Showing papers on "Integrable system published in 2021"

Finite-temperature transport in one-dimensional quantum lattice models

Abstract: Over the last decade impressive progress has been made in the theoretical understanding of transport properties of clean, one-dimensional quantum lattice systems. Many physically relevant models in one dimension are Bethe-ansatz integrable, including the anisotropic spin-1/2 Heisenberg (also called the spin-1/2 XXZ chain) and the Fermi-Hubbard model. Nevertheless, practical computations of correlation functions and transport coefficients pose hard problems from both the conceptual and technical points of view. Only because of recent progress in the theory of integrable systems, on the one hand, and the development of numerical methods, on the other hand, has it become possible to compute their finite-temperature and nonequilibrium transport properties quantitatively. Owing to the discovery of a novel class of quasilocal conserved quantities, there is now a qualitative understanding of the origin of ballistic finite-temperature transport, and even diffusive or superdiffusive subleading corrections, in integrable lattice models. The current understanding of transport in one-dimensional lattice models, in particular, in the paradigmatic example of the spin-1/2 XXZ and Fermi-Hubbard models, is reviewed, as well as state-of-the-art theoretical methods, including both analytical and computational approaches. Among other novel techniques, matrix-product-state-based simulation methods, dynamical typicality, and, in particular, generalized hydrodynamics are covered. The close and fruitful connection between theoretical models and recent experiments is discussed, with examples given from the realms of both quantum magnets and ultracold quantum gases in optical lattices.

New General Variants of Chebyshev Type Inequalities via Generalized Fractional Integral Operators

Abstract: In this study, new and general variants have been obtained on Chebyshev’s inequality, which is quite old in inequality theory but also a useful and effective type of inequality. The main findings obtained by using integrable functions and generalized fractional integral operators have generalized many existing results as well as iterating the Chebyshev inequality in special cases.

Complexity growth in integrable and chaotic models

Abstract: We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O( $$ \sqrt{N} $$ ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

Abundant optical soliton solutions for an integrable (2+1)-dimensional nonlinear conformable Schrödinger system

Abstract: The analytical solutions of the integrable generalized ( 2 + 1 ) -dimensional nonlinear conformable Schrodinger (NLCS) system of equations was explored in this paper with the aid of three novel techniques which consist of ( G ′ / G ) -expansion method, generalized Riccati equation mapping method and the Kudryashov method in the conformable sense. We have discovered a new and more general variety of exact traveling wave solutions by using the proposed methods with a variety of soliton solutions of several structures. With several plots illustrating the behavior of dynamic shapes of the solutions, the findings are highly applicable and detailed the physical dynamic of the considered nonlinear system.

The Folded Spin-1/2 XXZ Model: I. Diagonalisation, Jamming, and Ground State Properties

Abstract: We study an effective Hamiltonian generating time evolution of states on intermediate time scales in the strong-coupling limit of the spin-1/2 XXZ model. To leading order, it describes an integrable model with local interactions. We solve it completely by means of a coordinate Bethe Ansatz that manifestly breaks the translational symmetry. We demonstrate the existence of exponentially many jammed states and estimate their stability under the leading correction to the effective Hamiltonian. Some ground state properties of the model are discussed.

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

Abstract: The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r sT we introduce a scalar function S(i, j) = sT Kj (I + M)−1Kir which is defined as same as in discrete case. S(i, j) satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on r and s, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S(i, j) defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S(i, j) = S(j,i). The solution M provides t function by t = ∣I + M∣. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and...

Abundant closed-form solutions and solitonic structures to an integrable fifth-order generalized nonlinear evolution equation in plasma physics

Abstract: This paper investigates the evolution dynamics of closed-form solutions for a new integrable nonlinear fifth-order equation with spatial and temporal dispersion which describes shallow water waves moving in two directions. Some novel computational soliton solutions are obtained in the form of exponential rational functions, trigonometric and hyperbolic functions, and complex-soliton solutions. Some dynamical wave structures of soliton solutions are achieved in evolutionary dynamical structures of multi-wave solitons, double-solitons, triple-solitons, multiple solitons, breather-type solitons, Lump-type solitons, singular solitons, and Kink-wave solitons using the generalized exponential rational function (GERF) technique. All newly established solutions are verified by back substituting into the considered fifth-order nonlinear evolution equation using computerized symbolic computational work via Wolfram Mathematica. These newly formed results demonstrate that the considered fifth-order equation theoretically possesses very rich computational wave structures of closed-form solutions, which are also useful in obtaining a better understanding of the internal mechanism of other complex nonlinear physical models arising in the field of plasma physics and nonlinear sciences. The physical characteristics of some constructed solutions are also graphically displayed via three-dimensional plots by selecting the best appropriate constant parameter values to easily understand the complex physical phenomena of the nonlinear equations. Eventually, the results validate the effectiveness and trustworthiness of the used technique.

