Narrow Results

In this paper, we provide the ${ℰ}$-model formulation of the non-deformed Cherednik model as well as of its Poisson-Lie and Poisson-Lie-WZ deformed version. In all three cases, we solve the sufficient condition of integrability by using the ${ℰ}$-model formalism. We thus recover in an alternative way the known results for the non-deformed and the Poisson-Lie deformed models, while for the Poisson-Lie-WZ deformed one our results are new.

A mathematical representation that includes probability in describing a system’s development is called a stochastic process. A random variable or noise-containing function is generally employed to represent a stochastic term in a mathematical model. In this paper, we analyze optical soliton (OS) solutions for the well-known Stochastic Biswas–Milovic equation (SBME) with the parabolic law nonlinearity. The Sub-ODE approach is used for this purpose. A variety of new optical soliton solutions are generated, including hyperbolic function, periodic solitons, rational solitons, Jacobi elliptic function (JEF), Weierstrass Elliptic Function (WEF), positive solitons, bright solitons, kink type solitons and dark solitons. Localized solitons, such as bright (positive peaks) and dark (localized dips) solitons, are explained by hyperbolic functions. Localized algebraic waves are represented by rational solitons, even with periodic and doubly-periodic solitons are depicted by JEF and WEF, respectively. Kink solitons represent a variety of nonlinear phenomena by connecting different asymptotic states. Bose–Einstein condensation, fiber optic sensors, plasma physics, optical communication and other fields belong to applications for these solitons. Additionally, we will plot graphs to visually represent the system’s response. To plot some graphs for SBME, we will first import the necessary libraries into Jupyter as a machine learning tool, including matplotlib, scipy.integrate and numpy.

After formulating the notion of integrability for axially symmetric harmonic maps from ${ℝ}^3$ into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding $N$-solitonic harmonic maps into a non-compact Grassmann manifold $SU(p,q)/S(U(p)\times U(q))$ is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases $(p=1,q=1)$ and $(p=2,q=1)$ and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

Given a 2-vector field on a manifold, first we briefly discuss the complete integrability of the distribution which is the image of the 2-vector field. Then we show that a new Lie algebroid is defined on such a maniold which is coincident with the cotangent Lie algebroid when the 2-vector field is Poisson. The result is extended to the case of Lie algebroids.

By using moving frame theory, we obtain some necessary conditions involving curvatures for integrability of an almost Hermitian structure. As consequences, they are applied to S⁶.

We classify endomorphisms of the plane that preserve a pencil of curves.

We investigate second-order quasilinear equations of the form f_iju_xixj = 0, where u is a function of n independent variables x₁, …, x_n, and the coefficients f_ij depend on the first-order derivatives p¹ = u_x1, …, pⁿ = u_xn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective transformations on the space Pⁿ with coordinates p¹, …, pⁿ. The coefficient matrix f_ij defines on Pⁿ a conformal structure f_ij(p)dpⁱdp^j. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure f_ij(p) dpⁱdp^j is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical Frobenius theorem, which says that an involutive $C^1$ distribution is uniquely integrable.

In April 2003, Chern began a study of almost-complex structures on the six-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group ${\mathrm{\text{G}}}_2$. While he did not solve the (currently still open) problem of determining whether there exists an integrable almost-complex structure on $S^6$, he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on $S^6$.

The direct reduction method and computerized symbolic computation are extended to the Brusselator reaction–diffusion model, which describes a biochemical system and consists of two coupled nonlinear partial differential equations. New similarity reductions are obtained, which could be of biochemical interest. The resulting nonlinear ordinary differential equation in one dynamical variable cannot be transformed into any of the six standard forms which have solutions in terms of the Painlevé transcendents. Thus, the Brusselator model is not integrable according to the Painlevé conjecture.

