General Integrable Systems Research Articles

Finite-element simulation for X-ray volume diffractive optics based on the wave optical theory.

We developed a novel numerical simulation method for volume diffractive optics based on the Takagi-Taupin (TT) dynamical theory of diffraction. A general integral system of equations with a powerful and convenient distortion function was developed for finite-element analysis (FEA). The proposed framework is promising with regard to flexibility, robustness, and stability and has potential for solving dynamical X-ray diffraction problems related to diffractive optical elements of arbitrary shape and deformation. This FEA method was used for evaluating laterally graded multilayer (LGML) mirrors, and a general coordinate system was introduced to make the geometric optimization simple and effective. Moreover, the easily implemented boundary conditions inherent in FEA, combined with the analysis of the energy resolution derived from the TT theory, can make the simulation of volume diffractive optics, including LGML mirrors, more accurate. Thus, a comprehensive and highly efficient computation of LGML mirror diffraction problems was performed. The evaluation of the effects of the figure errors can provide practical guidance for the fabrication of X-ray optical elements.

Classification of solutions for an integral system with negative exponents on half space

ABSTRACT In this paper, we classify each pair of positive solutions of the following general integral system on half space The main technique we used is the method of moving sphere.

Classification of positive solutions for an integral system on the half space

In this paper, we consider the following general integral system on the half space R+nu(y)=∫R+nxnβf(v(x))|x−y|n−αdx,y∈∂R+n,v(x)=∫∂R+nxnβg(u(y))|x−y|n−αdy,x∈R+n.Under natural conditions on f and g, we obtain a classification of positive solutions by using a slight variant of the method of moving spheres. Some classifications results on the half space were established for β=1 by Dou, Guo and Zhu (2017) and for general β≥0 by Gluck (2018). These results require a whole integrability hypothesis on u and v, which is removed in this paper. Our results thus improve on the classification theorems in Dou, Guo and Zhu (2017), Gluck (2018).

Liouville Type Theorems for General Integral System with Negative Exponents

In this paper, we establish a Liouville type theorem for the following integral system with negative exponents \[ \begin{cases} u(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} f(u,v)(y) \, dy, & x \in \mathbb{R}^n, v(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} g(u,v)(y) \, dy, & x \in \mathbb{R}^n, \end{cases} \] where $n \geq 1$, $\nu \gt 0$, and $f$, $g$ are continuous functions defined on $\mathbb{R}_{+} \times \mathbb{R}_{+}$. Under nature structure conditions on $f$ and $g$, we classify each pair of positive solutions for above integral system by using the method of moving sphere in integral forms. Moreover, some other Liouville theorems are established for similar integral systems.

Darboux transformations for super-Schrödinger equation, super-Dirac equation and their exact solutions

The Darboux transformation (DT) for the super-integrable hierarchy has an essential difference from the general system. As we know, the super-integrable soliton equation hierarchies with four potentials are discussed. Starting from the spectral problems of super-AKNS hierarchy and super-Dirac hierarchy, a DT method for two super-integrable hierarchies is constructed, which is more complex than the general integrable system. Soliton solutions of super-Schrodinger equation and super-Dirac equation are presented by using DT, which contain some bright, dark and breather wave soliton solutions. Then, the properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients.

A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

We introduce a new rigorous method, based on Borel summability and asymptotic constants of motion generalizing \cite{invent} and \cite{ode1}, to analyze singular behavior of nonlinear ODEs in a neighborhood of infinity and provide global information about their solutions. In equations with the Painlev\'e-Kowalevski (P-K) property (stating that movable singularities are not branched) it allows for solving connection problems. The analysis in carried in detail for P$_1$, $y"=6y^2+z$, for which we find the Stokes multipliers in closed form and global asymptotics for solutions having power-like behavior in some direction in $\CC$, in particular for the tritronqu\'ees. Calculating the Stokes multipliers solely relies on the P-K property and does not use linearization techniques such as Riemann-Hilbert or isomonodromic reformulations. We discuss how the approach would work to calculate connection constants for a larger class of P-K integrable equations. We develop methods for finding asymptotic expansions in sectors where solutions have infinitely many singularities. These techniques do not rely on integrability and apply to more general second order ODEs which, after normalization, are asymptotically close to autonomous Hamiltonian systems.

Nonlinear integrable model of Frenkel-like excitations on a ribbon of triangular lattice

Following the considerable progress in nanoribbon technology, we propose to model the nonlinear Frenkel-like excitations on a triangular-lattice ribbon by the integrable nonlinear ladder system with the background-controlled intersite resonant coupling. The system of interest arises as a proper reduction of first general semidiscrete integrable system from an infinite hierarchy. The most significant local conservation laws related to the first general integrable system are found explicitly in the framework of generalized recursive approach. The obtained general local densities are equally applicable to any general semidiscrete integrable system from the respective infinite hierarchy. Using the recovered second densities, the Hamiltonian formulation of integrable nonlinear ladder system with background-controlled intersite resonant coupling is presented. In doing so, the relevant Poisson structure turns out to be essentially nontrivial. The Darboux transformation scheme as applied to the first general semidiscrete system is developed and the key role of Bäcklund transformation in justification of its self-consistency is pointed out. The spectral properties of Darboux matrix allow to restore the whole Darboux matrix thus ensuring generation one more soliton as compared with a priori known seed solution of integrable nonlinear system. The power of Darboux-dressing method is explicitly demonstrated in generating the multicomponent one-soliton solution to the integrable nonlinear ladder system with background-controlled intersite resonant coupling.

Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms

In this paper, we introduce polygonal fuzzy numbers to overcome the operational complexity of ordinary fuzzy numbers, and obtain two important inequalities by taking advantage of their fine properties. By presenting an actual example, we demonstrate that the approximation capability of polygonal fuzzy numbers is efficient. Furthermore, the concepts of K-quasi-additive integrals and K-integral norms are introduced. Whenever the polygonal fuzzy numbers space satisfies separability, the density problems for several functions spaces can be studied, by means of fuzzy-valued simple functions and fuzzy-valued Bernstein polynomials. We establish that the class of the integrally-bounded fuzzy-valued functions spans a complete and separable metric space in the K-integral norms. Finally, in the sense of K-integral norms, the universal approximation of fourlayer regular polygonal fuzzy neural networks for fuzzy-valued simple functions is discussed. Furthermore, we show that this type of networks also possesses universal approximation for the class of integrally-bounded fuzzyvalued functions. This result indicates that the approximation capability which regular polygonal fuzzy neural networks for continuous fuzzy systems can be extended as for general integrable systems.

Evolution of the unidirectional electromagnetic pulses in an anisotropic two-level medium

We find a general integrable system of reduced Maxwell-Bloch equations describing the interaction of a two-component electromagnetic field with the dipole transition of a two-level medium. The anisotropy of the anisotropic dipole momentum of the transition as a permanent dipole momentum is taken into account. A solution of the model is found for a particular case by using the inverse scattering transform. The method is based on a solution of the Riemann-Hilbert problem taking into account the symmetry properties of the corresponding fundamental solutions.

Decoherence in regular systems

We consider unitary evolution of finite bipartite quantum systems and study time dependence of purity for initial cat states -- coherent superpositions of Gaussian wave-packets. We derive explicit formula for purity in systems with nonzero time averaged coupling, a typical situation for systems where an uncoupled part of the Hamiltonian is Liouville integrable. Previous analytical studies have been limited to harmonic oscillator systems but with our theory we are able to derive analytical results for general integrable systems. Two regimes are clearly identified, at short times purity decays due to decoherence whereas at longer times it decays because of relaxation.

Integrable models for the dynamics of a longitudinal-transverse acoustic wave in a crystal with paramagnetic impurities

We theoretically study the evolution of longitudinal-transverse acoustic pulses propagating parallel to an external magnetic field in a system of resonant paramagnetic impurities with an effective spin S=1/2. For equal group velocities of the longitudinal and transverse waves, the pulse dynamics is shown to be described by evolution equations. In limiting cases, these equations reduce to equations integrable in terms of the inverse scattering transform method (ISTM). For the most general integrable system of equations that describes the dynamics of acoustic pulses outside the scope of the slow-envelope approximation, we derive the corresponding ISTM equations. These equations are used to find a soliton solution and a self-similar solution. The latter describes the leading edge of the packet of acoustic pulses generated when the initial unstable state of a spin system decays. Analysis of our solutions and models indicates that the presence of a longitudinal acoustic wave leads not only to a change in the amplitude and phase of the transverse wave but also to a qualitatively new dynamics of sound in such a medium.

Hamiltonian systems with orbits covering densely submanifolds of small codimension

The existence of a transition chain in a Hamiltonian system leads to the existence of orbits shadowing it, if some lambda lemma can be applied. This fact has been used to prove the existence of diffusion in perturbations of integrable a priori stable systems. We prove that, under suitable conditions, there are orbits which cover densely codimension two submanifolds of the (2 n−1)-dimensional energy level. We also construct examples, near general integrable systems, where the computations to verify the sufficient conditions for the appearance of such phenomenon can be performed.

The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations

We investigate the sequence of Boussinesq equations by the method of singular manifolds. For the Boussinesq equation, which is known to possess the Painlevé property, a Bäcklund transformation is defined. This Bäcklund transformation, which is formulated in terms of the Schwarzian derivative, obtains the system of modified Boussinesq equations and the resulting Miura-type transformation. The modified Boussinesq equations are found to be invariant under a discrete group of symmetries, acting on the dependent variables. By linearizing the Miura transformation (and modified equations) the Lax pair is readily obtained. Furthermore, by a result of Fokas and Anderson, the recursion operators defining the sequence of (higher-order) Boussinesq equations may be constructed from the Miura transformation. This allows the (recursive) definition of Bäcklund transformations for this sequence of equations. The recursion operator is found to preserve the discrete symmetries of the modified Boussinesq equations. This leads to the conclusion that the sequences of Boussinesq and modified Boussinesq equations identically possess the Painlevé property (are meromorphic). We also find that, by a simple reduction, the sequences of Caudrey–Dodd–Gibbon and Kuperschmidt equations are contained within the Boussinesq sequence. Rational solutions are iteratively constructed for the Boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems.