Integrable Research Articles

Complex Dynamics in Circular and Deformed Bilayer Graphene-Inspired Billiards with Anisotropy and Strain

While billiard systems of various shapes have been used as paradigmatic model systems in the fields of nonlinear dynamics and quantum chaos, few studies have investigated anisotropic billiards. Motivated by the tremendous advances in using and controlling electronic and optical mesoscopic systems with bilayer graphene (BLG), representing an easily accessible anisotropic material for electrons when trigonal warping is present, we investigate billiards of various anisotropies and geometries using a trajectory-tracing approach founded on the concept of ray–wave correspondence. We find that the presence of anisotropy can change the billiards’ dynamics dramatically from its isotropic counterpart. It may induce chaotic and mixed dynamics in otherwise integrable systems and may stabilize originally unstable trajectories. We characterize the dynamics of anisotropic billiards in real and phase space using Lyapunov exponents and the Poincaré surface of section as phase space representation.

Bethe Ansatz inside Calogero-Sutherland models

We study the trigonometric quantum spin-Calogero–Sutherland model, and the Haldane–Shastry spin chain as a special case, using a Bethe-Ansatz analysis. We harness the model’s Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero–Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe Ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero–Sutherland model that generalises the Yangian Gelfand–Tsetlin basis of Takemura–Uglov. The Bethe-Ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe Ansatz works in the presence of fusion, may be of independent interest.

Time almost periodicity for solutions of Toda lattice equation with almost periodic initial datum

Abstract This paper analyzes the initial value problem for the Toda lattice with almost periodic initial data: let $J(t; J_{0})$ denote the family of Jacobi matrices which are solutions of the Toda flow equation with initial condition $J(0; J_{0})=J_{0},$ then, the given almost periodic datum $J_{0}$ is a discrete linear Schrödinger operator with almost periodic potential, which plays a fundamental role in our considerations. We show that, under some given hypotheses, the spectrum of the Schrödinger operator is pure absolute continuous and homogeneous (measure-theoretically) by establishing exponential asymptotics on the size of spectral gaps. These two conclusions enable us to show the boundedness and almost periodicity in the time of solutions for Toda lattice equation with almost periodic initial data. As a consequence, our result presents a positive answer to the discrete Deift’s conjecture [Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices (Contemporary Mathematics, 458). American Mathematical Society, Providence, RI, 2008, pp. 419–430; Some open problems in random matrix theory and the theory of integrable systems. II. SIGMA Symmetry Integrability Geom. Methods Appl.13 (2017), Paper no. 016].

Quasi-Wronskian solitons in the non-commutative kuralay-IIA equation: analysis and simulations

Abstract In this study, we explore the non-commutative extension of the Kuralay-IIA equation. We treat real or complex-valued functions as non-commutative and employ the Lax pair corresponding to the evolution equation, similar to the commutative case. The quasi-Wronskian solution is derived using a Darboux transformation, with soliton solutions presented explicitly within the quasideterminant framework. Simulations illustrating the dynamics and solutions are provided for visualization. Additionally, we analyze the effects of non-commutativity on soliton profiles, revealing novel structures and dynamics distinct from commutative cases. This work paves the way for further exploration of non-commutative integrable systems and their applications in mathematical physics.

Diversity of soliton dynamics, positive multi-complexiton solutions and modulation instability for (3+1)-dimensional extended Kairat-X equation

This study explores the dynamics of a [Formula: see text]-dimensional extended Kairat-X equation, revealing its connections to differential geometry of curves and equivalence principles. Using the Hirota bilinear method and linear superposition principle (LSP), we derived lump-periodic solutions, higher-order solitons with bifurcations, resonant multi-soliton waves, and positive complexiton solutions. To highlight the significance of these results, we present 3D, density and contour plots for specific parameter values. Also, we analyzed the modulation instability (MI) by determining the stability region and gain spectrum. This novel work offers valuable insights into the integrable model introduced by Wazwaz, and provides algorithms for investigating newly developed systems related to plasma physics, optical communications, marine dynamics, and the geometry of curves.

Rogue wave patterns in the nonlinear Schrödinger-Boussinesq system

Abstract To the nonlinear Schr"{o}dinger--Boussinesq system, with the aid of Adler--Moser polynomials we predict the patterns of higher-order rogue wave solutions containing multiple large parameters. The new interesting rogue wave patterns of a number of true and predicted solutions are graphically illustrated, including fan-, heart-shaped structures and their skewed versions. The results are significant for both experimental and theoretical studies of rogue wave patterns of integrable systems.

