate that our physical picture of orbit-pairing domains has a precise correspondence in the spectral properties of a transfer matrix that acts in the space direction to generate the ensemble-averaged SFF. In addition, we find that domains of a second type control non-Gaussian fluctuations of the SFF. These domains are separated by walls which are related to the entanglement membrane, known to characterise the scrambling of quantum information., 22+7 pages

The versatile and exactly solvable Scarf II has been predicting, confirming and demonstrating interesting phenomena in complex PT-symmetric sector, most impressively. However, for the non-PT-symmetric sector it has gone underutilized. Here, we present most simple analytic forms for the scattering coefficients $(T(k),R(k),|\det S(k)|)$. On one hand, these forms demonstrate earlier effects and confirm the recent ones. On the other hand they make new predictions - all simply and analytically. We show the possibilities of both self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT sectors (potentials). The former one is not accompanied by time-reversed coherent perfect absorption (CPA) and gives rise to the parametrically controlled splitting of SS in to a finite number of complex conjugate pairs of eigenvalues (CCPEs). The latter ones (NSDSS) behave just oppositely: CPA but no splitting of SS. We demonstrate a one-sided reflectionlessness without invisibility. Most importantly, we bring out a surprising co-existence of both real discrete spectrum and a single SS in a fixed potential. Nevertheless, the complex Scarf II is not known to be pseudo-Hermitian ($\eta^{-1} H\eta=H^\dagger$) under a metric of the type $\eta(x)$, so far., Comment: 7 PAGES, 4 FIGURES AND NO TABLE. A new para after Eq. (6) has been added

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6-j symbols, about which almost nothing is known for $N>2$, with the exception of the tetrahedral symmetries, complex conjugation and transformation $q \longleftrightarrow q^{-1}$. In this paper we prove the eigenvalue hypothesis in $U_q(sl_2)$ case and show that it is equivalent to 6-j symbol symmetries (the Regge symmetry and two argument permutations). Then we apply the eigenvalue hypothesis to inclusive Racah matrices with 3 symmetric incoming representations of $U_q(sl_N)$ and an arbitrary outcoming one. It gives us 8 new additional symmetries that are not tetrahedral ones. Finally, we apply the eigenvalue hypothesis to exclusive Racah matrices with symmetric representations and obtain 4 tetrahedral symmetries., 22 pages

The directional properties of a dipole antenna are studied by placing it in the vicinity of a complex conjugate cylinder. An exact formal solution for the fields is obtained via the differential eq...

Light is a powerful tool to manipulate matter, but existing approaches often necessitate focused, high-intensity light that limits the manipulated object’s shape, material and size. The breakthrough development could lead to spacecraft without fuel. What if a spacecraft could travel through our solar system powered and accelerated using only light? That’s the goal of new research coming out of Caltech. More federal spending on directed energy weapon research & development has some stakeholders looking for operational systems to deploy in the next two or three years. The weapons which use focused energy in the forms of lasers, microwaves and other methods against targets ranging from drone swarms to ballistic missiles have long drawn the interest of the Department of Defense and its military services but have previously been relegated largely to the arena of the theoretical. They have artificially created a state that evolves in a direction opposite that of the thermodynamic arrow of time. Scientists have reversed the direction of time with a quantum computer.

We consider the second order homogeneous linear differential equation (H)$${ ay'' + by' + cy = 0 }$$ with real coefficients a, b, c, and a ≠ 0. The function y = emx is a solution if, and only if, m satisfies the auxiliary equation am2 + bm + c = 0. When the roots of this are the complex conjugates m = p ± iq, then y = e(p ± iq)x are complex solutions of (H). Nevertheless, real solutions are given by y = c1epx cos qx + c2epx sin qx.

A boundary value problem involving the scattering of electromagnetic waves from a sphere of complex conjugate medium (CCM) is studied. The sphere is placed in free space. The source of excitation for the sphere in our case is a plane wave. Incident, scattered and transmitted fields are formulated. The unknown coefficients in the scattered and transmitted fields are found using boundary conditions. From these electromagnetic fields, the Mie efficiencies are determined. The technique used in studying the scattering of electromagnetic waves from CCM is analytical and a closed form solution is obtained. It is shown by numerical results that the scattering is enhanced in case of CCM sphere as a target. Results for the limiting cases are also derived to compare the validity of our formulation with the published work.