On the Integrability of the Benjamin‐Ono Equation on the Torus

Abstract: In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are coordinates which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(\T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy.

Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes

Abstract: Symplectic integrators that preserve the geometric structure of Hamiltonian flows and do not exhibit secular growth in energy errors are suitable for the long-term integration of N-body Hamiltonian systems in the solar system. However, the construction of explicit symplectic integrators is frequently difficult in general relativity because all variables are inseparable. Moreover, even if two analytically integrable splitting parts exist in a relativistic Hamiltonian, all analytical solutions are not explicit functions of proper time. Naturally, implicit symplectic integrators, such as the midpoint rule, are applicable to this case. In general, these integrators are numerically more expensive to solve than same-order explicit symplectic algorithms. To address this issue, we split the Hamiltonian of Schwarzschild spacetime geometry into four integrable parts with analytical solutions as explicit functions of proper time. In this manner, second- and fourth-order explicit symplectic integrators can be easily made available. The new algorithms are also useful for modeling the chaotic motion of charged particles around a black hole with an external magnetic field. They demonstrate excellent long-term performance in maintaining bounded Hamiltonian errors and saving computational cost when appropriate proper time steps are adopted.

Nonlocal PT-symmetric integrable equations and related Riemann–Hilbert problems

Abstract: We aim to discuss about how to construct and classify nonlocal PT-symmetric integrable equations via nonlocal group reductions of matrix spectral problems. The nonlocalities considered are reverse-space, reverse-time and reverse-spacetime, each of which can involve either the transpose or the Hermitian transpose. The associated spectral problems are used to formulate a kind of Riemann–Hilbert problems and thus inverse scattering transforms. Soliton solutions are generated from specific Riemann–Hilbert problems with the identity jump matrix. We focus on two expository examples: nonlocal PT-symmetric matrix nonlinear Schrodinger and modified Korteweg–de Vries equations.

Physical properties for bidirectional wave solutions to a generalized fifth-order equation with third-order time-dispersion term

Abstract: Higher-order temporal-dispersion partial differential equations have its capability to visualize the evolution of steeper-waves for shorter wave-length better than the higher-order KdV does. In this work, we study the physical-structure propagations of generalized fifth-order nonlinear equation involving time-dispersion term. This model is recently proposed by Wazwaz (Xu and Wazwaz, 2020) and is considered to be the first type of integrable equations that involves a third-order time-dispersion term. We implement three functional-methods to seek solitary wave solutions to the proposed model. 2D-plots are provided to recognize the type of the obtained solutions. Finally, we propose some physical properties of the bidirectional waves that such model admits.

Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems

Abstract: We present a complete description of 2-dimensional equations that arise as symmetry reductions of four 3- dimensional Lax-integrable equations: (1) the universal hierarchy equation uyy = uzuxy− uyuxz; (2) the 3D rdDym equation uty = uxuxy− uyuxx; (3) the equation uty = utuxy− uyutx, which we call modified Veronese web equation; (4) Pavlov's equation uyy = utx + uyuxx− uxuxy.

A variety of multiple-soliton solutions for the integrable (4+1)-dimensional Fokas equation

Abstract: In this work, we study the integrable nonlinear (4+1)-dimensional Fokas equation. By means of systematic use of the simplified Hirota's method, we demonstrate the generation of a variety of multipl...

Correlation functions and transport coefficients in generalised hydrodynamics

Abstract: We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.

Constructing Integrable Lindblad Superoperators.

Abstract: We develop a new method for the construction of one-dimensional integrable Lindblad systems, which describe quantum many body models in contact with a Markovian environment. We find several new models with interesting features, such as annihilation-diffusion processes, a mixture of coherent and classical particle propagation, and a rectified steady state current. We also find new ways to represent known classical integrable stochastic equations by integrable Lindblad operators. Our method can be extended to various other situations and it establishes a structured approach to the study of solvable open quantum systems.