Complete Integrability of Subriemannian Geodesic Flows on 𝕊7

Complete Integrability of Subriemannian Geodesic Flows on 𝕊7

On the Klein–Gordon scalar field oscillators in a spacetime with spiral-like dislocations in external magnetic fields

We investigate the effects of two types of spiral dislocation (the distortion of the radial line, labeled as spiral dislocation I, and the distortion of a circle, labeled as spiral dislocation II) on the relativistic dynamics of the Klein–Gordon (KG) oscillator fields, both in the presence and absence of external magnetic fields. In this context, our investigations show that while spiral dislocation I affects the energies of the KG oscillators (with or without the magnetic field), spiral dislocation II has, interestingly, no effect on the KG oscillator’s energies unless a magnetic field is applied. However, for both types of spiral dislocations, we observe that the corresponding wave functions incorporate the effects of the dislocation parameter. Our findings are based on the exact solvability and conditional exact solvability (associated with the biconfluent Heun polynomials) of the KG oscillators (with or without the magnetic field, respectively) for spiral dislocation I, and the exact solvability of the KG oscillators (with or without the magnetic field) for spiral dislocation II. The exact solvability of the latter suggests that the oscillator’s frequency is solely determined by the magnetic field strength.

Klein–Gordon oscillators in traversable wormhole rainbow gravity spacetime: Conditional exact solvability via a throat radius and oscillator frequency correlation

In this study, we discuss an analytical solution for a set of the Klein–Gordon (KG) oscillators’ energies through a correlation between the frequency of the KG-oscillators and the traversable wormhole (TWH) throat radius. Under such restricted parametric correlation (hence the notion of conditionally exact solvability is unavoidable in the process), we report the effects of throat radius, rainbow parameter, disclination parameter and oscillator frequency on the spectroscopic structure of a vast number of [Formula: see text]-states (the radial and magnetic quantum numbers, respectively). In the process, we only use two loop quantum gravity motivated rainbow functions pairs. Only rainbow functions clearly and reliably fully adhere to the rainbow gravity model and secure Planck energy [Formula: see text] as the maximum possible energy for particles and anti-particles alike. Near the asymptotically flat upper and lower universes connected by the TWH, i.e. for the throat radius [Formula: see text], the energies tend to cluster around the rest mass energies, i.e. [Formula: see text]. Whereas, for [Formula: see text], the energies tend to approach [Formula: see text].

Exact solutions and their local structures of an inverse scattering integrable system with variable coefficients

An important manifestation of the integrability of nonlinear mathematical and physical models is their solvability through inverse scattering transform (IST). The problem to be investigated in this paper is a system of fifth-order nonlinear evolution equations (NLEEs) with variable coefficients related to time-varying spectral problems. The expected result is to derive the system of fifth-order NLEEs, obtain its exact solutions, verify its integrability in the sense IST solvability, and reveal some novel local structures of the obtained solutions. First, the system of fifth-order NLEEs is derived. Then, by combining the IST with a time-varying spectral parameter, the associated scattering data are determined for the reconstruction of potentials. Using the determined scattering data, exact solutions are ultimately obtained. Meanwhile, some local structures with new features are analyzed for two pairs of special solutions, including kink solitons, wide/narrow top bell solitons, and small-scale peaks in the time variable direction, as well as sine/cosine-like fluctuations in the spatial variable direction. This paper not only demonstrates that equipping the Ablowitz–Kaup–Newell–Segur (AKNS) problems with appropriate time-varying spectra can be used to derive some other inverse scattering integrable systems of NLEEs with variable coefficients, but also graphically illustrates the regulatory effect of time-varying spectral parameter on local solution structures.

On quasi-integrable deformation scheme of the KdV system

We propose a general approach to quasi-deform the Korteweg–De Vries (KdV) equation by deforming its Hamiltonian. The standard abelianization process based on the inherent sl(2) loop algebra leads to an infinite number of anomalous conservation laws, that yield conserved charges for definite space-time parity of the solution. Judicious choice of the deformed Hamiltonian yields an integrable system with scaled parameters as well as a hierarchy of deformed systems, some of which possibly are quasi-integrable. One such system maps to the known quasi-deformed nonlinear Schrödinger (NLS) soliton in the already known weak-coupling limit, whereas a generic scaling of the KdV amplitude also suggests quasi-integrability under an order-by-order expansion. In general, these deformed KdV solutions need to be parity-even for quasi-conservation that agrees with our analytical results. Following the recent demonstration of quasi-integrability in regularized long wave (RLW) and modified regularized long wave (mRLW) systems by ter Braak et al. (Nucl Phys B 939:49–94, 2019), that are particular cases of the present approach, general soliton solutions should numerically be accessible.

Bifurcations and exact solutions of the complex Ginzburg–Landau equation with Kudryashov's law

In this paper, we investigate the complex Ginzburg–Landau equation with Kudryashov's law using the method of dynamical systems. By using travelling-wave transformation, the model can be converted into a singular integrable travelling-wave system. Then we discuss the dynamical behaviour of the associated regular system and we obtain bifurcations of the phase portraits of the travelling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, homoclinic and peakon solutions.

