Nonintegrable Case Research Articles

A new approach for modelling the damped Helmholtz oscillator: applications to plasma physics and electronic circuits

In this paper, a new approach is devoted to find novel analytical and approximate solutions to the damped quadratic nonlinear Helmholtz equation (HE) in terms of the Weiersrtrass elliptic function. The exact solution for undamped HE (integrable case) and approximate/semi-analytical solution to the damped HE (non-integrable case) are given for any arbitrary initial conditions. As a special case, the necessary and sufficient condition for the integrability of the damped HE using an elementary approach is reported. In general, a new ansatz is suggested to find a semi-analytical solution to the non-integrable case in the form of Weierstrass elliptic function. In addition, the relation between the Weierstrass and Jacobian elliptic functions solutions to the integrable case will be derived in details. Also, we will make a comparison between the semi-analytical solution and the approximate numerical solutions via using Runge–Kutta fourth-order method, finite difference method, and homotopy perturbation method for the first-two approximations. Furthermore, the maximum distance errors between the approximate/semi-analytical solution and the approximate numerical solutions will be estimated. As real applications, the obtained solutions will be devoted to describe the characteristics behavior of the oscillations in RLC series circuits and in various plasma models such as electronegative complex plasma model.

Chaotic dynamics of the hydrogen atom in a space-dependent electric field

We study the chaotic dynamics of the hydrogen atom in a space-dependent gradient electric field. We use the Poincaré surface of section (PSS) and the Lyapunov exponents to characterize chaotic dynamics. The dynamical character of this system is studied and the regular-chaos transition is found when the system passes from the non-integrable case to the integrable one. It is found that for a given gradient electric field, this system has a critical scaled energy When the scaled energy the structure of the whole phase space is nearly regular. However, as the scaled energy chaotic structures appear in the PSS. As the scaled energy is very large, the whole PSS is chaotic. The chaotic property of this system is further verified by the positive value of the maximum Lyapunov exponent. In addition, we show that with increasing electric field gradient the dynamics of this system becomes increasingly chaotic. Our work provides an example to study the chaotic dynamics of the hydrogen atom by using the space-dependent electric field. We hope that our results can guide the future experimental researches about the dynamical property of the atoms or molecules in the non-uniform external fields.

On the Evolution of Orbits in the Outer Version of the Restricted Elliptic Doubly Averaged Three-Body Problem

The outer version of the restricted elliptic three-body problem, where a perturbed body of negligible mass is far from two bodies of finite masses (the main and perturbing ones) close to one another and its orbit is located entirely outside the orbit of the perturbing body, is considered. The terms up to the fourth order inclusive in the ratio of the orbital semimajor axes of the perturbing and perturbed bodies are retained in the expansion of the perturbing function for the problem. Explicit analytical expressions for the doubly averaged perturbing function and its derivatives with respect to the orbital elements on the right-hand sides of the evolution equations are presented. The integrable cases of the doubly averaged elliptic problem are investigated: planar and orthogonal apsidal orbits. Qualitative differences compared to the inner (satellite) version are pointed out. The evolution system is numerically integrated in the general (nonintegrable) case for a number of special initial conditions that hypothetically correspond to the orbital evolution of low-mass bodies (planetesimals, low-mass planets) in the exoplanetary system GJ 3512.

Some Peculiarities of the Evolution of Orbits in the Satellite Restricted Elliptic Doubly Averaged Three-Body Problem

The inner (or satellite) version of the restricted elliptic three-body problem is considered. The terms up to the fourth order inclusive in small parameter are retained in the expansion of the perturbing function for the problem. The ratio of the orbital semimajor axes of perturbed and perturbing bodies is such a parameter, while their mean longitudes are the fastest variables. The Gauss scheme of independent double averaging over fast variables is used to analyze the orbital evolution of a body of negligible mass. Explicit analytical expressions for the doubly averaged perturbing function and its derivatives with respect to the elements on the right-hand sides of the evolution equations are presented. The integrable cases of the doubly averaged problem are studied in detail: planar and orthogonal apsidal orbits. The evolution system is numerically integrated in the general (nonintegrable) case for some special values of the problem parameters and initial conditions, in particular, for a set of orbital elements in which the so-called “flips”, i.e., transitions of the orbit from prograde to retrograde and vice versa, manifest themselves. In the Sun–Jupiter–asteroid model using some special asteroid orbits as an example, we show the influence of the retained fourth-order terms and the ellipticity of Jupiter’s orbit on their evolution.

Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kahler case. Our main question is the existence of almost Kahler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.

Integrability and Chaos in Figure Skating

We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

Negative differential thermal resistance in one-dimensional hard-point gas models.

We study negative differential thermal resistance (NDTR) of one-dimensional hard-point gas models. The models are non-integrable unless all particles have the same mass. We show that NDTR can exist in both the integrable case and the non-integrable case. In the integrable case, the existence of NDTR is analytically predicted and numerically confirmed and a mechanism for NDTR different from that observed in lattice models is unveiled. In the non-integrable case, we show that the mechanism also works under certain conditions and the properties of NDTR are found to depend on the particle masses and the system size by the molecular dynamics simulations. These results shed new light on the mechanisms for NDTR and may help design new functional heat devices.

