Well-known Integrable Systems Research Articles

How close are integrable and nonintegrable models: A parametric case study based on the Salerno model.

In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable, nonintegrable one (the discrete nonlinear Schrödinger model). The question we ask is, for "generic" initial data, how close are the integrable to the nonintegrable models? Our more precise formulation of this question is, How well is the constancy of formerly conserved quantities preserved in the nonintegrable case? Upon examining this, we find that even slight deviations from integrability can be sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model. However, given that the knowledge of these quantities requires a deep physical and mathematical analysis of the system, we seek a more "generic" diagnostic towards a manifestation of integrability breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov exponents could be a sensitive diagnostic to that effect.

Deformation of Hamiltonian operator and its classifications. I

Based on the equivalence transformation, we study the classifications of the integrable generalizations of the KdV, the modified KdV and the Boussinesq equations obtained by the method introduced by Fuchssteiner and Fokas. The method is called “deformation of Hamiltonian operator” and generates generalized integrable systems with several parameters. Here the novel concept of the ‘border equation’ is introduced to isolate the special classes of integrable systems from the generalized ones. This enables us to position the well-known integrable systems. For example, the border equations of the generalized KdV are the KdV equation, the Camassa–Holm (CH) equation and the imaginary CH equation. The concept is powerful for the classification. As an example, new special integrable systems are found from the border equations of the general Boussinesq equation.

Integrable geometric flows for curves in pseudoconformal S3

We consider evolution equations for curves in the 3-dimensional sphere S3 that are invariant under the group SU(2,1) of pseudoconformal transformations, which preserves the standard contact structure on the sphere. In particular, we investigate how invariant evolutions of Legendrian and transverse curves induce both new and well-known integrable systems and hierarchies at the level of their geometric invariants.

Force Evolutionary Billiards and Billiard Equivalence of the Euler and Lagrange Cases

A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one).

Fifth-Order Alice-Bob Systems and Their Abundant Periodic and Solitary Wave Solutions**Sponsored by the National Natural Science Foundations of China under Grant No. 11435005 and K. C. Wong Magna Fund in Ningbo University

The study on the nonlocal systems is one of the hot topics in nonlinear science. In this paper, the well-known fifth-order integrable systems including the Sawada-Kotera (SK) equation, the Kaup-Kupershmidt (KK) equation and the fifth-order Koterweg-de Vrise (FOKdV) equation are extended to a generalized two-place nonlocal form, the generalised fifth-order Alice-Bob system. The Lax integrability of two sets of Alice-Bob systems for all the SK, KK and FOKdV type systems are explicitly given via matrix Lax pairs. The symmetry breaking and symmetry invariant periodic and solitary waves for one set of nonlocal SK, KK and FOKdV system are investigated via a special travelling wave solution ansatz.

Integrable nonlocal asymptotic reductions of physically significant nonlinear equations

Quasi-monochromatic complex reductions of a number of physically important equations are obtained. Starting from the cubic nonlinear Klein–Gordon (NLKG), the Korteweg–de Vries (KdV) and water wave equations, it is shown that the leading order asymptotic approximation can be transformed to the well-known integrable AKNS system (Ablowitz et al 1974 Stud. Appl. Math. 53 249) associated with second order (in space) nonlinear wave equations. This in turn establishes, for the first time, an important physical connection between the recently discovered nonlocal integrable reductions of the AKNS system and physically interesting equations. Reductions include the parity-time, reverse space-time and reverse time nonlocal nonlinear Schrödinger equations.

A new class of Euler equation on the dual of the N = 1 extended Neveu-Schwarz algebra

Let G be the N = 1 extended Neveu-Schwarz algebra and Greg* its regular dual. In this paper, we will study a super-Euler system with seven parameters (s1, s2, c1, …, c5) associated with Greg*. We will show that the super-Euler system is (1) local bi-superbihamiltonian if s1=14c1 and s2=12c2; (2) supersymmetric if s1 = c1 and s2 = c2; (3) local bi-superbihamiltonian and supersymmetric if s1 = c1 = 0 and s2 = c2 = 0. By choosing different parameters, we could obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the Ito equation, the 2-component Camassa-Holm equation, the 2-component Hunter-Saxton equation, and, especially, the Whitham-Broer-Kaup dispersive water-wave system.

On a Super Generalized X-Dependent Hirota Equation

AbstractHirota equation is a well-known integrable system which describes wave propagation in the ocean and optical fibres. The aim of this article is to construct a super generalized x-dependent Hirota equation which is the super extension of the Hirota equation. We investigate the supersymmetric integrable systems with two constraints and obtain their Lax representation. By means of the gauge transformation, we derive the gauge equivalent counterparts of the systems.

From Nothing to Something: Discrete Integrable Systems

Chinese ancient sage Laozi said that everything comes from ‘nothing’. Einstein believes the principle of nature is simple. Quantum physics proves that the world is discrete. And computer science takes continuous systems as discrete ones. This report is devoted to deriving a number of discrete models, including well-known integrable systems such as the KdV, KP, Toda, BKP, CKP, and special Viallet equations, from ‘nothing’ via simple principles. It is conjectured that the discrete models generated from nothing may be integrable because they are identities of simple algebra, model-independent nonlinear superpositions of a trivial integrable system (Riccati equation), index homogeneous decompositions of the simplest geometric theorem (the angle bisector theorem), as well as the Möbious transformation invariants.

Euler Equations Related to the Generalized Neveu-Schwarz Algebra

In this paper, we study supersymmetric or bi-superhamiltonian Euler equations related to the generalized Neveu-Schwarz algebra. As an application, we obtain several supersymmetric or bi-superhamiltonian generalizations of some well-known integrable systems including the coupled KdV equation, the 2-component Camassa-Holm equation and the 2-component Hunter-Saxton equation. To our knowledge, most of them are new.