On Superization of Nonlinear Integrable Dynamical Systems

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called "naive" approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis.

Krylov complexity and Trotter transitions in unitary circuit dynamics

We investigate many-body dynamics where the evolution is governed by unitary circuits through the lens of “Krylov complexity,” a recently proposed measure of complexity and quantum chaos. We extend the formalism of Krylov complexity to unitary circuit dynamics and focus on Floquet circuits arising as the Trotter decomposition of Hamiltonian dynamics. For short Trotter steps the results from Hamiltonian dynamics are recovered, whereas a large Trotter step results in different universal behavior characterized by the existence of local : operators with vanishing autocorrelation functions, as exemplified in dual-unitary circuits. These operators exhibit maximal complexity growth, act as a memoryless bath for the dynamics, and can be directly probed in current quantum computing setups. These two regimes are separated by a crossover in chaotic systems. Conversely, we find that free integrable systems exhibit a nonanalytic transition between these different regimes, where maximally ergodic operators appear at a critical Trotter step. Published by the American Physical Society 2025

Open-system eigenstate thermalization in a noninteracting integrable model

Open-system eigenstate thermalization in a noninteracting integrable model

Extensions of Riordan Arrays and Their Applications

The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions of the idea of a Riordan array have included almost Riordan arrays, double Riordan arrays, and their generalizations. After giving a brief overview of the Riordan group, we define two further extensions of the notion of Riordan arrays, and we give a number of applications for these extensions. The relevance of these applications indicates that these new extensions are worthy of study. The first extension is that of the reverse symmetrization of a Riordan array, for which we give two applications. The first application of this symmetrization is to the study of a family of Riordan arrays whose symmetrizations lead to the famous Robbins numbers as well as to numbers associated with the 20 vertex model of mathematical physics. We provide closed-form expressions for the elements of these arrays, and we also give a canonical Catalan factorization for them. We also describe an alternative family of Riordan arrays whose symmetrizations lead to the same integer sequences. The second application of this symmetrization process is to the area of the enumeration of lattice paths. We remain with the applications to lattice paths for the second extension of Riordan arrays that we introduce, which is the interleaved Riordan array. The methods used include generating functions, linear algebra, weighted compositions, and linear recurrences. In the case of the symmetrization process applied to Riordan arrays, we focus on the principal minor sequences of the resulting square matrices in the context of integrable lattice models.

Symmetry of traveling wave solutions for a Camassa–Holm type equation with higher-order nonlinearity

Abstract We are concerned with a Camassa–Holm type equation with higher-order nonlinearity including some integrable peakon models such as the Camassa–Holm equation, the Degasperis–Procesi equation and the Novikov equation. We show that all the horizontal symmetric waves for this equation must be traveling waves. This extends the previous results for the Camassa–Holm and Novikov equations.

Lattice Fermions with Solvable Wide Range Interactions

Abstract Exactly solvable (spinless) lattice fermions with wide range interactions are constructed explicitly based on exactly solvable stationary and reversible Markov chains $\mathcal {K}^R$ reported a few years earlier by Odake and myself. The reversibility of $\mathcal {K}^R$ with the stationary distribution $\pi$ leads to a positive classical Hamiltonian $\mathcal {H}^R$. The exact solvability of $\mathcal {H}^R$ warrants that of a spinless lattice fermion $c_x$, $c_x^\dagger$, $\mathcal {H}^R_f=\sum _{x,y\in \mathcal {X}}c_x^\dagger \mathcal {H}^R(x,y) c_y$ based on the principle advocated recently by myself. The reversible Markov chains $\mathcal {K}^R$ are constructed by convolutions of the orthogonality measures of the discrete orthogonal polynomials of the Askey scheme. Several explicit examples of the fermion systems with wide range interactions are presented.

Multidimensional integrable systems from contact geometry

Upon having presented a bird’s eye view of history of integrable systems, we give a brief review of certain recent advances in the longstanding problem of search for partial differential systems in four independent variables, often referred to as (3+1)-dimensional or 4D systems, that are integrable in the sense of soliton theory. Namely, we review a recent construction for a large new class of (3+1)-dimensional integrable systems with Lax pairs involving contact vector fields. This class contains inter alia two infinite families of such systems, thus establishing that there is significantly more integrable (3+1)-dimensional systems than it was believed for a long time. In fact, the construction under study yields (3+1)-dimensional integrable generalizations of many well-known dispersionless integrable (2+1)-dimensional systems like the dispersionless KP equation, as well as a first example of a (3+1)-dimensional integrable system with an algebraic, rather than rational, nonisospectral Lax pair. To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik–Veselov–Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.

Darboux integrability of a general circuit system

Darboux integrability of a general circuit system