CyberSec Research

We consider $\mathfrak{so}_4$ invariant matrix product states (MPS) in the $\mathfrak{so}_6$ symmetric integrable spin chain and prove their integrability. These MPS appear as fuzzy three-sphere solutions of matrix models with Yang-Mills-type interactions, and in particular they correspond to scalar defect sectors of $N=4$ SYM. We find that the algebra formed by the fuzzy three-sphere generators naturally leads to a boundary reflection algebra and hence a solution to the boundary Yang-Baxter equation for every representation of the fuzzy three-sphere. This allows us to find closed formula for the overlaps of Bethe states of $\mathfrak{so}_6$ symmetric chains with the fuzzy three-sphere MPS for arbitrary bond dimensions.

In this paper, we establishes a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlev\'e (dP) equations. We first apply nonisospectral deformations to noncommutative Laurent bi-OPs to obtain the noncommutative nonisospectral mixed relativistic Toda lattice and its Lax pair. Then, we perform a stationary reduction on this Lax pair to obtain a matrix dP-type equation. The validity of this reduction is demonstrated through a specific choice of weight function and the application of quasideterminant properties. In the scalar case, our matrix dP equation reduces to the known alternate dP II equation.

In a series of papers we have considered a non-stationary difference equation which was originally discovered for the deformed Virasoro conformal block. The equation involves mass parameters and, when they are tuned appropriately, the equation is regarded as a quantum KZ equation for $U_q(A_{1}^{(1)})$. We introduce a $\widehat{\mathfrak{gl}}_N$ generalization of the non-stationary difference equation. The Hamiltonian is expressed in terms of $q$-commuting variables and allows both factorized forms and a normal ordered form. By specializing the mass parameters appropriately, the Hamiltonian can be identified with the $R$-matrix of the symmetric tensor representation of $U_q(A_{N-1}^{(1)})$, which in turn comes from the 3D (tetrahedron) $R$-matrix. We conjecture that the affine Laumon partition function of type $A_{N-1}^{(1)}$ gives a solution to our $\widehat{\mathfrak{gl}}_N$ non-stationary difference equation. As a check of our conjecture, we work out the four dimensional limit and find that the non-stationary difference equation reduces to the Fuji-Suzuki-Tsuda system.

This brief review surveys recent progress driven by the gauge/Yang-Baxter equation (YBE) correspondence. This connection has proven to be a powerful tool for discovering novel integrable lattice spin models in statistical mechanics by exploiting dualities in supersymmetric gauge theories. In recent years, research has demonstrated the use of dual gauge theories to construct new lattice spin models that are dual to Ising-like models.

The Nijhoff-Quispel-Capel (NQC) equation is a general lattice quadrilateral equation presented in terms of a function $S(a,b)$ where $a$ and $b$ serve as extra parameters. It can be viewed as counterpart of Q3 equation which is the second top equation in the Adler-Bobenko-Suris list. In this paper, we review some known formulations of the NQC variable $S(a,b)$, such as the Cauchy matrix approach, the eigenfunction approach and via a spectral Wronskian. We also present a new perspective to formulate $S(a,b)$ from the eigenfunctions of a Lax pair of the lattice (non-potential) modified Korteweg de Vries equation. A new Dbar problem is introduced and employed in the derivation.

We review studies on the application of Lie group methods to delay ordinary differential equations (DODEs). For first- and second-order DODEs with a single delay parameter that depends on independent and dependent variables, the group classifications are performed. Classes of invariant DODEs for each Lie subgroup are written out. The symmetries allow us to construct invariant solutions to such equations. The application of variational methods to functionals with one delay yields DODEs with two delays. The Lagrangian and Hamiltonian approaches are reviewed. The delay analog of the Legendre transformation, which relates the Lagrangian and Hamiltonian approaches, is also analysed. Noether-type operator identities relate the invariance of delay functionals with the appropriate variational equations and their conserved quantities. These identities are used to formulate Noether-type theorems that give first integrals of second-order DODEs with symmetries. Finally, several open problems are formulated in the Conclusion.

The different forms of the tetrahedron equation appear when all possible ways to label the scattering process of infinitely long straight lines are considered in three dimensional spacetime. This is expected to lead to three dimensional integrability, analogous to the Yang-Baxter equation. Among the three possibilities, we consider two of them and their variants. We show that Clifford algebras solve both the constant and the spectral parameter dependent versions of all of them. We also present a scheme for canonically solving higher simplex equations using tetrahedron solutions.