In this paper, we presented a study on a non-smooth continuous system with emphasis on a special bifurcation. As the parameter varies, a series of concentric closed orbits appear near the equilibrium point. Moreover, the outermost closed orbit attracts all the trajectories outside. It is called as a semi-limit cycle as the trajectories at only one side of this orbit are attracted. By using the theory of generalized Jacobian matrix, it is revealed that this bifurcation can be featured by a pair of complex conjugate eigenvalues reaching exactly but not crossing the imaginary axis. The bifurcation can somewhat be considered to be a degenerate case of the Hopf bifurcation, in which the eigenvalues cross the imaginary axis totally. This study enriches the knowledge of bifurcation analysis for non-smooth dynamical systems.

The anomalous end modes are generated in a one-dimensional p-wave superconducting wire with second-neighbor couplings and a periodic driving in pairing potential. We show that the driving on the amplitude or the phase of the first-neighbor and second-neighbor pairing potential can both generate conventional end modes and anomalous end modes, with corresponding Floquet eigenvalues equal to ± 1 or appear in complex conjugate pairs. We have numerically studied the driving of the first-neighbor and second-neighbor pairing potential, and also analyzed the end modes, Floquet eigenvalues and the Fourier transform of these end modes.

This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter γ . We show that all the eigenvalues are real and finite for a small chirality γ . For a critical value γ = γ ⁎ , the GEP has 2 × 2 Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when γ increases from γ ⁎ , the 2 × 2 Jordan block will first split into a complex conjugate eigenpair, then rapidly collide with the real axis and bifurcate into positive (resonance) and negative eigenvalues with modulus smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material.

This study examines the instability and dynamical transitions of the two-layer western boundary currents represented by the Munk profile in the upper layer and a motionless bottom layer in a closed rectangular domain. First, a bound on the intensity of the Munk profile below which the western boundary currents are locally nonlinearly stable is provided. Second, by reducing the infinite dimensional system to a finite dimensional one via the center manifold reduction, non-dimensional transition numbers are derived, which determine the types of dynamical transitions both from a pair of simple complex eigenvalues as well as from a double pair of complex conjugate eigenvalues as the Reynolds number crosses a critical threshold. We show by careful numerical estimations of the transition numbers that the transitions in both cases are continuous at the critical Reynolds number. After the transition from a pair of simple complex eigenvalue, the western boundary layer currents turn into a periodic circulation, whereas a quasi-periodic or possibly a chaotic circulation emerges after the transition from a pair of double complex eigenvalues. Finally, a comparison between the transitions exhibited in one-layer and two-layer models is provided, which demonstrates the fundamental differences between the two models.

Understanding vanishing transmission in Fano resonances in quantum systems and metamaterials and perfect and ultralow transmission in disordered media has advanced the knowledge and applications of wave interactions. Here, we use analytic theory and numerical simulations to understand and control the transmission and transmission time in complex systems by deforming a medium and adjusting the level of gain or loss. Unlike the zeros of the scattering matrix, the position and motion of the zeros of the determinant of the transmission matrix (TM) in the complex plane of frequency and field decay rate have robust topological properties. In systems without loss or gain, the transmission zeros appear either singly on the real axis or as conjugate pairs in the complex plane. As the structure is modified, two single zeros and a complex conjugate pair of zeros may interconvert when they meet at a square root singularity in the rate of change of the distance between the transmission zeros in the complex plane with sample deformation. The transmission time is the spectral derivative of the argument of the determinant of the TM. It is a sum over Lorentzian functions associated with the resonances of the medium, which is the density of states, and with the zeros of the TM. Transmission vanishes, and the transmission time diverges as zeros are brought near the real axis. Monitoring the transmission and transmission time when two zeros are close may open new possibilities for ultrasensitive detection.