On the complex mixed dark-bright wave distributions to some conformable nonlinear integrable models

Abstract: In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kot...

Integrable nonunitary open quantum circuits

Abstract: 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.

From Gaudin Integrable Models to d -Dimensional Multipoint Conformal Blocks

Abstract: In this work, we initiate an integrability-based approach to multipoint conformal blocks for higher-dimensional conformal field theories. Our main observation is that conformal blocks for $N$-point functions may be considered as eigenfunctions of integrable Gaudin Hamiltonians. This provides us with a complete set of differential equations that can be used to evaluate multipoint blocks.

A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions

Abstract: With the advantages of fast calculating speed and high precision, the physics-informed neural network method opens up a new approach for numerically solving nonlinear partial differential equations. Based on conserved quantities, we devise a two-stage PINN method which is tailored to the nature of equations by introducing features of physical systems into neural networks. Its remarkable advantage lies in that it can impose physical constraints from a global perspective. In stage one, the original PINN is applied. In stage two, we additionally introduce the measurement of conserved quantities into mean squared error loss to train neural networks. This two-stage PINN method is utilized to simulate abundant localized wave solutions of integrable equations. We mainly study the Sawada-Kotera equation as well as the coupled equations: the classical Boussinesq-Burgers equations and acquire the data-driven soliton molecule, M-shape double-peak soliton, plateau soliton, interaction solution, etc. Numerical results illustrate that abundant dynamic behaviors of these solutions can be well reproduced and the two-stage PINN method can remarkably improve prediction accuracy and enhance the ability of generalization compared to the original PINN method.

Null infinity as an open Hamiltonian system

Abstract: When a system emits gravitational radiation, the Bondi mass decreases. If the Bondi energy is Hamiltonian, it can thus only be a time-dependent Hamiltonian. In this paper, we show that the Bondi energy can be understood as a time-dependent Hamiltonian on the covariant phase space. Our derivation starts from the Hamiltonian formulation in domains with boundaries that are null. We introduce the most general boundary conditions on a generic such null boundary, and compute quasi-local charges for boosts, energy and angular momentum. Initially, these domains are at finite distance, such that there is a natural IR regulator. To remove the IR regulator, we introduce a double null foliation together with an adapted Newman-Penrose null tetrad. Both null directions are surface orthogonal. We study the falloff conditions for such specific null foliations and take the limit to null infinity. At null infinity, we recover the Bondi mass and the usual covariant phase space for the two radiative modes at the full non-perturbative level. Apart from technical results, the framework gives two important physical insights. First of all, it explains the physical significance of the corner term that is added in the Wald-Zoupas framework to render the quasi-conserved charges integrable. The term to be added is simply the derivative of the Hamiltonian with respect to the background fields that drive the time-dependence of the Hamiltonian. Secondly, we propose a new interpretation of the Bondi mass as the thermodynamical free energy of gravitational edge modes at future null infinity. The Bondi mass law is then simply the statement that the free energy always decreases on its way towards thermal equilibrium.

Deep learning models for global coordinate transformations that linearise PDEs

Abstract: We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a non-linear partial differential equation (PDE) into a linear PDE. Our architecture is motivated by the linearising transformations provided by the Cole–Hopf transform for Burgers’ equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix K. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers’ equation, as well as the substantially more challenging Kuramoto–Sivashinsky equation, showing that our method provides a robust architecture for discovering linearising transforms for non-linear PDEs.

Inverse Scattering of the Zakharov-Shabat System Solves the Weak Noise Theory of the Kardar-Parisi-Zhang Equation.

Abstract: We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic, and integrable techniques We expand the program of the weak noise theory, which maps the large deviations onto a nonlinear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions

On the Discretization of the Coupled Integrable Dispersionless Equations

Abstract: We study the integrable discretization of the coupled integrable dispersionless equations. Two semi-discrete version and one full-discrete version of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

Abstract: This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space.,The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlene sense.,The paper reports new Painleve-integrable extended equations which belong to the shallow water wave medium.,The author addresses the integrability features of this model via using the Painleve analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method.,The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition.,The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions.,The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.