Classical Integrable Systems Research Articles

On the exploration of new solitary wave solutions for the classical integrable Kuralay-IIA system of equations

In this paper, we derive various new optical soliton solutions for the coupled Kuralay-IIA system of equations using an innovative solution approach known as the ϕ 6 − model expansion technique. This solution methodology employs a traveling wave transformation to reduce the considered problem into an easily solvable higher-order ordinary differential equation. Unlike other existing related methods, this solution approach adopted here allows us to extract a rich list of diverse exact soliton solutions for the considered problem. The obtained solutions incorporate the Jacobi elliptic functions which are shown to degenerate into trigonometric and hyperbolic function solutions. These solutions exhibit distinct wave structures consisting of dark, bright, rational, periodic, singular and mixed optical solitons profiles. In exploring the impact of spatial and temporal variables on the wave patterns of the considered model, physical structures of some of the obtained solitons solutions are characterized through 3D, contour and 2D wave profiles for selected parameter values. This not only ensures the validity of the solutions as well as the constraints arising from the solution technique but also offers researchers a deeper understanding of the properties of the considered problem. The outcomes here demonstrate the applicability, versatility and efficiency of the considered solution approach for deriving diverse new soliton solutions for even more complex systems of nonlinear evolution equations.

Progresses on Some Open Problems Related to Infinitely Many Symmetries

The quest to reveal the physical essence of the infinitely many symmetries and/or conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of n-wave solutions, such as the n-soliton solutions, multiple breathers, complexitons, and the n-periodic wave solutions (the algebro-geometric solutions with genus n), wherein n denotes an arbitrary integer that can potentially approach infinity. Each subwave comprising the n-wave solution may possess free parameters, including center parameters ci, width parameters (wave number) ki, and periodic parameters (the Riemann parameters) mi. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinitely many symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. This conjecture appears to hold true for all integrable systems with n-wave solutions. The conjecture intimates that the currently known infinitely many symmetries is not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned Korteweg–de Vries (KdV) equation and the Burgers equation as simple examples, the conjecture is substantiated for the n-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative, which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework. Notably, a ren-symmetric integrable Burgers hierarchy is explicitly derived. Both the supersymmetric and the classical integrable hierarchies are encompassed within the ren-symmetric integrable hierarchy. The results of this paper will make further progresses in nonlinear science: to find more infinitely many symmetries, to establish novel methods to solve nonlinear systems via symmetries, to find more novel exact solutions and new physics, and to open novel integrable theories such as the ren-symmetric integrable systems and the possible relations to fractional integrable systems.

Q-boson model and relations with integrable hierarchies

This work investigates the intricate relationship between the q-boson model, a quantum integrable system, and classical integrable systems such as the Toda and KP hierarchies. Initially, we analyze scalar products of off-shell Bethe states and explore their connections to tau functions of integrable hierarchies. Furthermore, we discuss correlation functions within this formalism, examining their representations in terms of tau functions, as well as their Schur polynomial expansions.

Exact solutions to macroscopic fluctuation theory through classical integrable systems

We give a short overview of recent developments in exact solutions for macroscopic fluctuation theory by using connections to classical integrable systems. A calculation of the cumulant generating function for a tagged particle is also given, agreeing with a previous result obtained from a microscopic analysis.

New Classical Integrable Systems from Generalized TT[over ¯]-Deformations.

We introduce and study a novel class of classical integrable many-body systems obtained by generalized TT[over ¯] deformations of free particles. Deformation terms are bilinears in densities and currents for the continuum of charges counting asymptotic particles of different momenta. In these models, which we dub "semiclassical Bethe systems" for their link with the dynamics of Bethe ansatz wave packets, many-body scattering processes are factorized, and two-body scattering shifts can be set to an almost arbitrary function of momenta. The dynamics is local but inherently different from that of known classical integrable systems. At short scales, the geometry of the deformation is dynamically resolved: either particles are slowed down (more space available), or accelerated via a novel classical particle-pair creation and annihilation process (less space available). The thermodynamics both at finite and infinite volumes is described by the equations of (or akin to) the thermodynamic Bethe ansatz, and at large scales generalized hydrodynamics emerge.

Three-stage thermalization of a quasi-integrable system

We consider a system of classical hard rods or billiard balls in one dimension, initially prepared in a Bragg-pulse state at a given temperature and subjected to external periodic fields. We show that at late times the system thermalizes in the thermodynamic limit via a three-stages process characterized by: an early phase where the dynamics is well described by Euler hydrodynamics, a subsequent phase where a (weak) turbulent phase is observed and where hydrodynamic gradient expansion can be broken, and a final one where the gas thermalizes according to viscous hydrodynamics. As the hard rod gas shares the same large-scale hydrodynamics as other quantum and classical integrable systems, we expect these features to universally characterize all many-body integrable systems in generic external potentials. Published by the American Physical Society 2024

Takiff algebras, Toda systems, and jet transformations

We study a family of completely integrable systems attached to Takiff algebras gN, extending Toda systems for split simple Lie algebras g. With respect to Darboux coordinates on coadjoint orbits O, the potentials of the hamiltonians are products of polynomial and exponential functions. Explicit general solutions for their equations of motion are obtained using differential operators called jet transformations. The classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, quantizing the Poisson algebra of functions on O. These results are illustrated with concrete examples, including a 3-body problem based on sl(2) and an extension of soliton solutions for A∞ to associated Takiff algebras.

