Integrable Research Articles

Flat limit of massless scalar scattering in AdS2

We explore the flat limit of massless scalar scattering in AdS2. We derive the 1→1S-matrix from the CFT 2-point function. We show a key property of the 2→2S-matrix in 2d, where the contact interaction in the flat limit gives momentum conserving delta function. We show the factorization of the n→nS-matrix for integrable models in the flat limit, focusing on contact interactions. We calculate the S-matrix by linking the CFT operator on the AdS boundary to the scattering state in flat-space. We use bulk operator reconstruction to study massless scalar scattering in the flat limit and solve the Klein-Gordon equation in global AdS2 for the massless scalar field. The solution is simple, involving a pure phase in global time and a sinusoidal function in the radial coordinate. This simplicity also extends to the smearing function, allowing us to map the scattering state to the CFT operator while taking AdS corrections into account.

Development and Analysis of novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Parametrically Driven Nonlinear System of Pseudo-Excitations on a Two-Leg Ladder Lattice

Following the main principles of developing the evolutionary nonlinear integrable systems on quasi-one-dimensional lattices, we suggest a novel nonlinear integrable system of parametrically driven pseudo-excitations on a regular two-leg ladder lattice. The initial (prototype) form of the system is derivable in the framework of semi-discrete zero-curvature equation with the spectral and evolution operators specified by the properly organized 3 × 3 square matrices. Although the lowest conserved local densities found via the direct recursive method do not prompt us the algebraic structure of system’s Hamiltonian function, but the heuristically substantiated search for the suitable two-stage transformation of prototype field functions to the physically motivated ones has allowed to disclose the physically meaningful nonlinear integrable system with time-dependent longitudinal and transverse inter-site coupling parameters. The time dependencies of inter-site coupling parameters in the transformed system are consistently defined in terms of the accompanying parametric driver formalized by the set of four homogeneous ordinary linear differential equations with the time-dependent coefficients. The physically meaningful parametrically driven nonlinear system permits its concise Hamiltonian formulation with the two pairs of field functions serving as the two pairs of canonically conjugated field amplitudes. The explicit example of oscillatory parametric drive is described in full mathematical details.

Statistics of Matrix Elements of Local Operators in Integrable Models

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨μ|O|λ⟩ in the same macrostate scale as exp(−cOLln(L)−LMμ,λO), where the probability distribution function for Mμ,λO is well described by Fréchet distributions and cO depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp(−dOL2), where dO depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections. Published by the American Physical Society 2024

Quantum next generation reservoir computing: an efficient quantum algorithm for forecasting quantum dynamics

Next Generation Reservoir Computing (NG-RC) is a modern class of model-free machine learning that enables an accurate forecasting of time series data generated by dynamical systems. We demonstrate that NG-RC can accurately predict full many-body quantum dynamics in both integrable and chaotic systems. This is in contrast to the conventional application of reservoir computing that concentrates on the prediction of the dynamics of observables. In addition, we apply a technique which we refer to as skipping ahead to predict far future states accurately without the need to extract information about the intermediate states. However, adopting a classical NG-RC for many-body quantum dynamics prediction is computationally prohibitive due to the large Hilbert space of sample input data. In this work, we propose an end-to-end quantum algorithm for many-body quantum dynamics forecasting with a quantum computational speedup via the block-encoding technique. This proposal presents an efficient model-free quantum scheme to forecast quantum dynamics coherently, bypassing inductive biases incurred in a model-based approach.

Integrable sigma model with generalized [formula omitted] structure, Yang-Baxter sigma model with generalized complex structure and multi-Yang-Baxter sigma model

We construct an integrable sigma model with a generalized F structure, which involves a generalized Nijenhuis structure J satisfying J3=−J. Utilizing the expression of the generalized complex structure on the metric Lie group manifold G in terms of operator relations on its Lie algebra g, we formulate a Yang-Baxter sigma model with a generalized complex structure. Additionally, we present multi-Yang-Baxter sigma models featuring two and three compatible Nijenhuis structures. Examples for each of these models are provided.

Periodic orbits in Fermi-Pasta-Ulam-Tsingou systems.