This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellar systems. Using the simplest model of the homogeneous stellar medium, we discuss nontrivial features of stellar systems. Taking them into account will make it possible to correctly interpret the results obtained earlier and will help to set up decisive numerical experiments in the future. In particular, we compare the initial value problem versus the eigenvalue problem. It turns out that in the unstable regime, the Landau-damped waves can be represented as a superposition of van Kampen modes {\it plus} a discrete damped mode, usually ignored in the stability study. This mode is a solution complex conjugate to the unstable Jeans mode. In contrast, the Landau-damped waves are not genuine modes: in modes, eigenfunctions depend on time as $\exp (-{\rm i} \omega t)$, while the waves do not have eigenfunctions on the real $v$-axis at all. However, `eigenfunctions' on the complex $v$-contours do exist. Deviations from the Landau damping are common and can be due to singularities or cut-off of the initial perturbation above some fixed value in the velocity space., Comment: 9 pages, 10 figures

The aim is to model a system by an equivalent one which satisfactorily reproduces the system vibration behaviour over the frequency range of interest. This new system is described by relevant modal parameters which are to be identified from motion measurements when the system is subjected to periodic force excitation at select excitation frequencies. It is assumed that the new system can be represented by symmetric mass, damping and stiffness matrices. General viscous damping is assumed. Earlier work was unable to satisfactorily identify a simple two degree of freedom system when the motion measurement data was accurate to only two digits. Whereas theory requires that the system eigenvalues and eigenvectors be complex conjugates, this did not turn out to be the case. This paper uses two different identification procedures to investigate this anomaly and shows that for this simple system, the upper half plane eigenvalues can be identified far more accurately than their complex conjugates.

Suppose that in the theory of quantum mechanics the potential V is chosen in such a way that the Hamiltonian $$H = \frac{-\hbar ^2}{2m} \Delta + V$$ (\(\hbar \) being Planck’s constant) is self-adjoint in the Hilbert space \(\mathcal H= L^2(\mathbb R^n)\). Let \(\varphi _\hbar \) be the solution of the corresponding Schrodinger equation, i.e. $$ i\hbar \, \frac{\partial \varphi }{\partial t} = H \varphi , $$ and \(\varphi ^*_\hbar \) its complex conjugate.

In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the $SU(3)$ colour, the $SU(2)$ flavour, and broken $SU(3)$ providing the family triplets. \noindent In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quark's degrees of freedom which do appear in the Standard Model. Because the $SU(3)$ colour algebra is endowed with natural $Z_3$-graded discrete automorphisms, in order to introduce entanglement the $Z_3$-graded version of Lorentz and Poincar\'e algebras with their realizations are considered. The colour multiplets of quarks are described by $12$-component colour Dirac equations, with a $Z_3$-graded triplet of masses (one real and a Lee-Wick complex conjugate pair). We argue that all quarks in the Standard Model can be described by the $72$-component master quark sextet of $12$-component coloured Dirac fields., Comment: 39 pages, 3 figures. This version is conformal with the revised version published in the journal (Nucl. Phys. B)

Starting from the lattice Landau gauge gluon and ghost propagator data we use a sequence of Padé approximants, identify the poles and zeros for each approximant and map them into the analytic structure of the propagators. For the Landau gauge gluon propagator the Padé analysis identifies a pair of complex conjugate poles and a branch cut along the negative real axis of the Euclidean p2 momenta. For the Landau gauge ghost propagator the Padé analysis shows a single pole at p2=0 and a branch cut also along the negative real axis of the Euclidean p2 momenta. The method gives precise estimates for the gluon complex poles that agree well with other estimates found in the literature. For the branch cut the Padé analysis gives, at least, a rough estimate of the corresponding branch point.

Higher-spin theory contains a complex coupling parameter η. Different higher-spin vertices are associated with different powers of η and its complex conjugate $$ \overline{\eta} $$ η ¯ . Using Z-dominance Lemma of [1], that controls spin-locality of the higher-spin equations, we show that the third-order contribution to the zero-form B(Z; Y; K) admits a Z-dominated form that leads to spin-local vertices in the η2 and $$ {\overline{\eta}}^2 $$ η ¯ 2 sectors of the higher-spin equations. These vertices include, in particular, the η2 and $$ {\overline{\eta}}^2 $$ η ¯ 2 parts of the ϕ4 scalar field vertex.