Rapidity distribution within the defocusing non-linear Schrödinger equation model

We consider the classical field integrable system whose evolution equation is the nonlinear Schrödinger equation with defocusing non-linearities, which is the classical limit of the quantum Lieb-Liniger model. We propose a simple derivation of the relation between two sets of conserved quantities: on the one hand the trace of the monodromy matrix, parameterized by the spectral parameter and introduced in the inverse-scattering framework, and on the other hand the rapidity distribution, a concept imported from the Lieb-Liniger model. To do so we use the definition of the rapidity distribution as the asymptotic momentum distribution after a very large expansion. We propose two different ways to derive the result, each one using a thought experiment that implements an expansion.

The algebraic structure of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy

We prove that each member of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine–Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Pöppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Pöppe product—the product rule for Hankel operators pioneered by Ch. Pöppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a ‘Pöppe polynomial’ expansion for basic compositions in terms of ‘linear signature expansions’ representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.

Multiphase solutions and their reductions for a nonlocal nonlinear Schrödinger equation with focusing nonlinearity

AbstractA nonlocal nonlinear Schrödinger equation with focusing nonlinearity is considered, which has been derived as a continuum limit of the Calogero–Sutherland model in an integrable classical dynamical system. The equation is shown to stem from the compatibility conditions of a system of linear partial differential equations (PDEs), assuring its complete integrability. We construct a nonsingular N‐phase solution (N: positive integer) of the equation by means of a direct method. The features of the one‐ and two‐phase solutions are investigated in comparison with the corresponding solutions of the defocusing version of the equation. We also provide an alternative representation of the N‐phase solution in terms of solutions of a system of nonlinear algebraic equations. Furthermore, the eigenvalue problem associated with the N‐phase solution is discussed briefly with some exact results. Subsequently, we demonstrate that the N‐soliton solution can be obtained simply by taking the long‐wave limit of the N‐phase solution. The similar limiting procedure gives an alternative representation of the N‐soliton solution as well as the exact results related to the corresponding eigenvalue problem.

Symplectic and inverse spectral geometry of integrable systems: A glimpse and open problems

We first give a glimpse of finite dimensional classical integrable Hamiltonian systems from the point of view of symplectic geometry and briefly discuss their quantum counterparts, with an emphasis on recent progress on inverse spectral geometry. Then we propose several open problems about the geometry, topology and dynamics of these systems. The problems are largely motivated by the works of a number of authors, including Arnold, Atiyah, Colin de Verdière, Delzant, Duistermaat, Eliasson, Fomenko, Guillemin, Kolmogorov, Kostant, Moser and Sternberg.

Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields

In this paper, we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of theorem 1 in Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203).

Chaos In 2D Bohmian Trajectories of Commensurate Harmonics Oscillators

Particle trajectories guided by the wave function are well-defined through Bohmian mechanics, which is a causal interpretation of quantum mechanics. Periodic and chaotic behaviours could be exhibited from the certain classical integrable systems that have been shown within this framework. In this study, we developed Mathematica programs to plot the Bohmian trajectories and Lyapunov exponents. These programs serve as computer experiments for numerical generation and illustration of the results. We show that the behaviours of commensurate two-dimensional harmonic oscillator systems are dependent on ratios of frequency.

Topological adiabatic dynamics in classical mass-spring chains with clamps

The path dependence of adiabatic evolution in classical harmonic chains with clamps is examined. It is shown that cutting and joining a chain may braid adiabatic normal mode frequencies. Accordingly, different adiabatic paths with the same endpoints may transport a normal mode to a different one, and an adiabatic cycle pumps action variables, i.e., the adiabatic invariants of integrable classical systems. Another adiabatic pump for artificial edge modes induced by clamps is shown as an application. Extensions to completely integrable systems and quantum systems are outlined.

Long‐diagonal pentagram maps

AbstractThe pentagram map on polygons in the projective plane was introduced by R. Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. In the present paper we introduce and prove integrability of long‐diagonal pentagram maps on polygons in , by now the most universal pentagram‐type map encompassing all known integrable cases. We also establish an equivalence of long‐diagonal and bi‐diagonal maps and present a simple self‐contained construction of the Lax form for both. Finally, we prove that the continuous limit of all these maps is equivalent to the ‐KdV equation, generalizing the Boussinesq equation for .

Centralizers of Jacobian derivations

Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.

Lax equations for relativistic Gaudin models on elliptic curve

We describe the most general GL NM classical elliptic finite-dimensional integrable system, which Lax matrix has n simple poles on elliptic curve. For M = 1 it reproduces the classical inhomogeneous spin chain, for N = 1 it is the Gaudin type (multispin) extension of the spin Ruijsenaars–Schneider model, and for n = 1 the model of M interacting relativistic GL N tops emerges in some particular case. In this way we present a classification for relativistic Gaudin models on GL-bundles over elliptic curve. As a by-product we describe the inhomogeneous Ruijsenaars chain. We show that this model can be considered as a particular case of multispin Ruijsenaars–Schneider model when residues of the Lax matrix are of rank one. An explicit parametrization of the classical spin variables through the canonical variables is obtained for this model. Finally, the most general GL NM model is also described through R-matrices satisfying associative Yang–Baxter equation. This description provides the trigonometric and rational analogues of GL NM models.

Generalized hydrodynamics of the KdV soliton gas

We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg–de Vries equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport.

Rigid fibers of integrable systems on cotangent bundles

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.

Thermalization without chaos in harmonic systems

Recent numerical results showed that thermalization of Fourier modes is achieved in short time-scales in the Toda model, despite its integrability and the absence of chaos. Here we provide numerical evidence that the scenario according to which chaos is irrelevant for thermalization is realized even in the simplest of all classical integrable system: the harmonic chain. We study relaxation from an atypical condition given with respect to random modes, showing that a thermal state with equilibrium properties is attained in short times. Such a result is independent from the orthonormal basis used to represent the chain state, provided it is a random basis.