Integrable Research Articles

On the multiple soliton and lump solutions to the (3+1)-dimensional Painlevé integrable Boussinesq-type and KP-type equations

This study aims to construct and investigate a novel (3+1)-dimensional model that combines both the Boussinesq-type and the Kadomtsev-Petviashvili (KP) equations, which is called a (3+1)-dimensional Boussinesq-KP-type (B-KP-type) equation. We demonstrate that the combined model does not exhibit Painleve integrability. However, it does provide two separate integrable equations: one of the (3+1)-dimensional Boussinesq-type equation and the other of the (3+1)-dimensional KP-type equation. The simplified Hirota’s direct approach can formally admit multiple soliton solutions for each integrable model. In addition, we employ Maple symbolic computation with the Hirota bilinear form to derive a class of lump solutions for each equation. Moreover, the (3+1)-dimensional B-KP-type equation is analyzed using the families of tanh and tan methods for deriving traveling wave solutions, including shock wave solutions, periodic solutions, and others. Due to the significant correlation and wide range of applications of the Boussinesq-type and KP equations, the derived equations will play a crucial role in elucidating and interpreting various nonlinear phenomena observed in fluid mechanics and other fields of nonlinear physics and engineering issues.

Non-Gaussian diffusive fluctuations in Dirac fluids

Dirac fluids-interacting systems obeying particle-hole symmetry and Lorentz invariance-are among the simplest hydrodynamic systems; they have also been studied as effective descriptions of transport in strongly interacting Dirac semimetals. Direct experimental signatures of the Dirac fluid are elusive, as its charge transport is diffusive as in conventional metals. In this paper, we point out a striking consequence of fluctuating relativistic hydrodynamics: The full counting statistics (FCS) of charge transport is highly non-Gaussian. We predict the exact asymptotic form of the FCS, which generalizes a result previously derived for certain interacting integrable systems. A consequence is that, starting from quasi-one-dimensional nonequilibrium initial conditions, charge noise in the hydrodynamic regime is parametrically enhanced relative to that in conventional diffusive metals.

Integrability of special quadratic systems with invariant hyperbolas

In Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ.45(2021), 1–90, the authors investigate about the integrability of the family QSH (the whole class of non-degenerate planar quadratic systems possessing at least one invariant hyperbola). However, some very difficult cases are left open in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ.45(2021), 1–90, and the main aim of this article is to study the Liouvillian integrability some of the systems that were left behind in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ.45(2021), 1–90.

Study on a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in nonlinear physics: Multiple soliton solutions, lump solutions, and breather wave solutions

In this work, we study an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equations that appear in many nonlinear physics applications. We show that this extended equation retains its complete integrability via Painlevé analysis. We explore multiple soliton solutions by using the Hirota bilinear method. Moreover, we derive lump solutions where two numerical examples are tested. Breather wave solutions were also explored by using a variety of distinct schemes. We also determine other traveling wave solutions, rational solutions, periodic solutions, exponential solutions, ratio of trigonometric or hyperbolic functions, and others.

Rotations and Integrability

We discuss some families of integrable and superintegrable systems in -dimensional Euclidean space which are invariant under rotations. The invariant Hamiltonian is integrable with integrals of motion and an additional integral ofmotion , which are first- and fourth-order polynomials in momenta, respectively.

Non-equilibrium dynamics of symmetry-resolved entanglement and entanglement asymmetry: exact asymptotics in Rule 54*

Abstract Symmetry resolved entanglement and entanglement asymmetry are two measures of quantum correlations sensitive to symmetries of the system. Here we discuss their non-equilibrium dynamics in the Rule 54 cellular automaton, a simple, yet interacting, integrable model. Both quantities can be expressed in terms of the more analytically tractable ‘charged moments’, i.e. traces of powers of a suitably deformed density matrix, via a replica trick. We express them in terms of a tensor network, which we contract in space using a system of local algebraic relations. This gives the asymptotic form for the charged moments, valid in the regime of large but finite time that is shorter than all the relevant subsystem sizes. In this regime the charge moments decay exponentially with the rate given by the leading solution to a cubic equation.

From Chaos to Integrability in Double Scaled Sachdev-Ye-Kitaev Model via a Chord Path Integral

We study thermodynamic phase transitions between integrable and chaotic dynamics. We do so by analyzing models that interpolate between the chaotic double scaled Sachdev-Ye-Kitaev (SYK) and the integrable p-spin systems, in a limit where they are described by chord diagrams. We develop a path integral formalism by coarse graining over the diagrams, which we use to argue that the system has two distinct phases: one is continuously connected to the chaotic system, and the other to the integrable. They are separated by a line of first order transition that ends at some finite temperature. Published by the American Physical Society 2024

Classical and Bohmian Trajectories in Integrable and Nonintegrable Systems

In the present paper, we study both classical and quantum Hénon–Heiles systems. In particular, we make a comparison between the classical and quantum trajectories of integrable and nonintegrable Hénon–Heiles Hamiltonians. From a classical standpoint, we study both theoretically and numerically the form of invariant curves in the Poincaré surfaces of section for several values of the coupling parameter in the integrable case and compare them with those in the nonintegrable case. Then, we examine the corresponding Bohmian trajectories, and we find that they are chaotic in both cases, but with chaos emerging at different times.