We explicitly construct an integrable and strongly interacting dissipative quantum circuit via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneous transfer matrix, from which conserved superoperator charges can be derived, in particular, the circuit's dynamical generator. After showing the trace preservation and complete positivity of local maps, we reinterpret them as the Kraus representation of the local dynamics of free fermions with single-site dephasing. The integrability of the map is broken by adding interactions to the local coherent dynamics or by removing the dephasing. In particular, even circuits built from convex combinations of local free-fermion unitaries are nonintegrable. Moreover, the construction allows us to explicitly build circuits belonging to different non-Hermitian symmetry classes, which are characterized by the behavior under transposition instead of complex conjugation. We confirm all our analytical results by using complex spacing ratios to examine the spectral statistics of the dissipative circuits., 10 pages, 2 figures. v2: some additional details, as published

This paper presents a bi-directional N-type locally-active memristor (LAM) model, which has two symmetrical locally-active regions with respect to the origin. For this memristor, the locally-active regions coincide with the edge of chaos regimes, where complex dynamic behaviors may occur. By deriving the small-signal admittance of the memristor, it is found that the LAM may be pure resistive, inductive or capacitive in terms of biasing voltages. When the LAM operates at the edge of chaos and connects with an external inductor in series, a second-order periodic oscillator is established. In this case, it is possible to move the poles of the system from the left-half plane to the right-half plane, thereby generating periodic oscillations. Complex dynamics of the second-order memristor-based oscillator is analyzed via Hopf bifurcation analysis and based on the theories of local activity and edge of chaos. Using the parasitic capacitor of the LAM in the second-order oscillator, a third-order chaotic oscillator is designed, which has a negative real pole and a pair of complex conjugate poles with positive real parts, satisfying the Shilnikov criterion. It is found that both periodic oscillation and chaos appear only at the edge of chaos of the LAM, thus verifying the complexity theorem, no-complexity theorem and edge of chaos theorem. Finally, circuit experiments are performed, which verify the dynamics and the effectiveness of the LAM model, the periodic oscillator and the chaotic oscillator.

We investigate the parity-time (PT) phase transition in photonic crystals with $$C_{6v}$$ C 6 v symmetry, with balanced gain and loss on dielectric rods in the triangular lattice. A two-level non-Hermitian model that incorporates the gain and loss in the tight-binding approximation was employed to describe the dispersion of the PT symmetric system. In the unbroken PT phase, the double Dirac cone feature associated with the $$C_{6v}$$ C 6 v symmetry is preserved, with a frequency shift of second order due to the presence of gain and loss. The helical edge states with real eigenfrequencies can exist in the common band gap for two topologically distinct lattices. In the broken PT phase, the non-Hermitian perturbation deforms the dispersion by merging the frequency bands into complex conjugate pairs and forming the exceptional contours that feature the PT phase transition. In this situation, the band gap closes and the edge states are mixed with the bulk states.

We assess the calculation of hyperfine coupling (HFC) tensors by different variants of Projected Hartree-Fock (PHF) theory. For a set of small main-group S = ½ radicals (BO, CO+, CN, AlO, vinyl, me...

We perform complex analyses of the gluon propagator at nonzero quark chemical potential in the long-wavelength limit, using an effective model with a gluon mass term of the Landau-gauge Yang-Mills theory, which is a Landau-gauge limit of the Curci-Ferrari model with quantum corrections being included within the one-loop level. We mainly investigate complex poles of the gluon propagator, which could be relevant to confinement. Around typical values of the model parameters, we show that the gluon propagator has one or two pairs of complex conjugate poles depending on the value of the chemical potential. In addition to a pair similar to that in the case of zero chemical potential, a new pair appears near the real axis when the chemical potential is roughly between the effective quark mass and the effective gluon mass of the model. We discuss possible interpretations of these poles. Additionally, we prove the uniqueness of analytic continuation of the Matsubara propagator to a class of functions that vanish at infinity and are holomorphic except for a finite number of complex poles and singularities on the real axis., 18 pages, 14 figures, final version published in PRD

In Fourier domain optical coherence tomography (FDOCT), the depth profile is mirrored about the zero delay between the sample and reference optical paths, limiting the imaging depth to half of the entire ranging space and undermining the optimal sensitivity window. We present a new method, to the best of our knowledge, to remove the complex conjugate artifact by using circularly polarized light as reference. Quadrature detection of the complex fringe is achieved by utilizing the intrinsic λ / 4 delay between two polarization channels. We use passive broadband polarization optics to control the polarization state of the light in the reference and sample arms and a balanced polarization diversity detection unit to simultaneously detect phase-shifted fringes. We demonstrate a 40 dB artifact suppression ratio with a swept-source optical coherence tomography system. Our proposed method is immune to sample motion and laser phase noise, and imposes no restrictions to the source bandwidth, imaging speed, or computational power. In vivo images of the human finger, as well as the cornea and retina of a non-human primate, were demonstrated.