This work investigates the long-time asymptotic behaviors of solutions to the initial value problem of the two-component nonlinear Klein-Gordon equation by inverse scattering transform and Riemann-Hilbert formulism. Two reflection coefficients are defined and their properties are analyzed in detail. The Riemann-Hilbert problem associated with the initial value problem is constructed in term of the two reflection coefficients. The Deift-Zhou nonlinear steepest descent method is then employed to analyze the Riemann-Hilbert problem, yielding the long-time asymptotics of the solution in different regions. Specifically, a higher-order asymptotic expansion of the solution inside the light cone is provided, and the leading term of this asymptotic solution is compared with results from direct numerical simulations, showing excellent agreement. This work not only provides a comprehensive analysis of the long-time behaviors of the two-component nonlinear Klein-Gordon equation but also offers a robust framework for future studies on similar nonlinear systems with third-order Lax pair.

The influence matrix (IM) provides a powerful framework for characterizing nonequilibrium quantum many-body dynamics by encoding multitime correlations into tensor-network states. Understanding how its computational complexity relates to underlying dynamics is crucial for both theoretical insight and practical utility, yet remains largely unexplored despite a few case studies. Here, we address this question for a family of brickwork quantum circuits ranging from integrable to chaotic regimes. Using tools from geometric group theory, we identify three qualitatively distinct scalings of temporal entanglement entropy, establishing a hierarchy of computational resources required for accurate tensor-network representations of the IM for these models. We further analyze the memory structure of the IM and distinguish between classical and quantum temporal correlations. In particular, for certain examples, we identify effectively classical IMs that admit an efficient Monte Carlo algorithm for computing multitime correlations. In more generic settings without an explicit classical description of the IM, we introduce an operational measure of quantum memory with an experimental protocol, and discuss examples exhibiting long-time genuinely quantum correlations. Our results establish a new connection between quantum many-body dynamics and group theory, providing fresh insights into the complexity of the IM and its intricate connection to the physical characteristics of the dynamics.

We study a certain type of multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. We show that all the coefficients in the multiple commutation relations between the $L$-operator elements are given in terms of the trigonometric weight functions for the vector representation, independent of the representation of the $L$-operator. For rank one case, our proof also gives a conceptual understanding why the coefficients can also be expressed using the Izergin-Korepin determinants. As a related result, by specializing expressions for the universal nested Bethe vector by Pakuliak-Ragoucy-Slavnov, we also find a construction of the Gelfand-Tsetlin basis for the vector representation using different $L$-operator elements from the constructions by Nazarov-Tarasov or Molev. We also present corresponding results for the Yangian $Y_h(\mathfrak{gl}_N)$.

We show that every fusion category containing a non-invertible, self-dual object $a$ gives rise to an integrable anyonic chain whose Hamiltonian density satisfies the Temperley-Lieb algebra. This spin chain arises by considering the projection onto the identity channel in the fusion process $a\otimes a$. We relate these models to Pasquier's construction of ADE lattice models. We then exploit the underlying Temperley-Lieb structure to discuss the spectrum of these models and argue that these models are gapped when the quantum dimension of $a$ is greater than 2. We show that for fusion categories where the dimension is close to 2, such as the Fib$\times$Fib and Haagerup fusion categories, the finite size effects are large and they can obscure the numerical analysis of the gap.

This paper investigates the algebraic reduction of the infinite-dimensional symmetries of the Ablowitz-Kaup-Newell-Segur system when restricted to multi-soliton solution. By systematically analyzing, we demonstrate that the entire $K$-symmetry hierarchy collapses into a finite-dimensional module over the field of wave parameters, spanned by elementary center-translation generators. Higher order $K$-symmetries are explicitly reconstructed as linear combinations of these basis vectors. In contrast, $\tau$-symmetries resist such decomposition on pure soliton backgrounds, however, they become finite-dimensional once the underlying solution space is extended to the full multi-wave manifold, which carries more independent wave parameters. We construct an explicit basis consisting of four fundamental symmetry vector fields, two lowest $K$-symmetries and two primary $\tau$-symmetries, in terms of which all higher $\tau$-symmetries are uniquely expressible as linear combinations of these symmetry vector fields. These findings not only clarify the algebraic structure of infinite symmetries on special solution, but also provide an algorithmic framework for deriving exact multi-wave solutions of integrable systems.

