Recursion Operator Research Articles

On Sawada-Kotera and Kaup-Kuperschmidt integrable systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the M2 -extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.

Reduction of the Laplace sequence and sine-Gordon type equations

Abstract In this work, we continue the development of methods for constructing Lax pairs and recursion operators for nonlinear integrable hyperbolic equations of soliton type, previously proposed in the work of Habibullin et al. (2016 {\it J. Phys. A: Math. Theor.} {\bf 57} 015203). This approach is based on the use of the well-known theory of Laplace transforms. The article completes the proof that for any known integrable equation of sine-Gordon type, the sequence of Laplace transforms associated with its linearization admits a third-order finite-field reduction. It is shown that the found reductions are closely related to the Lax pair and recursion operators for both characteristic directions of the given hyperbolic equation. Previously unknown Lax pairs and recursion operators were constructed.

A combined Kaup–Newell type integrable hierarchy with four potentials and its bi-Hamiltonian formulation

This paper aims to introduce a Kaup–Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct its associated Liouville integrable Hamiltonian hierarchy, through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian formulation. An illustrative example of combined derivative nonlinear Schrödinger equations with two arbitrary constants is explicitly presented.

A combined integrable hierarchy with four potentials and its recursion operator and bi-Hamiltonian structure

A combined integrable hierarchy with four potentials and its recursion operator and bi-Hamiltonian structure

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

An extended AKNS eigenvalue problem and its affiliated integrable Hamiltonian hierarchies

This paper aims to study an extended 4 × 4 AKNS eigenvalue problem and construct its affiliated integrable hierarchies of bi-Hamiltonian models over the real field. The Lax pair framework serves as the fundamental tool, ensuring integrability through bi-Hamiltonian structures with hereditary recursion operators. We compute illustrative examples of lower-order equations to demonstrate the affiliated integrable hierarchies.

A combined generalized Kaup–Newell soliton hierarchy and its hereditary recursion operator and bi-Hamiltonian structure

A combined generalized Kaup–Newell soliton hierarchy and its hereditary recursion operator and bi-Hamiltonian structure

[formula omitted]-dressing approach and [formula omitted]-soliton solutions of the general reverse-space nonlocal nonlinear Schrödinger equation

Using the ∂̄-dressing method, we study the general reverse-space nonlocal nonlinear Schrödinger (nNLS) equation. Beginning with a 3 × 3 matrix ∂̄-problem, the associated spatial and time spectral problems are obtained through two linear constraint equations. Furthermore, the gauge equivalence between the Heisenberg chain equation and the general reverse-space nNLS equation is established. By employing a recursive operator, a hierarchy for the general reverse-space nNLS equation is proposed. Moreover, by selecting a suitable spectral transformation matrix, the N-soliton solutions of the general reverse-space nNLS equation are calculated, yielding the explicit one-soliton and two-soliton solutions.

A Hierarchical Neural Network for Point Cloud Segmentation and Geometric Primitive Fitting.

Automated generation of geometric models from point cloud data holds significant importance in the field of computer vision and has expansive applications, such as shape modeling and object recognition. However, prevalent methods exhibit accuracy issues. In this study, we introduce a novel hierarchical neural network that utilizes recursive PointConv operations on nested subdivisions of point sets. This network effectively extracts features, segments point clouds, and accurately identifies and computes parameters of regular geometric primitives with notable resilience to noise. On fine-grained primitive detection, our approach outperforms Supervised Primitive Fitting Network (SPFN) by 18.5% and Cascaded Primitive Fitting Network (CPFN) by 11.2%. Additionally, our approach consistently maintains low absolute errors in parameter prediction across varying noise levels in the point cloud data. Our experiments validate the robustness of our proposed method and establish its superiority relative to other methodologies in the extant literature.

Nonlinear stability of smooth multi-solitons for the Dullin-Gottwald-Holm equation

In this work, the stability of exact smooth multi-solitons for the Dullin-Gottwald-Holm (DGH) equation is investigated for the first time. Firstly, through the inverse scattering approach, we derive the conservation laws in terms of scattering data and multi-solitons related to the discrete spectrum based on the Lax pair of the DGH equation. Notably, a new series Hamiltonian derived from bi-Hamilton structure are equivalent to the conservation laws, and can also be expressed in terms of scattering data. Thus we successfully connect scattering data (multi-solitons) to the recursion operator between Hamiltonian, and to the Lyapunov functional constructed from Hamiltonian. With this Lyapunov functional, it is identified that these smooth multi-solitons serve as non-isolated constrained minimizers, adhering to a suitable variational nonlocal elliptic equation. Moreover, the integrable properties of the recursion operator are presented, which is the crucial for spectral analysis in this work. Consequently, the investigation of dynamical stability transforms into a problem on the spectrum of explicit linearized systems. It is worth noting that, compared to previous works on Camassa-Holm equation and KdV-type equations, recursion operator derived from existing Hamiltonian for the DGH equation do not possess good integrable properties. We construct a new series of Hamiltonians with both analysable recursion operator and simple initial variation derivative (δH1/δu=m) to overcome this problem. Furthermore, orbital stability of the smooth double solitons is proved at last of the work.

Two-component integrable extension of general heavenly equation

We introduce an integrable two-component extension of the general heavenly equation and prove that the solutions of this extension are in one-to-one correspondence with 4-dimensional hyper-para-Hermitian metrics. Furthermore, we demonstrate that if the metrics in question are hyper-para-Kähler, then our system reduces to the general heavenly equation. We also present an infinite hierarchy of nonlocal symmetries, as well as a recursion operator, for the system under study.