In this article, we study two extended higher-order KdV-type models, namely, the extended Sawada-Kotera (eSK) and the extended Lax (eLax) equations. These models successfully describe propagation of dimly nonlinear long waves in fluids, ion-acoustic waves in harmonic sparklers. We firstly derive multi-soliton solutions of the models. We then construct interection solutions in-terms of hyperbolic and sinusoidal functions using the multi-soliton solutions with appropriate complex conjugate parameters. Such parameters influence and control the phase shifts, propagation direction and energies of the waves. In particularly, we present their collision solutions in the identical plane with different parametric constraints, which degenerate to the line rogue waves, x-shaped rogue waves, cnoidal periodic waves, interactions of rogue and bell waves, line breather and double breather waves. The dynamical characteristics of the wave solutions has been plotted by choosing particular values of the parameters in graphically.

In this paper, we consider a new generalized KP equation which is obtained by adding the extra term $$u_{tz}$$ in the previous equation. Based on these detailed discussions in the previous reference documentations, we know that solitons, breathers, lump waves, and rogue waves are four typical local waves. Therefore, we mainly focus on investigating the multi-soliton solutions, high-order breather solutions, and high-order rational solutions. The high-order breather solutions can be derived by taking complex conjugate parameters in the multi-soliton solutions. Applying the long wave limit method to the multi-soliton solutions, we conclude Theorem 3.1 which can be used directly to obtain high-order rational solutions. Meanwhile, for the case of three-soliton and five-soliton, the elastic interaction solutions among two parallel breathers and one soliton as well as between one breather and one soliton also can be derived, respectively. For all these types of exact solutions, we provide corresponding graphics to illustrate their dynamical characteristics in the end.

In the field of meteor study using radio techniques, the transmitting antenna of the radio beacons is most of the time a classical turnstile antenna design. As an alternative, it is proposed a pyramidal shaped omnidirectional antenna with complex conjugate impedance, which exhibits similar performance. By design, this type of antenna does not require additional electrical components for obtaining the circular polarization and the impedance matching. Furthermore, the pyramidal shape allows the practical implementation of the antenna using a simple and light mechanical structure.

The study of the optical properties of plasmonic nanostructures in the stationary regime has greatly benefited from the development of numerical methods, among which Finite Difference Time Domain (FDTD) is popular. In contrast, the use of these numerical tools for assessing the transient plasmonic optical response triggered by ultrashort laser pulses is hampered by the difficulty to address small variations of the material optical properties with reasonable computational time. Yet, many of the developments based on this ultrashort response rely on the dynamics of the near-field topography around the nanostructures. In this article, we present a way to bridge this gap with the complex-conjugate pole-residue pair (CCPRP) approach. A CCPRP-based FDTD simulator has been developed. First, a simple methodology to check the end-to-end accuracy of the FDTD simulation is provided. Then, in conjunction with a three-temperature model, the approach enables us to calculate the ultrafast transient near field inside and around a gold nanoparticle (AuNP) upon absorption of a subpicosecond laser pulse. The transient variation of the field intensity inside and around the AuNP is compared with the one determined by the Mie theory. The dependence of the transient field intensity on the distance away from the nanoparticle surface and on the delay time after laser pulse absorption is finally analyzed.

In Fourier domain OCT, the depth profile is mirrored about the zero delay, limiting the imaging depth to half of the entire ranging space. We present a novel configuration for OCT to robustly remove the complex conjugate artifact. Our method utilizes the intrinsic delay of circularly polarized light in two polarization channels, using only passive broadband polarization optics and conventional polarization diversity detection unit. Our method is immune to sample motion and adds no restrictions to source bandwidth, imaging speed or computational load. 45 dB suppression of the mirror artifact is demonstrated by an SSOCT with some in-vivo images.