This paper studies the computational power of quantum computers to explore as to whether they can recognize properties which are in nondeterministic polynomial-time class (NP) and beyond. To study the computational power, we use the Feynman's path integral (FPI) formulation of quantum mechanics. From a computational point of view the Feynman's path integral computes a quantum dynamical analogue of the k-ary relation computed by an Alternating Turing machine (ATM) using AND-OR Parallelism. Hence, if we can find a suitable mapping function between an instance of a mathematical problem and the corresponding interference problem, using suitable potential functions for which FPI can be integrated exactly, the computational power of a quantum computer can be bounded to that of an alternating Turing machine that can solve problems in NP (e.g, factorization problem) and in polynomial space. Unfortunately, FPI is exactly integrable only for a few problems (e.g., the harmonic oscillator) involving quadratic potentials; otherwise, they may be only approximately computable or noncomputable. This means we cannot in general solve all quantum dynamical problems exactly except for those special cases of quadratic potentials, e.g., harmonic oscillator. Since there is a one to one correspondence between the quantum mechanical problems that can be analytically solved and the path integrals that can be exactly evaluated, we can say that the noncomputability of FPI implies quantum unsolvability. This is the analogue of classical unsolvability.

The Feynman's path graph can be considered as a semantic parse graph for the quantum mechanical sentence. It provides a semantic valuation function of the terminal sentence based on probability amplitudes to disambiguate a given quantum description and obtain an interpretation in a linear time. In Feynman's path integral, the kernels are partially ordered over time (different alternate paths acting concurrently at the same time) and multiplied. The semantic valuation is computable only if the FPI is computable. Thus both the expressive power and complexity aspects quantum computing are mirrored by the exact and efficient integrability of FPI.

The important requirements are stated for the success of quantum computation. These requirements involve coherent preserving Hamiltonians as well as exact integrability of the corresponding Feynman path integrals. Also we explain the role of metric entropy in dynamical evolutionary system and outline some of the open problems in the design of quantum computational systems. Finally, we observe that unless we understand quantum nondemolition measurements, quantum integrability, quantum chaos and the direction of time arrow, the quantum control and computational paradigms will remain elusive and the design of systems based on quantum dynamical evolution may not be feasible.

We show that a nonlinear dynamical system in Poincaré–Dulac normal form (in ℝⁿ) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the "parent" linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincaré condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind the classical integration method of Horn, Lyapounov and Dulac.

We study the asymptotic behavior of a singular potential that arises under several frequently occurring analytic behaviors of the eigenfunctions (of the Schrödinger eigenvalue problem) within the C²-class of functions. We find that the asymptotic behavior of the singular potential crucially depends on the analytic property of the eigenfunction near the singular point.

We consider NSR superstring in AdS₃⊗S³⊗R₄ background. The action is expressed in terms of the variables on the group manifolds SL(2, R) and SU(2) as (1+1)-dimensional σ-models with Wess–Zumino terms, associated with AdS₃ and S³ respectively. R₄ is the flat Euclidean space. We show the existence of classical nonlocal conserved currents in the superspace formulation for the nonlinear σ-model with WZ term in the context of NSR string. We propose Ward identities utilizing existence of the nonlocal conserved currents.

We consider non-Hermitian but PT-symmetric extensions of Calogero models, which have been proposed by Basu-Mallick and Kundu for two types of Lie algebras. We address the question of whether these extensions are meaningful for all remaining Lie algebras (Coxeter groups) and if in addition one may extend the models beyond the rational case to trigonometric, hyperbolic and elliptic models. We find that all these new models remain integrable, albeit for the non-rational potentials one requires additional terms in the extension in order to compensate for the breaking of integrability.

The exchange operator formalism in polar coordinates, previously considered for the Calogero–Marchioro–Wolfes problem, is generalized to a recently introduced, infinite family of exactly solvable and integrable Hamiltonians H_k, k = 1, 2, 3,…, on a plane. The elements of the dihedral group D_2k are realized as operators on this plane and used to define some differential-difference operators D_r and D_φ. The latter serve to construct D_2k-extended and invariant Hamiltonians , from which the starting Hamiltonians H_k can be retrieved by projection in the D_2k identity representation space.

The quantum H₄ integrable system is a 4D system with rational potential related to the non-crystallographic root system H₄ with 600-cell symmetry. It is shown that the gauge-rotated H₄ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H₄, is in algebraic form: it has polynomial coefficients in front of the derivatives. Any eigenfunction is a polynomial multiplied by ground-state function (factorization property). Spectra correspond to one of the anisotropic harmonic oscillators. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector α = (1, 5, 8, 12).

Lax representation is elaborated for the equations of motion of massless superparticle on the AdS₄×ℂℙ³ superbackground that proves their classical integrability.

In this paper, we define non-relativistic string in the background with nontrivial metric. Then we analyze its properties when we focus on integrability of this string in $Ad{S}_5\times S^5$ background.