Recursion Operator Research Articles

A (3+1)-dimensional integrable Calogero-Bogoyavlenskii-Schiff equation and its inverse operator: lump solutions and multiple soliton solutions

"In this work, we built a (3+1)-dimensional integrable equation. We started by reformulating the main equation of our model by combining the recursion operator of the Calogero-Bogoyavlenskii-Schiff equation with its inverse recursion op- erator. We confirm the complete integrability of our new developed equation by demon- strating that it satisfies the Painlev´e property. We get a variety of lump solutions that are obtained under specific constraints. Furthermore, we used the simplified Hirota’s direct approach to find multiple soliton solutions to the new evolution equation. In ad- dition, other techniques are used to solve the new evolution equation, in order to get some physically relevant solutions."

Adaptive motion planning for CFETR multipurpose overload robot based on iterative tractrix

In this paper, an adaptive motion planning method is proposed for the control requirements of the China Fusion Engineering Test Reactor multipurpose overload robot (CMOR). Firstly, a kinematic model of CMOR is established by Denavit-Hartenberg method, and the working process of entering and leaving the vacuum chamber to complete the maintenance operation is analyzed. An iterative tractrix-based motion planning method is proposed by expanding the basic principle of tractrix. Assuming that the CMOR any link has two mutually perpendicular degrees of freedom to achieve spatial motion capability, a more natural and smooth movement of the CMOR is achieved by recursive operations on the tractrix traction. To accelerate the calculation process and avoid each joint rotation angle of CMOR exceeding the set value, a bidirectional iterative tractrix method and a joint limit rotation angle optimization method based on the bidirectional iterative tractrix method are proposed to improve the convergence speed and motion safety. Finally, a CMOR visualization simulation system is built, and the results show that the CMOR adaptive motion planning method can effectively track the target trajectory and avoid joint rotation angle overrun.

Generalized conditional symmetries and pre-Hamiltonian operators

In this paper, we consider the connection between generalized conditional symmetries (GCSs) and pre-Hamiltonian operators. The set of GCSs of an evolutionary partial differential equations system is divided into a union of many linear subspaces by different characteristic operators, and we consider the mappings between two of them, which generalize the recursion operators of symmetries and the pre-Hamiltonian operators. Finally, we give a systematic method to construct infinitely many GCSs for integrable systems, including the Gelfand–Dickey hierarchy and the AKNS-D hierarchy. All time flows in one integrable hierarchy, admitting infinitely many common GCSs.

Applications of the R-Matrix Method in Integrable Systems

Based on work related to the R-matrix theory, we first abstract the Lax pairs proposed by Blaszak and Sergyeyev into a unified form. Then, a generalized zero-curvature equation expressed by the Poisson bracket is exhibited. As an application of this theory, a generalized (2+1)-dimensional integrable system is obtained, from which a resulting generalized Davey–Stewartson (DS) equation and a generalized Pavlov equation (gPe) are further obtained. Via the use of a nonisospectral zero-curvature-type equation, some (3+1) -dimensional integrable systems are produced. Next, we investigate the recursion operator of the gPe using an approach under the framework of the R-matrix theory. Furthermore, a type of solution for the resulting linearized equation of the gPe is produced by using its conserved densities. In addition, by applying a nonisospectral Lax pair, a (3+1)-dimensional integrable system is generated and reduced to a Boussinesq-type equation in which the recursion operators and the linearization are produced by using a Lie symmetry analysis; the resulting invertible mappings are presented as well. Finally, a Bäcklund transformation of the Boussinesq-type equation is constructed, which can be used to generate some exact solutions.

The tail-recursive fragment of timed recursive CTL

Timed Recursive CTL (TRCTL) was recently proposed as a merger of two extensions of the well-known branching-time logic CTL: Timed CTL on one hand is interpreted over real-time systems like timed automata, and Recursive CTL (RecCTL) on the other hand obtains high expressiveness through the introduction of a recursion operator. Model checking for the resulting logic is known to be 2-EXPTIME-complete.The aim of this paper is to investigate the possibility to obtain a fragment of lower complexity without losing too much expressive power. It is obtained by a syntactic property called “tail-recursiveness” that restricts the way that recursive formulas can be built. This restriction is known to decrease the complexity of model checking by half an exponential in the untimed setting. We show that this also works in the real-time world: model checking for the tail-recursive fragment of TRCTL is EXPSPACE-complete already in its data complexity, i.e. for a fixed formula. The upper bound is obtained via a model-checking procedure on region graphs combining standard untiming constructions with a tailored algorithm. The lower bound is established by a reduction from a suitable tiling problem.