The acoustical radiosity model (ARM) is a typical algorithm in geometrical acoustics to simulate the sound field in a room of ideally diffusely reflecting boundary. Even in such a room, the sound field decay is a complex process, as the relaxation of the sound field observed in simulations has shown. Based on the Laplace transform of the acoustical radiosity equation, this paper gives a set of properties of the ARM. It shows the system has a series of real or complex conjugate L-eigenvalues and corresponding L-eigenfunctions. Under the relaxation condition, the sound energy decay on the room boundary, generated by finite initial excitation, can be expanded as a summation of decay components, which are composed of real and/or complex conjugate decay modes. Each decay mode is a decaying and oscillating L-eigenfunction corresponding to an L-eigenvalue. The real part of the L-eigenvalue is the exponential decay rate, and the image part is the angular frequency of the oscillation. The reverberant sound field inside the room space has a similar decay structure to the boundary. As an example, the decay structure in a sphere is analyzed. The relaxation of the sound field is explained by the geometrical significance of the sound field decay.

This paper extends the concept of Complex Conjugate Control (CCC) of linear wave energy converters (WECs) to nonlinear WECs by designing optimal limit cycles with Hamiltonian Surface Shaping and Power Flow Control (HSSPFC). It will be shown that CCC for a regular wave is equivalent to a power factor of one in electrical power networks, equivalent to mechanical resonance in a mass-spring-damper (MSD) system, and equivalent to a linear limit cycle constrained to a Hamiltonian surface defined in HSSPFC. Specifically, the optimal linear limit cycle is defined as a second-order center in the phase plane projection of the constant energy orbit across the Hamiltonian surface. This concept of CCC described by a linear limit cycle constrained to a Hamiltonian surface will be extended to nonlinear limit cycles constrained to a Hamiltonian surface for maximum energy harvesting by the nonlinear WEC. The case studies presented confirm increased energy harvesting which utilizes nonlinear geometry realization for reactive power generation.

In view of the expectation that the existence of complex poles is a signal of confinement, we investigate the analytic structure of the gluon, quark, and ghost propagators in the Landau gauge QCD and QCD-like theories by employing an effective model with a gluon mass term of the Yang-Mills theory, which we call the massive Yang-Mills model. In this model, we particularly investigate the number of complex poles in the parameter space of the model consisting of gauge coupling constant, gluon mass, and quark mass for the gauge group $SU(3)$ and various numbers of quark flavors $N_F$ within the asymptotic free region. Both the gluon and quark propagators at the best-fit parameters for $N_F=2$ QCD have one pair of complex conjugate poles, while the number of complex poles in the gluon propagator varies between zero and four depending on the number of quark flavors and quark mass. Moreover, as a general feature, we argue that the gluon spectral function of this model with nonzero quark mass is negative in the infrared limit. In sharp contrast to gluons, the quark and ghost propagators are insensitive to the number of quark flavors within the current approximations adopted in this paper. These results suggest that details of the confinement mechanism may depend on the number of quark flavors and quark mass., 22 pages, 15 figures, revised version, ref. 54 corrected

Photonic pseudospin-1/2 systems, which exhibit Dirac cone dispersion at Brillouin zone corners in analogy to graphene, have been extensively studied in recent years. However, it is known that a linear band crossing of two bands cannot emerge at the center of Brillouin zone in a two-dimensional photonic system respecting time reversal symmetry. Using a square lattice of elliptical magneto-optical cylinders, we constructed an unpaired Dirac point at the Brillouin zone center as the intersection of the second and third bands corresponding to the monopole and dipole excitations. Effective medium theory can be applied to the two linearly crossed bands with the effective constitutive parameters numerically calculated using the boundary effective medium approach. It is shown that only the effective permittivity approaches zero while the determinant of the nonzero effective permeability vanishes at the Dirac point frequency, showing a different behavior from the double-zero index metamaterials obtained from the pseudospin-1 triply degenerate points for time reversal symmetric systems. Exotic phenomena, such as the Klein tunneling and Zitterbewegung, in the pseudospin-1/2 system can be well understood from the effective medium description. When the Dirac point is lifted, the edge state dispersion near the $\Gamma$ point can be accurately predicted by the effective constitutive parameters. We also further realized magneto-optical complex conjugate metamaterials for a wide frequency range by introducing a particular type of non-Hermittian perturbations which make the two linear bands coalescence to form exceptional points at the real frequency., Comment: 48 pages, 14 figures