Canonical Quantization of Systems with Time-Dependent Constraints

The Hamilton-Jacobi method of constrained systems is discussed. The equations of motion for a singular system with time dependent constraints are obtained as total differential equations in many variables. The integrability conditions for the relativistic particle in a plane wave lead us to obtain the canonical phase space coordinates without using any gauge fixing condition. As a result of the quantization, we get the Klein-Gordon theory for a particle in a plane wave. The path integral quantization for this system is obtained using the canonical path integral formulation method.     

Integrability of two coupled Kadomtsev–Petviashvili equations

The integrability of two coupled KP equations is studied. The simplified Hereman form of Hirota’s bilinear method is used to examine the integrability of each coupled equation. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation.     

Integro-differential non-linear equations and continual Lie algebras

The integrability problem of integro-differential equations with, generally speaking, singular kernels is discussed after an example of new continual analogs of the two-dimensional Toda lattices. These equations are associated with new infinite-dimensional Lie algebras via zero curvature type representation. The structural constants of these algebras are distributions. A formal solution of the Goursát problem is obtained. For the case with the kernel of the integral operator being _±-distribution an explicit expression in quadratures for the solutions is given.     

Integrable and nonintegrable classical spin clusters

The nonlinear dynamics is investigated for a system ofN classical spins. This represents a Hamiltonian system withN degrees of freedom. According to the Liouville theorem, the complete integrability of such a system requires the existence ofN independent integrals of the motion which are mutually in involution. As a basis for the investigation of regular and chaotic spin motions, we have examined in detail the problem of integrability of a two-spin system. It represents the simplest autonomous spin system for which the integrability problem is nontrivial. We have shown that a pair of spins coupled by an anisotropic exchange interaction represents a completely integrable system for any values of the coupling constants. The second integral of the motion (in addition to the Hamiltonian), which ensures the complete integrability, turns out to be quadratic in the spin variables. If, in addition to the exchange anisotropy also singlesite anisotropy terms are included in the two-spin Hamiltonian, a second integral of the motion quadratic in the spin variables exists and thus guarantees integrability, only if the model constants satisfy a certain condition. Our numerical calculations strongly suggest that the violation of this condition implies not only the nonexistence of a quadratic integral, but the nonexistence of a second independent integral of motion in general. Finally, as an example of a completely integrableN-spin system we present the Kittel-Shore model of uniformly interacting spins, for which we have constructed theN independent integrals in involution as well as the action-angle variables explicitly.     

Integrability of coupled KdV equations

The integrability of coupled KdV equations is examined. The simplified form of Hirota’s bilinear method is used to achieve this goal. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation. The resonance phenomenon of each model will be examined.     

INTEGRABILITY OF LIE SYSTEMS THROUGH RICCATI EQUATIONS

Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyze a geometric method to construct integrability conditions for Riccati equations following these approaches. Our procedure provides us with a unified geometrical viewpoint that allows us to analyze some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalized to describe integrability conditions for any Lie system. Finally we show the usefulness of our treatment in order to study the problem of the linearizability of Riccati equations.     

A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables

We propose a geometrical approach to the problem of integrability of Hamiltonian systems of low dimensions using the Hamilton–Jacobi method of separation of variables, based on the method of moving frames. As an illustration we present a complete classification of all separable Hamiltonian systems defined in two-dimensional Riemannian manifolds of arbitrary curvature and a criterion for separability. Connections to bi-Hamiltonian theory are also found.     

On integrable rational potentials of the Dirac equation

The one-dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.     

A new high-order spectral problem of the mKdV and its associated integrable decomposition

A new approach to formulizing a new high-order matrix spectral problem from a normal 2× 2 matrix modified Korteweg--de Vries (mKdV) spectral problem is presented. It is found that the isospectral evolution equation hierarchy of this new higher-order matrix spectral problem turns out to be the well-known mKdV equation hierarchy. By using the binary nonlinearization method, a new integrable decomposition of the mKdV equation is obtained in the sense of Liouville. The proof of the integrability shows that r-matrix structure is very interesting.     

Trajectory tracking control of parallel robots in the presence of joint drive flexibility

Trajectory tracking control of parallel manipulators is aimed in the presence of flexibility at the joint drives. Joint structural damping is also considered in the dynamic model. The system is first converted into an open-tree structure by disconnecting a sufficient number of unactuated joints. The closed loops are then expressed by constraint equations. It is shown that, in a parallel robot with flexible joint drives, the acceleration level inverse dynamics equations are singular because the control torques do not have an instantaneous effect on the end-effector accelerations due to the elastic media. Eliminating the Lagrange multipliers and the intermediate variables, a fourth-order input–output relation is obtained between the actuator torques and the end-effector position variables. The proposed control law decouples and linearizes the system and achieves asymptotic stability by feedback of positions and velocities of the actuated joints and rotors. As a case study, a three degree of freedom, two legged planar parallel manipulator is simulated to illustrate the performance of the method. The end-effector desired trajectory is chosen such that the kinematic and drive singular positions are avoided.     