The Calogero–Bogoyavlenskii–Schiff breaking soliton equation: Recursion operators and higher symmetries

We find two one-parametric families of recursion operators and use them to construct higher symmetries for the Calogero–Bogoyavlenskii–Schiff breaking soliton equation. Then we prove that the recursion operators from the first family pair-wise commute with respect to the Nijenhuis bracket (are compatible).

A new look into the concept of separable system of Classical Mechanics

The theory of integrable systems of Classical Mechanics has been deeply affected, in the second half of the last century, by the discovery of the integrability of the Korteweg–de Vries equation. New ideas appeared at that time (such as, for instance, the concepts of Lax and bihamiltonian representations, and the notions of Poisson manifold, Poisson pencil, and recursion operator) which have changed our way of seeing the geometry of integrable systems. They also changed our way of seeing the geometry of separable systems. The paper is an attempt to explain why and in what form the idea of recursion operator leads to reshape the definition of separable system, and to discover a new set of separability conditions called “Kowalevski separability conditions”.

AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials

The aim of this paper is to derive a kind of integrable hierarchies with four potentials, which possess bi-Hamiltonian structures, from reduced AKNS matrix spectral problems. The associated recursion operators are worked out explicitly. The Lax pair formulation and the trace identity are basic tools in the analysis. Two nonlinear examples in the resulting integrable hierarchies are integrable nonlinear Schrödinger type equations and integrable modified Korteweg–de Vries type equations.

Optical quantum longitudinal conformable normalization energy of timelike spherical magnetic fibers

In this work, we calculate new concept for their Fermi–Walker conformable derivatives with spherical timelike magnetic fiber in spherical de-Sitter space [Formula: see text]. The description is advanced for timelike magnetic trajectory. First, we find conformable fractional derivatives of spherical magnetic Lorentz fields. Moreover, we compute the Fermi–Walker conformable derivatives of the normalization and recursion operators for these spherical vector fields. Furthermore, we give some characterizations of Lorentz fields of electromagnetic fields of associated with spherical magnetic conformable particle. Finally, we obtain the energy of the conformable derivative of the normalization function of the Lorentz forces and we have energy relations of some vector fields associated with spherical frame in the de-Sitter space [Formula: see text].

Cyclic Convolutional Neural Network Model Based on Artificial Intelligence

Abstract This paper mainly discusses the internal correlation between meshless discrete data and learning samples, meshless dynamic analysis recursive operation and information transmission mode in cyclic convolutional neural networks. This paper establishes a cyclic convolutional neural network based on the meshless method. This paper demonstrates an agent model of cyclic convolutional neural network based on dynamic characteristics. This method combines the advantages of the flexible configuration of meshless nodes in the discrete model. The universality and adaptability of cyclic convolutional neural networks are improved. In addition, because of the unique historical memory characteristics of the periodic module, it can analyze continuous data efficiently. The solution of dynamic analysis is accelerated without affecting the calculation accuracy. Based on a group of examples, the accuracy and effectiveness of this method are studied experimentally.

Symmetry analysis and hidden variational structure of Westervelt’s equation in nonlinear acoustics

Westervelt’s equation is a nonlinear wave equation that is widely used to model the propagation of sound waves in a compressible medium, with one important application being ultra-sound in human tissue. Two fundamental aspects of the general dissipative version of Westervelt’s equation – symmetries and conservation laws – are studied in the present work by modern methods. Numerous results are obtained: new conserved integrals; potential systems yielding hidden symmetries and nonlocal conservation laws; mapping of Westervelt’s equation in the undamped case into a linear wave equation; hidden variational structures, including a Lagrangian and a Hamiltonian; a recursion operator and a Noether operator; contact symmetries; higher-order symmetries and higher-order conservation laws.

New conserved integrals and invariants of radial compressible flow in n > 1 dimensions

Conserved integrals and invariants (advected scalars) are studied for the equations of radial compressible fluid/gas flow in n > 1 dimensions. Apart from entropy, which is a well-known invariant, three additional invariants are found from an explicit determination of invariants up to first order. One holds for a general equation of state, and the two others hold only for entropic equations of state. A recursion operator on invariants is presented, which produces two hierarchies of higher-order invariants. Each invariant yields a corresponding integral invariant, describing an advected conserved integral on transported radial domains. In addition, a direct determination of kinematic conserved densities uncovers two ‘hidden’ non-advected conserved integrals: one describes enthalpy-flux, holding for barotropic equations of state; the other describes entropy-weighted energy, holding for entropic equations of state. A further explicit determination of a class of first-order conserved densities shows that the corresponding non-kinematic conserved integrals on transported radial domains are equivalent to integral invariants, modulo trivial densities.

Frequent Itemset Mining Algorithm Based on Linear Table

Aiming at the speed of frequent itemset mining, a new frequent itemset mining algorithm based on a linear table is proposed. The linear table can store more shared information and reduce the number of scans to the original dataset. Furthermore, operations such as pruning and grouping are also used to optimize the algorithm. For different datasets, the algorithm shows different mining speeds. (1) In sparse datasets, the algorithm achieves an average 45% improvement in mining speed over the bit combination algorithm, and a 2-3 times improvement for the classic FP-growth algorithm. (2) In dense datasets, the average improvement over the classic FP-growth algorithm is 50-70%. For the bit combination algorithm, there are dozens of times of improvement. In fact, the algorithm that integrates bit combinations with bitwise AND operation can effectively avoid recursive operations and it is beneficial to the parallelization. Further analysis shows that the linear table is easy to split to facilitate the data batch mining processing.

On the elliptical solutions of models connected to the short pulse equation

In the present paper, we consider a special hierarchy of equations comprising the short pulse equation, the sine-Gordon integrable hierarchy and the elastic beam equation. These equations are highly non-linear and rely on transformations to arrive at solutions. Previously, recursion operators and hodograph mappings were successful in reducing these equations. However, we show that via the conservation laws or the one-parameter Lie group, the special hierarchy may be integrated and will admit the exact solutions that feature elliptical functions.

Sustainable Analysis of Insulator Fault Detection Based on Fine-Grained Visual Optimization

Insulators of the kind used for overhead transmission lines institute important kinds of insulation control, namely, electrical insulation and mechanical fixing. Because of their large exposure to the environment, they are affected by factors such as climate, temperature, durability, the easy occurrence of explosions, damage, the threat of going missing, and other faults. These seriously influence the safety of the power transmission, so insulation monitoring must be conducted. With the development of unmanned technology, the staff used unmanned aircraft to take aerial photos of the detected insulators, and the insulator images were obtained by naked eye observation. Although this method looks very reliable, in practice, due to the large quantity of insulator-collected seismic data, and the complex background, workers are usually relying on their experience to make judgements, so it is easy for mistakes to appear. In recent years, with the rapid development of computer technology, more and more attention has been paid to fault detection and identification in insulators by computer-aided workers. In order to improve the detection accuracy of self-exploding insulators, especially in bad weather environments, and to overcome the influence of fog on target detection, a regression attention convolutional neural network is used for optimization. Through the recursive operation of multi-scale attention, multi-scale feature information is connected in series, the regional focus is recursively generated from coarse to fine, and the target region is detected to achieve optimal results. The experimental results show that the proposed method can effectively improve the fault diagnosis ability of insulators. Compared with the accuracy of other basic models, such as FCAN and MG-CNN, the accuracy of RA-CNN in multi-layer cascade optimization is higher than that in the previous two models, which is 74.9% and 75.6%, respectively. In addition, the results of the ablation experiments at different scales showed that the identification results of different two-level combinations were 78.2%, 81.4%, and 83.6%, and the accuracy of selecting three-level combinations was up to 85.3%, which was significantly higher than the other models.