Integrable couplings stemming from three-dimensional unital algebras

This study aims to explore the connection between integrable couplings and three-dimensional unital algebras. Expansions in these algebras generate integrable couplings, including nonlinear integrable couplings, and their corresponding Lax pairs and hereditary recursion operators possess specific structures, associated with non-semisimple Lie algebras. Applications to the KdV equation are explicitly presented.

Integrable Couplings and Two-Dimensional Unital Algebras

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored.

Wave behaviors for fractional generalized nonlinear Schrödinger equation via Riemann–Hilbert method

This paper aims to study the explicit fractional generalized nonlinear Schrödinger (fGNLS) equation by the Riemann–Hilbert (RH) method and to explore the impact of the order of fractional derivatives ϵ on solitons. Firstly, utilizing the recursion operator of the generalized nonlinear Schrödinger (GNLS) equation, the anomalous dispersion relation is constructed. Secondly, the explicit form of the fGNLS equation is obtained by the anomalous dispersion relation and the completeness. Then, the N-soliton solutions are acquired through RH problems. We found that the energy of the solitons decreases with the increase of the order of fractional derivatives ϵ. Specifically, we demonstrate that the fractional one-soliton solution constitutes a valid solution of the fGNLS equation by the Darboux transform.

A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation. A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out, which exhibits the Liouville integrability of each model in the resulting hierarchy. Two specific examples, consisting of novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations, are given.

Fractional optical normalization operator of magnetic field and electroosmotic optimistic energy

In this manuscript, we introduce optical fractional normalization and fractional recursive operators for a particle. The significance of this research lies in obtaining optical fractional recursive operators and normalized operators of magnetic fields through the utilization of a fractional spherical frame within spherical space. Throughout the study, we employ the general form of fractional derivatives, as it enables us to derive more generalized results compared to classical derivatives. Additionally, we derive the microfluidic optical fractional electroosmotic magnetic optimistic fractional energy. We propose a design for optical recursion-based fractional electroosmotic magnetic optimistic fractional energy. This innovative study holds significant potential for applications across various fields, including fluid dynamics, optics, and energy conversion.

On the generalization of cognitive optical networking applications using composable machine learning

Model generalization characterizes the sustainability of machine learning (ML) designs applied to novel system states and therefore plays a vital role toward the realization of cognitive networking. In this paper, we present a composable ML framework (namely, CompML), aiming at generalizing ML-aided cognitive applications for optical networks. CompML makes use of three basic functional modules, i.e., the Loading, Recursion, and Readout modules, to model the loading/initialization processes (e.g., the launch of a signal), extract cumulative features by recursive operations, and produce model inferences, respectively. By the composition of the three modules and adoption of an end-to-end training mechanism, CompML allows for generalizing multiple tasks of the same domain [e.g., quality-of-transmission (QoT) estimation for different lightpaths]. We perform case studies of CompML on QoT estimation and nonlinearity compensation using both simulation and experimental data. Results show the superior generalization ability of CompML compared with the baselines, achieving mean absolute error (MAE) for generalized signal-to-noise ratio (GSNR) prediction error of below 1.06 dB for unseen lightpaths and up to 3 dB Q-factor improvement for nonlinearity compensation.

A generalization of the optical quantum model using fractional normalization and recursion

In this paper, we present a comprehensive investigation into the construction of electromagnetic timelike particles within the frame of Minkowski space, employing the Bishop model. Our study involves deriving the fractional derivatives of key Lorentz forces namely Ω(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (\ extbf{t})$$\\end{document}, Ω(v1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (v_1)$$\\end{document}, and Ω(v2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (v_2)$$\\end{document}. Furthermore, we delve into the computation of essential normalizing and recursion operators tailored to magnetic vector fields, drawing from the principles outlined in the Bishop model. Additionally, we extend our analysis to determine the Fermi–Walker (FW) fractional derivatives, which play a crucial role in understanding the behavior and dynamics of the normalizing and recursion operators. Through this thorough exploration, we aim to provide valuable insights into the electromagnetic properties of timelike particles within the context of Minkowski space.

WITHDRAWN: Construction of a new [formula omitted]-dimensional KdV equation and its closed-form solutions with solitary wave behaviour and conserved vectors

This paper discusses the construction of a new (3+1)-dimensional Korteweg–de Vries (KdV) equation. By employing the KdV’s recursion operator, we extract two equations, and with elemental computation steps, the obtained result is 3uxyt+3uxzt−(ut−6uux+uxxx)yz−2ux∂x−1uyxz−2ux∂x−1uzxy=0. We then transform the new equation to a simpler one to avoid the appearance of the integral in the equation. Thereafter, we apply the Lie symmetry technique and gain a 7-dimensional Lie algebra L7 of point symmetries. The one-dimensional optimal system of Lie subalgebras is then computed and used in the reduction process to achieve seven exact solutions. These obtained solutions are graphically illustrated as 3D and 2D plots that show different propagations of solitary wave solutions such as breather, periodic, bell shape, and others. Finally, the conserved vectors are computed by invoking Ibragimov’s method.

Research on Fault Diagnosis Algorithm of Power Cable Based on Deep Learning

The self-learning method is used to realize the online monitoring of optical cable after the fault occurs. According to the fuzzy characteristic of cable fault, a fuzzy basis function network is constructed which is similar to nonlinear function. Thus, the dynamic characteristics of the cable system are obtained. Then, this paper iterates each additional sample to realize the learning of the center point and radial of the network. A normal multivariable fuzzy distribution is used to determine the expected output of the new sample, and recursive operations are performed on the matrix and covariance matrix of the original sample. In this way, all failure probability distributions at the sampling points are obtained. Finally, the hardware module of the system is designed from two aspects: the distributed short-circuit fault collecting and processing equipment and the on-line temperature measuring equipment for grounding fault. In this way, the functions of software module reset, storage and communication are realized.