Cohomologies on almost complex manifolds and the $\partial \overline{\partial}$-lemma

We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as vector-valued forms on $M$. We show how one of these, the $N$-cohomology $H^{\bullet}_N (M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H^{\bullet}_J (M)$, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The $J$-cohomology encodes whether a complex manifold satisfies the $\partial \bar{\partial}$-lemma, and more generally in the non-integrable case the $J$-cohomology encodes whether $(M, J)$ satisfies the $\mathrm{d} \mathcal{L}_J$-lemma, which we introduce and motivate in this paper. We discuss several explicit examples in detail, including a non-integrable example. We also show that $H^k_J$ is finite-dimensional for compact integrable $(M, J)$, and use spectral sequences to establish partial results on the finite-dimensionality of $H^k_J$ in the compact non-integrable case.

Interference of the signal from a local dynamical process with the quantum state propagation in spin chains

The effect of a local instantaneous quantum dynamical process (QDP), either unitary or non-unitary, on the quantum state transfer through a unitary Hamiltonian evolution is investigated for both integrable and non-integrable dynamics. There are interference effects of the quantum state propagation and the QDP signal propagation. The state transfer fidelity is small for further sites, from the site where the information is coded, indicating a finite speed for the propagation of the quantum correlation. There is a small change in the state transfer fidelity for the case of the non-unitary QDP intervening in the the background of unitary dynamics. In the case of the unitary QDP, the change is more pronounced, with a substantial increase in the fidelity for appropriate sites and times. For the non-integrable case, namely a kicked Harper model, the state transfer fidelity is quite large for further sites for short times, indicating that a finite speed for the propagation of the quantum correlation cannot be defined.

Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

In this paper we study the phase transition of continuum Widom-Rowlinson measures in $\mathbb{R}^d$ with $q$ types of particles and random radii. Each particle $x_i$ of type $i$ is marked by a random radius $r_i$ distributed by a probability measure $Q_i$ on $\mathbb{R}^+$. The particles of same type do not interact each other whereas particles $x_i$ and $x_j$ with different type $i \neq j$ interact via an exclusion hardcore interaction forcing $r_i+r_j$ to be smaller than $|x_i-x_j|$. In the integrable case (i.e. $\int r^d Q_i(dr)<+\infty$, $1\le i\le q$), we show that the Widom-Rowlinson measures exhibit a standard phase transition providing uniqueness, when the activity is small, and co-existence of $q$ ordered phases, when the activity is large. In the non-integrable case (i.e. $\int r^d Q_i(dr)=+\infty$, $1\le i \le q$), we show another type of phase transition. We prove, when the activity is small, the existence of at least $q+1$ extremal phases and we conjecture that, when the activity is large, only the $q$ ordered phases subsist. We prove a weak version of this conjecture by showing that the symmetric Widom-Rowlinson measure with free boundary condition is a mixing of the $q$ ordered phases if and only if the activity is large.

On possible relations between an increasing process and its compensator in the non-integrable case

On possible relations between an increasing process and its compensator in the non-integrable case

Dynamics of a lattice gauge theory with fermionic matter—minimal quantum simulator with time-dependent impurities in ultracold gases

We propose a minimal model to study the real-time dynamics of a lattice gauge theory (LGT) coupled to fermionic matter in a cold atom quantum simulator setup. We show that dynamical correlators of the gauge fields can be measured in experiments studying the time-evolution of two pairs of impurities, and suggest the protocol for implementing the model in cold atom experiments. Further, we discuss a number of unexpected features found in the integrable limit of the model, as well as its extensions to a non-integrable case. A potential experimental implementation of our model in the latter regime would allow one to simulate strongly-interacting LGT beyond current capabilities of classical computers.

Optimal control of dark solitons

We consider optimal control methods for dark solitons and their higher-order states based on the completely integrable nonautonomous nonlinear Schrödinger equation (NLSE) models with varying gravitational-like and harmonic oscillator potentials, and in the presence of gain or absorption. In view of the fact that the time-dependence of the dark soliton phase (chirp) represents a major difference between bright and dark solitons, our main goal here is to gain a deeper understanding at a fundamental level based on the Lax pairs of how this time-dependent phase appears in the framework of the nonautonomous NLSE models with varying dispersion and nonlinearity. Specifically, we consider dynamics of dark solitons and their higher-order states in a variety of different optical scenarios. The main features of dark solitons are supplied by numerical calculations performed both for the exactly integrable and nonintegrable cases. We reveal nontrivial dynamics of black and gray solitons under the optimal control conditions. Since the experimental realization of dark solitons is possible only if a finite background pedestal is used, we consider dynamics of dark solitons and their higher-order states superimposed on the ground state of nonlinear harmonic oscillator. Direct numerical experiments performed in the framework of the Gross-Pitaevskii (GP) model in a broad range of parameters reveal many fundamental features of the dark solitons dynamics in confining harmonic oscillator potential, including the background BEC cloud spreading near the Feshbach resonance (when repulsive nonlinearity has a dispersive form). One of the most important effects is related to the “saturation” of the maximum value of self-compression near the Feshbach resonance. This effect exists both for the central black soliton and for the gray solitons formed during the higher-order dark soliton oscillations in the parabolic trap. The soliton becomes black at the turning points in the harmonic potential and gray at the trap center. This makes possible to observe the nontrivial dynamics of the periodic transformation of the black soliton to the gray one.

Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds

We give the global mathematical formulation of a class of generalized four-dimensional theories of gravity coupled to scalar matter and to Abelian gauge fields. In such theories, the scalar fields are described by a section of a surjective pseudo-Riemannian submersion π over space–time, whose total space carries a Lorentzian metric making the fibers into totally-geodesic connected Riemannian submanifolds. In particular, π is a fiber bundle endowed with a complete Ehresmann connection whose transport acts through isometries between the fibers. In turn, the Abelian gauge fields are “twisted” by a flat symplectic vector bundle defined over the total space of π. This vector bundle is endowed with a vertical taming which locally encodes the gauge couplings and theta angles of the theory and gives rise to the notion of twisted self-duality, of crucial importance to construct the theory. When the Ehresmann connection of π is integrable, we show that our theories are locally equivalent to ordinary Einstein-Scalar-Maxwell theories and hence provide a global non-trivial extension of the universal bosonic sector of four-dimensional supergravity. In this case, we show using a special trivializing atlas of π that global solutions of such models can be interpreted as classical “locally-geometric” U-folds. In the non-integrable case, our theories differ locally from ordinary Einstein-Scalar-Maxwell theories and may provide a geometric description of classical U-folds which are “locally non-geometric”.

On possible relations between an increasing process and its compensator in the non-integrable case

On possible relations between an increasing process and its compensator in the non-integrable case

Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator

The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, but also diagnoses the chaotic behavior of many-body quantum systems and characterizes the information scrambling. Based on the OTOCs, three different concepts -- quantum chaos, holographic duality, and information scrambling -- are found to be intimately related to each other. Despite of its theoretical importance, the experimental measurement of the OTOC is quite challenging and so far there is no experimental measurement of the OTOC for local operators. Here we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and non-integrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for non-intgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.

Out-of-Time-Ordered Density Correlators in Luttinger Liquids

Information scrambling and the butterfly effect in chaotic quantum systems can be diagnosed by out-of-time-ordered (OTO) commutators through an exponential growth and large late time value. We show that the latter feature shows up in a strongly correlated many-body system, a Luttinger liquid, whose density fluctuations we study at long and short wavelengths, both in equilibrium and after a quantum quench. We find rich behavior combining robustly universal and nonuniversal features. The OTO commutators display temperature- and initial-state-independent behavior and grow as t^{2} for short times. For the short-wavelength density operator, they reach a sizable value after the light cone only in an interacting Luttinger liquid, where the bare excitations break up into collective modes. This challenges the common interpretation of the OTO commutator in chaotic systems. We benchmark our findings numerically on an interacting spinless fermion model in 1D and find persistence of central features even in the nonintegrable case. As a nonuniversal feature, the short-time growth exhibits a distance-dependent power.

Yang–Baxter deformations of W2,4 × T1,1 and the associated T-dual models

Recently, for principal chiral models and symmetric coset sigma models, Hoare and Tseytlin proposed an interesting conjecture that the Yang–Baxter deformations with the homogeneous classical Yang–Baxter equation are equivalent to non-abelian T-dualities with topological terms. It is significant to examine this conjecture for non-symmetric (i.e., non-integrable) cases. Such an example is the W2,4×T1,1 background. In this note, we study Yang–Baxter deformations of type IIB string theory defined on W2,4×T1,1 and the associated T-dual models, and show that this conjecture is valid even for this case. Our result indicates that the conjecture would be valid beyond integrability.

Effects of integrated designs of alarm and process information on diagnosis performance in digital nuclear power plants

In the main control rooms of nuclear power plants (NPPs), operators frequently switch between alarm displays and system-information displays to incorporate information from different screens. In this study, we investigated two integrated designs of alarm and process information – integrating alarm information into process displays (denoted as Alarm2Process integration) and integrating process information into alarm displays (denoted as Process2Alarm integration). To analyse the effects of the two integration approaches and time pressure on the diagnosis performance, a laboratory experiment was conducted with ninety-six students. The results show that compared with the non-integrated case, Process2Alarm integration yields better diagnosis performance in terms of diagnosis accuracy, time required to generate correct hypothesis and completion time. In contrast, the Alarm2Process integration leads to higher levels of workload, with no improvement in diagnosis performance. The diagnosis performance of Process2Alarm integration was consistently better than that of Alarm2Process integration, regardless of the levels of time pressure.Practitioner Summary: To facilitate operator’s synthesis of NPP information when performing diagnosis tasks, we proposed to integrate process information into alarm displays. The laboratory validation shows that the integration approach significantly improves the diagnosis performance for both low and high time-pressure levels.

Classical perturbation theory and resonances in some rigid body systems

We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincare sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincare sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincare sets are similar to the ones of the Kovalevskaya case.