A novel integrability condition for the Riccati equation, the simplest form of nonlinear ordinary differential equations, is obtained by using elementary quadrature method. Under this condition, the analytical general solutions containing free parameters are presented in this paper, which can be extended to second-order linear ordinary differential equation. These results provide valuable mathematical criteria for analyzing nonlinear phenomena in many disciplines.

For PT-symmetric periodic Schrodinger operator, which is a small perturbation of the zero potential, we calculate the spectrum and the divisor of zeroes of the Bloch function in the leading order of the perturbation theory. In particular, we show that the analogs of lacunae of the Bloch spectrum are ellipses, and their focal points coincide with the branch points of the spectral curve.

We present a study of a quasi-integrable deformation of the three-particle open Toda chain, constructed by introducing a translation-invariant three-body interaction terms. Although this modification explicitly breaks the exact integrability of the standard Toda model, it retains fundamental structural properties, including energy and momentum conservation. Furthermore, we show that under a specific time-reflection and discrete symmetry among the chain coordinates, the system admits a quasi-conserved higher-order integral. Through analytic and numerical analysis of the deformed dynamics, we demonstrate the emergence and long-time persistence of quasi-conserved quantities, thereby establishing a controlled realization of quasi-integrability in a minimal nonlinear chain. Given the central role of integrable systems in elucidating the dynamics of classical and quantum models, this framework provides a concrete setting to investigate the mechanisms underlying the gradual breakdown of integrability and the onset of quasi-integrability in few-body systems.

The purpose of this article is to provide a perspective - admittedly, a rather subjective one - of recent developments at the interface of machine learning/data-driven methods and nonlinear wave studies. We review some recent pillars of the rapidly evolving landscape of scientific machine learning, including deep learning, data-driven equation discovery, and operator learning, among others. We then showcase these methods in applications ranging from learning lattice dynamical models and reduced order modeling of effective dynamics to discovery of conservation laws and potential identification of integrability of ODE and PDE models. Our intention is to make clear that these machine learning methods are complementary to the preexisting powerful tools of the nonlinear waves community, and should be integrated into this toolkit to augment and enable mathematical discoveries and computational capabilities in the age of data.

We investigate the asymmetric integrable turbulence and rogue waves (RWs) emerging from the modulation instability (MI) of plane waves for the DNLS equation. The \(n\)-th moments and ensemble-averaged kinetic and potential energy exhibit oscillatory convergence towards their steady-state values. Specifically, the amplitudes of oscillations for these indexes decay asymptotically with time as \(t^{-1.36}\), while the phase shifts demonstrate a nonlinear decay with a rate of \(t^{-0.78}\). The frequency of these oscillations is observed to be twice the maximum growth rate of MI. These oscillations can be classified into two distinct types: one is in phase with ensemble-averaged potential energy modulus $|\langle H_4\rangle|$, and the other is anti-phase. At the same time, this unity is also reflected in the wave-action spectrum \( S_k(t) \) for a given \( k \), the auto-correlation function \( g(x,t) \) for a given \( x \), as well as the PDF \( P(I,t) \). The critical feature of the turbulence is the wave-action spectrum, which follows a power-law distribution of \( |k+3|^{-\alpha} \) expect for $k=-3$. Unlike the NLS equation, the turbulence in the DNLS setting is asymmetric, primarily due to the asymmetry between the wave number of the plane wave from the MI and the perturbation wave number.. As the asymptotic peak value of \( S_k \) is observed at \( k = -3 \), the auto-correlation function exhibits a nonzero level as \( x \to \pm L/2 \). The PDF of the wave intensity asymptotically approaches the exponential distribution in an oscillatory manner. However, during the initial stage of the nonlinear phase, MI slightly increases the occurrence of RWs. This happens at the moments when the potential modulus is at its minimum, where the probability of RWs occurring in the range of \( I\in [12, 15] \) is significantly higher than in the asymptotic steady state.

In this paper we derive bilinear forms and solutions in Casoratians for some fourth-order lattice Gel'fand-Dikii (lattice GD-4) type equations. These equations were recently formulated from the direct linearization approach and exhibit multidimensionally consistent property in multi-component form. The obtained solitons and Casoratian forms enable us to extend these equations by introducing a parameter $\delta$. These $\delta$-extended lattice GD-4 type equations are still consistent around the cube, and their bilinear forms together with Casoration solutions are presented.