Separation of variables in the classical integrableSL(3) magnetic chain

There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. A third problem, however, should be added to the list: separation of variables. Though much simpler thant the two others, it has important relations to quantum integrability. Separation of variables is constructed for theSL(3) magnetic chain—an example of an integrable model associated to a nonhyperelliptic algebraic curve.Supported by the Academy of Finland     

Partial field decomposition of multi-source cyclostationary sound field

The cyclostationary sound field is a kind of special non-stationary field which has obvious modulation phenomena. Conventional planar near-field acoustic holography (PNAH) technique cannot exactly reflect its modulation characteristics. If the cyclic spectral density (CSD) instead of the complex sound pressure is adopted as the reconstruction variable, the modulating wave and carrier wave components of the cyclostationary sound field can be extracted effectively. A new technique called cyclostationary PNAH utilizing the CSD is proposed in this paper. Based on this technique, the problem of partial field decomposition by singular value decomposition (SVD) of multiple incoherent cyclostationary sound sources is researched. The results of numerical simulation and experiments show that the CPNAH technique and SVD decomposition method are effective.     

A new solving method of non-uniform dielectric planar waveguide

A new solving method of non-unifrom dielectric planar waveguide is presented in this paper. By applying this method to solving the problem of planar dielectric waveguide with arbitrary refractive index variation, the advantage of this new method is more evident. By dividing the waveguide into proper numbers and thickness of layers, we can transform the problem of non-uniform dielectric planar waveguide into the multilayer dielectric waveguide to calculate its propagation constantβ.     

Decomposable differential operators in a cosmological context

The integrability conditions for a certain second order ordinary differential equation in two variables are studied via the concept of decomposability of the associated differential operator. The results are applied to regain in a unified manner the known exact solutions for locally rotationally symmetric, spatially homogeneous cosmological models. In addition, new solutions are obtained.     

奇异值非线性修正的低对比度红外图像实时增强

由于场景中目标与背景的温差相对较小,红外图像会存在对比度低、视觉效果差的问题,针对这一问题,提出一种基于奇异值非线性修正的红外图像对比度实时增强方法。该方法首先对红外图像进行奇异值分解得到其原始奇异值,然后采用一个对数型非线性变换对图像奇异值进行优化,最后根据修正的奇异值重构出对比度增强的红外图像。利用对数型非线性变换修正图像奇异值不仅能够有效拉伸奇异值的动态范围,同时可优化奇异值的变化梯度,使图像的能量信息得到更充分地表达,改善红外图像不良的视觉效果。实验结果表明,该方法较几种对比方法在视觉效果和客观评价方面均具有更优的增强性能;同时体现出良好的实时性,为实现红外图像的实时增强提供了新途径。     

A Lie algebra structure on variation vector fields along curves in 2-dimensional space forms

A Lie algebra structure on variation vector fields along an immersed curve in a 2-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation are finally discussed from this point of view.     

Soliton multidimensional equations and integrable evolutions preserving Laplace's equation

The KP equation, which is an integrable nonlinear evolution equation in 2+1, i.e., two spatial and one temporal dimensions, is a physically significant generalization of the KdV equation. The question of constructing an integrable generalization of the KP equation in 3+1, has been one of the central open problems in the field of integrability. By complexifying the independent variables of the KP equation, I obtain an integrable nonlinear evolution equation in 4+2. The requirement that real initial conditions remain real under this evolution, implies that the dependent variable satisfies a nonlinear evolution equation in 3+1 coupled with Laplace's equation. A reduction of this system of equations to a single equation in 2+1 contains as particular cases certain singular integro-differential equations which appear in the theory of water waves.     

New integrable cases of a Cremona transformation: a finite-order orbits analysis

We analyse the properties of a particular birational mapping of two variables (Cremona transformation) depending on two free parameters ( and ), associated with the action of a discrete group of non-linear (birational) transformations on the entries of a q × q matrix. This mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. It has been seen to yield weak chaos and integrability. We have found new integrable cases of this Cremona transformation, corresponding to the values of = 0 when , besides the already known values = 0 and = −1, and also arbitrary when = 0. For these cases, one has a foliation of the parameter space in elliptic curves. We give the equations of these elliptic curves. Based on this very example we show how one can find these integrability cases of the Cremona transformation and actually integrate it using a method based on the systematic study of the finite-order conditions of the Cremona transformation. The method is shown to be efficient and straightforward. The various integrability cases are revisited using many different representations of this very mapping (birational transformations, recursion in one variable, …).     

Integrability of dynamical systems and the singular-point analysis

Various aspects of the integrability of dynamical systems are discussed with the help of the singular point analysis. In particular the connection with the Painlevé property is described. Several examples will serve as illustrations.     

Integrability of Calogero–Coulomb problems

In this short review we describe the integrability properties of the Calogero-type perturbations of one- and two-center Coulomb problems and of the Stark–Coulomb problem. We present the explicit expressions of their constants of motion and show that these systems admit partial separation of variables.