Separation of variables for the generalized Henon–Heiles system and system with quartic potential

We consider two well-known integrable systems on the plane using the concept of natural Poisson bivectors on Riemannian manifolds. A geometric approach to construction of variables of separation and separated relations for the generalized Henon–Heiles system and the generalized system with quartic potential is discussed in detail.

Gross-Neveu models, nonlinear Dirac equations, surfaces and strings

Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model (GN2), and its chiral cousin, the NJL2 model, have shown that there are phases with inhomogeneous crystalline condensates. These (static) condensates can be found analytically because the relevant Hartree-Fock and gap equations can be reduced to the nonlinear Schr\"odinger equation, whose deformations are governed by the mKdV and AKNS integrable hierarchies, respectively. Recently, Thies et al have shown that time-dependent Hartree-Fock solutions describing baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation, and can be mapped directly to classical string solutions in AdS3. Here we propose a geometric perspective for this result, based on the generalized Weierstrass spinor representation for the embedding of 2d surfaces into 3d spaces, which explains why these well-known integrable systems underlie these various Gross-Neveu gap equations, and why there should be a connection to classical string theory solutions. This geometric viewpoint may be useful for higher dimensional models, where the relevant integrable hierarchies include the Davey-Stewartson and Novikov-Veselov systems.

1/fαnoise and integrable systems

In this paper, we show that there is a family of well-known integrable systems whose spectral fluctuations decay as 1/f(4), and thus do not follow the 1/f(2) law recently conjectured for integrable systems. We present a simple theoretical justification of this fact, and propose an alternative characterization of quantum chaos versus integrability formulated directly in terms of the power spectrum of the spacings of the unfolded spectrum.

Multi-Hamiltonian structures for r-matrix systems

For the rational, elliptic, and trigonometric r-matrices, we exhibit the links between three “levels” of Poisson spaces: (a) some finite-dimensional spaces of matrix-valued meromorphic functions on the complex line, (b) spaces of spectral curves and sheaves supported on them, and (c) symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to the Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family and which are such that the Lagrangian leaves are the intersections of the symplectic leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems.

The Finite Discrete KP Hierarchy and the Rational Functions

The set of all rational functions with any fixed denominator that simultaneously nullify in the infinite point is parametrized by means of a well-known integrable system: a finite dimensional version of the discrete KP hierarchy. This type of study was originated in Y. Nakamura's works who used others integrable systems. Our work proves that the finite discrete KP hierarchy completely parametrizes the spaceRatΛ(n)of rational functions of the formf(x)=q(x)/zn, whereq(x)is a polynomial of ordern−1with nonzero independent coefficent. More exactly, it is proved that there exists a bijection fromRatΛ(n)to the moduli space of solutions of the finite discrete KP hierarchy and a compatible linear system.

New integrable couplings and Hamiltonian structure of the KN hierarchy and the DLW hierarchy

Two new loop algebras F ∼ and G ∼ are constructed, which are devoted to establishing the resulting isospectral problems. By taking use of the compatibility of Lax pairs, the two corresponding zero curvature equations are presented from which the integrable couplings of the KN hierarchy and the dispersive long wave hierarchy (briefly called DLW hierarchy). As far as we can see, the above results are new. Again via employing the quadratic identity, the Hamiltonian structures of the two well-known integrable systems are obtained, respectively, and they are Liouville integrable.

A numerical integrator for the two-fixed-centres problem conserving all constants of motion

The two-fixed-centres problem describes the motion of a particle influenced by the gravitational pull of two fixed particles. This is a well-known integrable Hamiltonian system having three constants of motion. This paper presents a discrete two-dimensional two-fixed-centres problem which conserves two of these constants of motion. Moreover, this discrete system preserves the discrete analogue of the third constant of motion. A canonical transformation is introduced to separate the variables and remove singularities of the system. A combination of this transformation and energy-preserving method plays an important role in deriving the discrete system with a variable time step.

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra $U_q(\hat{sl}(n))$, where the rank $n$ coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the $sl(n)$-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.

On deformations of standard R-matrices for integrable infinite-dimensional systems

Simple deformations, with a parameter ϵ, of classical R-matrices which follow from decomposition of appropriate Lie algebras, are considered. As a result nonstandard Lax representations for some well-known integrable systems are presented and integrable evolution equations are constructed.

On the integrability of large N plane-wave matrix theory

We show the three-loop integrability of large N plane-wave matrix theory in a subsector of states comprised of two complex light scalar fields. This is done by diagonalizing the theory's Hamiltonian in perturbation theory and taking the large N limit. At one-loop level the result is known to be equal to the Heisenberg spin-1/2 chain, which is a well-known integrable system. Here, integrability implies the existence of hidden conserved charges and results in a degeneracy of parity pairs in the spectrum. In order to confirm integrability at higher loops, we show that this degeneracy is not lifted and that (corrected) conserved charges exist. Plane-wave matrix theory is intricately connected to N=4 super-Yang–Mills, as it arises as a consistent reduction of the gauge theory on a three-sphere. We find that after appropriately renormalizing the mass parameter of the plane-wave matrix theory the effective Hamiltonian is identical to the dilatation operator of N=4 super-Yang–Mills theory in the considered subsector. Our results therefore represent a strong support for the conjectured three-loop integrability of planar N=4 SYM and are in disagreement with a recent dual string theory finding. Finally, we study the stability of the large N integrability against nonsupersymmetric deformations of the model.