Geometry of Integrable Systems Related to the Restricted Grassmannia

A hierarchy of differential equations on a Banach Lie-Poisson space related to the restricted Grassmannian is studied. Flows on the groupoid of partial isometries and on the restricted Grassmannian are described, and a momentum map picture is presented.

5d 2-Chern-Simons Theory and 3d Integrable Field Theories

The 4-dimensional semi-holomorphic Chern-Simons theory of Costello and Yamazaki provides a gauge-theoretic origin for the Lax connection of 2-dimensional integrable field theories. The purpose of this paper is to extend this framework to the setting of 3-dimensional integrable field theories by considering a 5-dimensional semi-holomorphic higher Chern-Simons theory for a higher connection (A, B) on R3×CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3 \ imes \\mathbb {C}P^1$$\\end{document}. The input data for this theory are the choice of a meromorphic 1-form ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}P^1$$\\end{document} and a strict Lie 2-group with cyclic structure on its underlying Lie 2-algebra. Integrable field theories on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} are constructed by imposing suitable boundary conditions on the connection (A, B) at the 3-dimensional defects located at the poles of ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} and choosing certain admissible meromorphic solutions of the bulk equations of motion. The latter provides a natural notion of higher Lax connection for 3-dimensional integrable field theories, including a 2-form component B which can be integrated over Cauchy surfaces to produce conserved charges. As a first application of this approach, we show how to construct a generalization of Ward’s (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional integrable chiral model from a suitable choice of data in the 5-dimensional theory.

Constructing integrable spin-boson model with open boundary conditions

Constructing integrable spin-boson model with open boundary conditions

On Sawada-Kotera and Kaup-Kuperschmidt integrable systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the M2 -extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.

New integrable chiral cosmological models with two scalar fields

New integrable chiral cosmological models with two scalar fields

Multi-component decompositions, linear superpositions, and new nonlinear integrable coupled KdV-type systems

In the realm of nonlinear integrable systems, the presence of decompositions facilitates the establishment of linear superposition solutions and the derivation of novel coupled systems exhibiting nonlinear integrability. By focusing on single-component decompositions within the potential BKP hierarchy, it has been observed that specific linear superpositions of decomposition solutions remain consistent with the underlying equations. Moreover, through the implementation of multi-component decompositions within the potential BKP hierarchy, successful endeavors have been undertaken to formulate linear superposition solutions and novel coupled KdV-type systems that resist decoupling via alterations in dependent variables.

Exactly solvable inhomogeneous fermion systems

Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on one-dimensional lattices are constructed explicitly based on the discrete orthogonal polynomials of Askey scheme, e.g. the Krawtchouk, Hahn, Racah, Meixner, q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground state two point correlation functions are derived explicitly. The multi point correlation functions are obtained by Wick’s Theorem. Corresponding 15 exactly solvable XX spin systems are also displayed. They all have nearest neighbour interactions. The exact solvability of Schrödinger equation means that of the corresponding Fokker-Planck equation. This leads to 15 exactly solvable Birth and Death fermions and 15 Birth and Death spin models. These provide plenty of materials for calculating interesting quantities, e.g. entanglement entropy, etc.

One loop effective actions in Kerr-(A)dS black holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024

New cases of integrable ninth-order conservative and dissipative dynamical systems

New cases of integrable dynamical systems of the ninth order homogeneous in terms of variables are presented, in which a system on a tangent bundle to a four-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of a different sign. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

Controlling Energy Storage Crossing Quantum Phase Transitions in an Integrable Spin Quantum Battery.

We investigate the performance of a one-dimensional dimerized XY chain as a spin quantum battery. Such integrable model shows a rich quantum phase diagram that emerges through a mapping of the spins onto auxiliary fermionic degrees of freedom. We consider a charging protocol relying on the double quench of an internal parameter, namely the strength of the dimerization, and address the energy stored in the systems. We observe three distinct regimes, depending on the timescale characterizing the duration of the charging: a short-time regime related to the dynamics of the single dimers, a long-time regime related to the recurrence time of the system at finite size, and a thermodynamic limit time regime. In the latter, the energy stored is almost unaffected by the charging time and the precise values of the charging parameters, provided the quench crosses a quantum phase transition. Finally, we analytically prove that the three-timescale behavior and the strong dependence of the energy stored on the quantum phase diagram also hold in the quantum Ising chain in a transverse field. Our results can play a relevant role in the design of stable solid-state quantum batteries.

Fermionic integrable models and graded Borchers triples

We provide an operator-algebraic construction of integrable models of quantum field theory on 1+1-dimensional Minkowski space with fermionic scattering states. These are obtained by a grading of the wedge-local fields or, alternatively, of the underlying Borchers triple defining the theory. This leads to a net of graded-local field algebras, of which the even part can be considered observable, although it is lacking Haag duality. Importantly, the nuclearity condition implying nontriviality of the local field algebras is independent of the grading, so that existing results on this technical question can be utilized. Application of Haag–Ruelle scattering theory confirms that the asymptotic particles are indeed fermionic. We also discuss connections with the form factor programme.

On Integrability of Certain Classes of Variable Dissipation Systems

We study the non-conservative systems for which the methods for studying, for example, Hamiltonian systems is not applicable in general. Therefore, for such systems, it is necessary, in some sense, to “directly” integrate the main equation of dynamics. Herewith, we offer more universal interpretation of both obtained cases and new ones of complete integrability in transcendental functions in dynamics in a non-conservative force field.