The Fermi-Pasta-Ulam-Tsingou (FPUT) paradox is the phenomenon whereby a one-dimensional chain of oscillators with nonlinear couplings shows long-lived nonergodic behavior prior to thermalization. The trajectory of the system in phase space, with a long-wavelength initial condition, closely follows that of the Toda model over short times, as both systems seem to relax quickly to a non-thermal, metastable state. Over longer times, resonances in the FPUT spectrum drive the system toward equilibrium, away from the Toda trajectory. Similar resonances are observed in q-breather spectra, suggesting that q-breathers are involved in the route toward thermalization. In this article, we first review previous important results related to the metastable state, solitons, and q-breathers. We then investigate orbit bifurcations of q-breathers and show that they occur due to resonances, where the q-breather frequencies become commensurate as mΩ1=Ωk. The resonances appear as peaks in the breather energy spectrum. Furthermore, they give rise to new "composite periodic orbits," which are nonlinear combinations of multiple q-breathers that exist following orbit bifurcations. We find that such resonances are absent in integrable systems, as a consequence of the (extensive number of) conservation laws associated with integrability.

Forty years: Geometric numerical integration of dynamical systems in China

Kang Feng proposed the symplectic geometric algorithms for Hamiltonian systems at Beijing “International Symposium on Differential Geometry and Differential Equations” in August 1984, opening up a new field of research. Over the past 40 years, geometric numerical integration for dynamical systems has been irreplaceably applied in numerous fields, including celestial mechanics, plasma physics, fluid mechanics, biochemistry, meteorological forecasting, seismic exploration, data science and artificial intelligence. This paper mainly summarizes the achievements of geometric numerical integration for dynamical systems in China over the past 40 years.

Chaotic roots of the modular multiplication dynamical system in Shor's algorithm

Shor's factoring algorithm, believed to provide an exponential speedup over classical computation, relies on finding the period of an exactly periodic quantum modular multiplication operator. This exact periodicity is the hallmark of an integrable system, which is paradoxical from the viewpoint of quantum chaos, given that the classical limit of the modular multiplication operator is a highly chaotic system that occupies the “maximally random” Bernoulli level of the classical ergodic hierarchy. In this work, we approach this apparent paradox from a quantum dynamical systems viewpoint, and consider whether signatures of ergodicity and chaos may indeed be encoded in such an “integrable” quantization of a chaotic system. We show that Shor's modular multiplication operator, in specific cases, can be written as a superposition of quantized A-baker's maps exhibiting more typical signatures of quantum chaos and ergodicity. This work suggests that the integrability of Shor's modular multiplication operator may stem from the interference of other “chaotic” quantizations of the same family of maps, and paves the way for deeper studies on the interplay of integrability, ergodicity, and chaos in and via quantum algorithms. Published by the American Physical Society 2024

FateNet: an integration of dynamical systems and deep learning for cell fate prediction.

Understanding cellular decision-making, particularly its timing and impact on the biological system such as tissue health and function, is a fundamental challenge in biology and medicine. Existing methods for inferring fate decisions and cellular state dynamics from single-cell RNA sequencing data lack precision regarding decision points and broader tissue implications. Addressing this gap, we present FateNet, a computational approach integrating dynamical systems theory and deep learning to probe the cell decision-making process using scRNA-seq data. By leveraging information about normal forms and scaling behavior near bifurcations common to many dynamical systems, FateNet predicts cell decision occurrence with higher accuracy than conventional methods and offers qualitative insights into the new state of the biological system. Also, through in-silico perturbation experiments, FateNet identifies key genes and pathways governing the differentiation process in hematopoiesis. Validated using different scRNA-seq data, FateNet emerges as a user-friendly and valuable tool for predicting critical points in biological processes, providing insights into complex trajectories. github.com/ThomasMBury/fatenet.