Legendre Polynomials-Based Displacement Field Reconstruction of Plate-Type Structures With Quasi-Distributed FBG Sensors

Structural deformation estimation is of great interest for many engineering fields, such as aerospace structures, antenna panels, and so on. During operation, however, the direct measurement of global displacements is often difficult, due to the limits of installation space and electromagnetic interference. This article proposed a displacement field reconstruction method for deformed plate-type structures using discrete curvature measurements and a polynomial superposition approach. Specifically, the displacement field is expressed as a linear superposition of the Legendre polynomial basis functions, where the weight coefficients are calculated using the discrete curvatures captured by a quasi-distributed fiber Bragg grating (FBG) sensor array and the given boundary conditions. This method overcomes the drawback of error accumulation in other curvature-based methods due to the use of recursion operations. In addition, the curvature fitting and the determination of the number of polynomial basis functions were also presented to obtain the best reconstruction quality. Simulations and experiments have been conducted to validate the proposed method. The simulation results show that the displacement reconstruction precisions are more than 99% for two types of complex deformations. Experiments were carried out on an aluminum plate and a phased-array antenna panel, respectively. The results show that the reconstructed displacements of the reference points match well with those measured displacements, and the reconstruction precision exceeds 96%. This proposed curvature-based approach has high potential to global displacement monitoring of plate-type structures used in aerospace and antenna industrial fields.

On fully-nonlinear symmetry-integrable equations with rational functions in their highest derivative: Recursion operators

We report a class of symmetry-intergable third-order evolution equations in 1+1 dimensions under the condition that the equations admit a second-order recursion operator that contains an adjoint symmetry (integrating factor) of order six. The recursion operators are given explicitly.

Spectral stability of multi-solitons for generalized Hamiltonian system I: The Caudrey–Dodd–Gibbon–Sawada–Kotera equation

We consider a generalized Hamiltonian system called the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, which is a natural extension of the Korteweg–de Vries (KdV) equation. Our main results are two-fold. Firstly, spectral stability of solitons and multi-solitons are presented by employing some inverse scattering transform techniques. The main difficulty arises from the third order eigenvalue problem of the equation. The second result is a Hamilton–Krein index identity, which verifies a stability criterion established by Maddocks and Sachs (1993) in showing the Lyapunov stability of KdV N-solitons. However, due to generalized Hamiltonian system nature of the CDGSK equation, it does not possess the L2 conservation law. The linearized operators around the multi-solitons have no spectra gap any more. The main ingredient of the proof is new operator identities for higher order linearized Hamiltonians motivated by the Sylvester Law of Inertia. Such operator identities are shown by employing the recursion operators of the CDGSK equation.

Memristive Recurrent Neural Network Circuit for Fast Solving Equality-Constrained Quadratic Programming With Parallel Operation

Equality-constrained quadratic programming (QP) has been one of the most basic and typical problems in the Internet of Things domain. In big data scenarios, how to quickly and accurately solve the problem in hardware has not been realized. Therefore, in this article, a memristive recurrent neural circuit that can parallel solve the QP problem in real time is proposed. First, a new memristive synaptic array is designed that can simultaneously implement parallel reading and writing. On the basis of this structure, a new neural network circuit based on memristor is designed that can perform large-scale recursive operations by parallel methods. This circuit can solve the equality-constrained QP problem in different situations by using such real-time programmable memristor arrays processing in memory. The PSpice simulation results show that the problem can be solved with 99.8% precision. Based on practical verification, the neural circuit experiment on PCB is presented with 97.34% precision. Moreover, the circuit has good robustness under the interference of weight value. And, it has an advantage in processing time compared with FPGA.

A generalized nonlinear fifth-order KdV-type equation with multiple soliton solutions: Painlevé analysis and Hirota Bilinear technique

In present work, we formulate a new generalized nonlinear KdV-type equation of fifth-order using the recursion operator. This equation generalizes the Sawada-Kotera equation and the Lax equation that study the vibrations in mechanical engineering, nonlinear waves in shallow water, and other sciences. To determine the integrability, we use Painlevé analysis and construct solutions for multiple solitons by employing the Hirota bilinear technique to the established equation. It produces a bilinear form for the driven equation and utilizes the Lagrange interpolation to create a dependent variable transformation. We construct the solutions for multiple solitons and show the graphics for these built solutions. The mathematical software program Mathematica employs symbolic computation to obtain the multiple solitons and various dynamical behavior of the solutions for newly generated equation The Sawada-Kotera equation and Lax equation have various applications in mechanical engineering, plasma physics, nonlinear water waves, soliton theory, mathematical physics, and other nonlinear fields.