Tautological relations and integrable systems

We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Gu\'er\'e and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case $n=1$ and arbitrary $g$ using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hern\'andez Iglesias. We also prove all the above mentioned relations in the case $g=0$ and arbitrary $n$.Comment: v3: final journal version, 44 pages

One-dimensional Fermi polaron after a kick: Two-sided singularity of the momentum distribution, Bragg reflection and other exact results

A mobile impurity particle immersed in a quantum fluid forms a polaron - a quasiparticle consisting of the impurity and a local disturbance of the fluid around it. We ask what happens to a one-dimensional polaron after a kick, i.e. an abrupt application of a force that instantly delivers a finite impulse to the impurity. In the framework of an integrable model describing an impurity in a one-dimensional gas of fermions or hard-core bosons, we calculate the distribution of the polaron momentum established when the post-kick relaxation is over. A remarkable feature of this distribution is a two-sided power-law singularity. It emerges due to one of two processes. In the first process, the whole impulse is transferred to the polaron, without creating phonon-like excitations of the fluid. In the second process, the impulse is shared between the polaron and the center-of-mass motion of the fluid, again without creating any fluid excitations. The latter process is, in fact, a Bragg reflection at the edge of the emergent Brillouin zone. We carefully analyze the conditions for each of the two processes. The asymptotic form of the distribution in the vicinity of the singularity is derived.

Multi-pole soliton of discrete integrable equations and modified Riemann-Hilbert approach: discrete Hirota equation

In this study, the Riemann-Hilbert approach was developed and applied to the discrete Hirota equation. We constructed a modified Riemann-Hilbert problem compatible with the discrete Hirota equation and derived a reconstruction formula for its solutions. Because the characteristic function contains a potential, we modify the Riemann-Hilbert approach to make the Riemann-Hilbert matrix have good asymptotic properties. We believe that the modified Riemann-Hilbert approach can also be applied to other discrete integrable models. By using the direct method of Laurent series, we obtained the expression of multi-pole solutions for the discrete Hirota equation and demonstrated the dynamic behavior of some solutions.

Intermediate spectral statistics of rational triangular quantum billiards.

Triangular billiards whose angles are rational multiples of π are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle, which permits one to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion, and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short-range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.

Two mass-imbalanced atoms in a hard-wall trap: Deep learning integrability of many-body systems.

The study of integrable systems has led to significant advancements in our understanding of many-body physics. We design a series of numerical experiments to analyze the integrability of a mass-imbalanced two-body system through energy-level statistics and deep learning of wave functions. The level spacing distributions are fitted by a Brody distribution and the fitting parameter ω is found to separate the integrable and nonintegrable mass ratios by a critical line ω=0. The convolutional neural network built from the probability density images could identify the transition points between integrable and nonintegrable systems with high accuracy, yet in a much shorter computation time. A brilliant example of the network's ability is to identify a new integrable mass ratio 1/3 by learning from the known integrable case of equal mass, with a remarkable network confidence of 98.22%. The robustness of our neural networks is further enhanced by adversarial learning, where samples are generated by standard and quantum perturbations mixed in the probability density images and the wave functions, respectively.

Hirota Bilinear Approach to Multi-Component Nonlocal Nonlinear Schrödinger Equations

Nonlocal nonlinear Schrödinger equations are among the important models of nonlocal integrable systems. This paper aims to present a general formula for arbitrary-order breather solutions to multi-component nonlocal nonlinear Schrödinger equations by using the Hirota bilinear method. In particular, abundant wave solutions of two- and three-component nonlocal nonlinear Schrödinger equations, including periodic and mixed-wave solutions, are obtained by taking appropriate values for the involved parameters in the general solution formula. Moreover, diverse wave structures of the resulting breather and periodic wave solutions with different parameters are discussed in detail.

Exact closed-form solution for the completely integrable helmholtz-duffing oscillator with applications to real-world problems

The Helmholtz-Duffing oscillator is an important vibration model that can be used to study many engineering systems and physical phenomena. In this article, we derived new exact closed-form solutions for the completely integrable Helmholtz Duffing oscillator subject to a constant force. The exact solution was derived naturally from the first integral of the governing differential equation. This approach is completely different from the ansatz method, used in previously published studies, where the exact solution was forced to take the form an initially assumed solution. Through various algebraic transformations and with the aid of standard elliptic integral tables, the present exact period was derived in terms of the complete elliptic integral of the first kind while the exact displacement was derived in terms of the Jacobi cosine function. Unlike the exact ansatz solutions, which have limited range of validity, the present exact solutions are applicable to all bounded periodic states of the Helmholtz-Duffing oscillator with constant force. The validity of the present exact solutions was verified using numerical solutions and published exact solutions in the form of the Weierstrass elliptic function. It was found that the numerical differentiation method produced significant errors for some system inputs and cannot be relied on as a benchmark solution. However, the present exact solutions are benchmark solutions that could be used to check the accuracy of new and existing approximate solutions. Finally, the application of the exact solutions to analyze some real-world problems was demonstrated.

Novel hybrid waves solutions of Sawada–Kotera like integrable model arising in fluid mechanics

The purpose of the paper is to uncover more advanced soliton solutions for the three-component Sawada–Kotera (SK)-like equation. The Hirota bilinear (HB) method is exploited to obtain a bilinear form and general solution to the three-component SK like equation. The study explores various soliton solutions by considering multiple dispersion coefficients. The focus is on multiple bifurcated solitons, hybrid breathers, periodic lumps, and periodic rogue waves, as well as the resonance in soliton solutions. The fission or fusion processes of solitons are analyzed for specific values of parameters. From the literature, real Y-shaped and X-shaped solitons in water waves are provided. The results are presented through graphs generated using MATLAB 2021. The important feature of the proposed study is to show different behavior of the soliton in each component. The behavior of solitons, their interactions, and their transformations are all governed by the fundamental concept of energy conservation in all three cases. The amplitudes and forms of the solitons are impacted by the energy redistribution that occurs during collisions, fission, or other interactions. Understanding these energy dynamics aids in predicting how these nonlinear wave events would behave in complex systems and sheds light on the fundamental physics of these phenomena.

Dynamical Magic Transitions in Monitored Clifford+ T Circuits

The classical simulation of highly entangling quantum dynamics is conjectured to be generically hard. Thus, recently discovered measurement-induced transitions between highly entangling and low-entanglement dynamics are phase transitions in classical simulability. Here, we study simulability transitions beyond entanglement: noting that some highly entangling dynamics (e.g., integrable systems or Clifford circuits) are easy to classically simulate, thus requiring “magic”—a subtle form of quantum resource—to achieve computational hardness, we ask how the dynamics of magic competes with measurements. We study the resulting “dynamical magic transitions” focusing on random monitored Clifford circuits doped by T gates (injecting magic). We identify dynamical “stabilizer purification”—the collapse of a superposition of stabilizer states by measurements—as the mechanism driving this transition. We find cases where transitions in magic and entanglement coincide, but also others with a magic and simulability transition in a highly (volume-law) entangled phase. In establishing our results, we use Pauli-based computation, a scheme distilling the quantum essence of the dynamics to a magic state register subject to mutually commuting measurements. We link stabilizer purification to “magic fragmentation” wherein these measurements separate into disjoint, O(1)-weight blocks, and relate this to the spread of magic in the original circuit becoming arrested. Published by the American Physical Society 2024

Painlevé analysis and Hirota direct method for analyzing three novel physical fluid extended KP, Boussinesq, and KP-Boussinesq equations: Multi-solitons/shocks and lumps

Due to the extensive and diverse connections of the Kadomtsev-Petviashvili (KP) equation with various physical phenomena, such as the propagation of waves in sea and ocean water and the propagation of nonlinear waves in plasmas when two-dimensional perturbations are considered, thus in this study and based on the original KP equation, we construct and investigate three extended models, which are called the extended KP (eKP) equation, the extended Boussinesq (eBO) equation, and the extended KP-Boussinesq (eKP-BO) equation. These equations are commonly observed in many physical contexts involving nonlinear and dissipative media. Our research begins with thoroughly verifying the complete integrability of the three extended models. Once this crucial step is accomplished, we will examine these three extended models using the simplified Hirota approach (SHA), leading us to derive multiple soliton/shock solutions. Furthermore, the Hirota bilinear form of each model will be employed to derive a set of lump solutions for these three extended models, utilizing different parameter values. We also analyze some derived solutions graphically by presenting some two- and three-dimensional graphics to understand the dynamic behavior of the waves described by these solutions. The obtained results are anticipated to benefit a broad spectrum of academics interested in studying fluid mechanics, water wave characterization, and other interdisciplinary domains. Additionally, these models are anticipated to be essential in describing and understanding the dynamics of several nonlinear phenomena that arise and propagate in different plasma systems when perturbations are considered in multidimensions.

A family of integrable maps associated with the Volterra lattice

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the g = 2 case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus g⩾1 . The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case (g = 1), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices.