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Due to variation of working conditions and influence of noise in vibration data, rolling bearing intelligent diagnosis based on deep learning faces challenges in efficient utilization of monitoring data and scientific extraction of fault features. This study proposes a one-dimensional convolution neural network (1D-CNN)-based intelligent diagnosis method for a rolling bearing, which fuses the horizontal and the vertical vibration signals, makes full use of spectral order features by full-spectrum analysis, and achieves accurate classification of fault pattern by 1D-CNN model. The experimental datasets of constant and variable working conditions of rolling bearing are constructed. The test results of the proposed method show that spectral order features are extracted effectively by full-spectrum analysis and high diagnostic accuracy is obtained by the constructed 1D-CNN model on both datasets. The comparison with the other four similar methods indicates that the diagnostic accuracy of the proposed method outperforms the comparative methods significantly in the case of variable operating conditions. [ABSTRACT FROM AUTHOR]

We investigate a variable-coefficient fractional nonlinear Schrödinger(vc-FNLS) equation with Wadati potential and PT-symmetric potential. We find the Lévy index can be used to transition from a breather wave to a soliton as the fractional order derivative is increasing. The influences of fractional α and β on the breather, dark and bright solitons of space–time vc-FNLS equation are analyzed in detail. Some different propagation dynamics are generated via the different parameters α and β. We find that the value of α decreases, the number of oscillations or singularities increases for small time values and decreases for large time values. And the stability of fundamental soliton and asymmetric soliton is explored via the linear stability analysis and direct propagations. Interestingly stability of the soliton is also explored against collisions with Wadati potential and PT-symmetric potential. To additionally explore robustness of the solitons in the vc-FNLS model, it is also relevant to show that they keep its integrity against "bombardment" by impinging waves. These states indicate that the interaction acts both repulsively and attractively, and the analysis can be extended for vc-FNLS equations with more than two different fractional-diffraction terms. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media with PT-symmetric potentials. • We investigate a variable-coefficient fractional nonlinear Schrödinger equation. • The fractional order Lévy index can be used to transition from a breather wave to a solito. • Some stabilities and interactions of the novel fractional solutions are considered via numerical simulation. • The novel features of interaction dynamics are different from these usual features of solitons. [ABSTRACT FROM AUTHOR]

Some bright wave solutions of the coupled derivative nonlinear Schrödinger(CDNLS) equations are derived with Riemann–Hilbert(RH) method. We observe some novel non-autonomous bright soliton structures by choosing the different nonlinear perturbation function g (t). It is shown that there are some possibilities to manipulate the interactions of non-autonomous wave solution through choosing nonlinear and gain/loss functions. Some dynamic behaviors of interactions between two bright solitons have been asymptotically analyzed, the two parabolic-type bright solitons propagate with the opposite directions after the interaction. And some interactions of the linear-, parabolic- and periodic-type bright two-solitons are elastic. A general approach is presented to manipulate soliton waves, which has the potential application for the optical self-routing, non-Kerr media and Bose–Einstein condensates(BEC). [ABSTRACT FROM AUTHOR]

The generalized nonlocal nonlinear Hirota (GNNH) equation has been widely concerned, it can be regarded as the generalization of the nonlocal Schrödinger equation, and can be reduced to a nonlocal Hirota equation. In this paper, we mainly study a GNNH equation and its determinant representation of the N-fold Darboux transformation. Then we derive some novel exact solutions including the breather wave solitons, bright solitons, some characteristics of solitary wave and interactions are considered. In particularly, the dynamic features of one-soliton, two-soliton solutions and the elastic interactions between the two solitons are displayed. We find that unlike the local case, the q(x,t) and q ∗ (− x , t) $q^{*}(-x,t)$ of the GNNH equation have some novel characteristics of solitary wave, which are different form the classical Hirota equation. [ABSTRACT FROM AUTHOR]

Abstract A generalized nonlocal nonlinear Gross–Pitaevskii(GP) equation is presented, which can be reduced to the nonlocal GP equation with self-induced PT -symmetric potential. We derive some novel non-autonomous soliton solutions of nonlocal GP equation via inverse scattering and similarity transformations. These obtained solutions have singularities and are not the standard soliton solutions, here they can be called soliton-like(SL) solutions. Then we consider some controllable behaviors of these non-autonomous wave solutions. The obtained results are different from the solutions of the local nonlinear GP equation. Some propagation phenomena are produced through manipulating non-autonomous SL waves, which can present the potential applications to the soliton wave phenomena in nonlocal wave models. [ABSTRACT FROM AUTHOR]

Abstract We consider the Gross–Pitaevskii(GP) equation with the combination of periodic and harmonic external potentials. In particular, the method of inverse scattering transformation is applied to the GP equation with external potentials. Furthermore, some exact soliton solutions are obtained for the GP equation by using inverse scattering transformation, in which some physically relevant bright solutions are described. The stabilities of the obtained matter-wave solutions are addressed numerically such that some stable solutions are found, and some solitons can be stable in a wide region. These results may raise the possibility of relative experiments and potential applications. [ABSTRACT FROM AUTHOR]

We present the generalized Gross–Pitaevskii (GP( p , q )) equation with two kinds of specific space–time modulated potentials, including the generalized Rosen–Morse potential, the combination of harmonic and Gaussian (harmonic–Gaussian) potential. Some analytical bright soliton solutions are derived from the generalized GP( p , q ) equation via the complex similarity transformation and the generalized stationary nonlinear Schrödinger (NLS) equation. The soliton solutions can be reduced to Jacobi elliptic and spikon-like solutions by choosing the different parameters, and describe some physically relevant phenomenons. Furthermore, some stabilities of the obtained matter-wave solutions are addressed numerically. We find that some solutions are stable and can be observed over a broad range of parameters through analyzing the effect of the phase noise on these solutions. The obtained results may give a certain theoretical guiding significance of relative experiments in Bose–Einstein condensates. [ABSTRACT FROM AUTHOR]

The working state of machinery can be reflected by vibration signals. Accurate classification of these vibration signals is helpful for the machinery fault diagnosis. A novel classification method for vibration signals, named Transform Domain Sparse Representation-based Classification (TDSRC), is proposed. The method achieves high classification accuracy by three steps. Firstly, time-domain vibration signals, including training samples and test samples, are transformed to another domain, e.g. frequency-domain, wavelet-domain etc. Then, the transform coefficients of the training samples are combined as a dictionary and the transform coefficients of the test samples are sparsely coded on the dictionary. Finally, the class label of the test samples is identified by their minimal reconstruction errors. Although the proposed method is very similar to the Sparse Representation-based Classification (SRC), experimental results illustrates its performance is far superior to SRC in the classification of vibration signals. These experiments include: frequency-domain classification of bearing vibration data from the Case Western Reserve University (CWRU) Bearing Data Center and wavelet-domain classification of six fault-types gearbox vibration data from our rotating machinery experimental platform. [ABSTRACT FROM AUTHOR]

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Starting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics. [ABSTRACT FROM AUTHOR]

How to construct a variable coefficient integrable coupling equation hierarchy is an important problem. In this paper, we present new Lax pairs with some arbitrary functions and generate a variable coefficient integrable coupling of Ablowitz-Kaup-Newell-Segur hierarchy from a zero-curvature equation. Then the Hamiltonian structure of the variable coefficient coupling equation hierarchy is derived from the variational trace identity. It is also indicated that this method is an efficient and straightforward way to construct the variable coefficient integrable coupling equation hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, the exact solutions of generalized nonlinear Schrödinger (GNLS) equation are obtained by using Darboux transformation(DT). We derive some expressions of the 1-solitons, 2-solitons and n-soliton solutions of the GNLS equation via constructing special Lax pairs. And we choose different seed solutions and solve the GNLS equation to obtain the soliton solutions, breather solutions and rational wave solutions. Based on these obtained solutions, we consider the elastic interactions and dynamics between two solitons. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate the reverse-space–time nonlocal nonlinear Fokas–Lenells equation by the idea of Ablowitz and Musslimani. The reverse-space–time Fokas–Lenells equation, associated with a 2 × 2 matrix Lax pair, is the important integrable system, which can be reduced to the nonlocal Fokas–Lenells equation. Based on its Lax pair, we construct nonlocal version of N-fold Darboux transformation (DT) for the Fokas–Lenells equation, and obtain two kinds of soliton solutions from vanishing and plane wave backgrounds. Further some novel one-soliton and two-soliton are derived with the zero and nonzero seed solutions through complex computations, including the bright soliton, kink soliton and breather wave soliton. Moreover, various graphical analyses on the presented solutions are made to reveal the dynamic behaviors, which display the elastic interactions between two solitons and their amplitudes keeping unchanged after the interactions except for the phase shifts. It is clearly shown that these solutions have new properties which differ from ones of the classical Fokas–Lenells equation. [ABSTRACT FROM AUTHOR]

A high-order dispersive cubic-quintic Gross–Pitaevskii (HDCQGP) equation (a generalized variable coefficients nonlinear Schrödinger equation with the third and fourth-order and the cubic-quintic nonlinear terms) is considered, and is transformed into a standard cubic-quintic nonlinear Schrödinger equation (NLSE). By using the generalized tanh-function method, we study exact solutions of the HDCQGP equation with time-modulated potential and nonlinearity. In particular, based on the similarity transformation, we report several families of non-autonomous wave solutions of the HDCQGP equation with snaking behaviors and different amplitude surfaces. At last, we consider the numerical simulation of two solitons collision for the NLSE with different parameters. These results may raise the possibility of relative experiments and potential applications. [ABSTRACT FROM AUTHOR]

Abstract: A kind of nonlinear integrable couplings of C-KdV soliton hierarchy is constructed, then we present the infinitely many conservation laws for the nonlinear integrable couplings of the C-KdV soliton hierarchy. [Copyright &y& Elsevier]

Abstract: In this paper, we construct explicit rogue wave solutions and dark-bright solutions for the inhomogeneous coupled nonlinear Schrödinger equation with variable coefficients by means of similarity transformations. Some explicit solutions to the coupled nonlinear Schrödinger equation with potentials and nonlinearities depending on both time and spatial coordinates are considered. We present the general approach and use it to calculate rogue wave solutions and dark-bright solutions for nonlinearities and potentials of physical interest in applications to Bose–Einstein condensates and nonlinear optics. [Copyright &y& Elsevier]

It is shown that the Kronecker product can be applied to construct new nonlinear integrable coupling system of soliton equation hierarchy in this paper. A direct application to the KN spectral problem leads to a novel KN soliton equation hierarchy of nonlinear integrable coupling system, which is different from any one obtained before. Furthermore, we present the Hamiltonian structures of nonlinear integrable couplings of KN hierarchy by using a pairs from special non-semisimple matrix Lie algebras and variational identities over the associated loop algebras. It is also indicated that the study of nonlinear integrable couplings using the Kronecker product is an efficient and straightforward method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the rational solutions with arbitrary constants of a Hirota equation are presented employing the Darboux transformations. Then, the similarity transformation allows us to relate certain class of multi-rogue wave solutions of the higher-order nonlinear Schrödinger equation (NLSE) to the solutions of (1+1)-dimensional integrable Hirota equation. The multi-rogue wave solutions with arbitrary constants are considered. The obtained rogue wave solutions can be used to describe the possible formation mechanisms for optical and Bose–Einstein condensates. [Copyright &y& Elsevier]

Abstract: A three-dimensional Gross–Pitaevskii (3D-GP) equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrödinger (NLS) equation. We study exact solutions of the 3D-GP equation with (space, time)-modulated potential and nonlinearity. In particular, based on the similarity transformation, we report several families of exact solutions of the 3D-GP equation with different amplitude surfaces, in which some physically relevant solutions are described. These results may raise the possibility of relative experiments and potential applications. [Copyright &y& Elsevier]

Abstract: In this paper, we present an integrable coupling of lattice hierarchy and its continuous limits by using Lie algebra . By introducing a complex discrete spectral problem, the integrable coupling of Kac–Van Moerbeke lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable coupling of discrete soliton equation hierarchy, which has the integrable coupling system of MKdV hierarchy as a new kind of continuous limit. [Copyright &y& Elsevier]

Abstract: We present a new method to construct two (2+1)-dimensional lattice hierarchies. By considering a kind of spectral problems with two shift operators and , a new (2+1)-dimensional discrete soliton hierarchy and a new lattice hierarchy are constructed by using matrix spectral problem. [Copyright &y& Elsevier]

In this paper, we consider the discrete soliton hierarchy with self-consistent sources by using Lie algebra sl(3). Then a Blaszak-Marciniak equations hierarchy with self-consistent sources is obtained through the discrete zero curvature equation. [ABSTRACT FROM AUTHOR]

Abstract: A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders. We construct the generalized fractional trace identity through the Riemann–Liouville fractional derivative. An example of the fractional KN soliton equation hierarchy and Hamiltonian structure is presented, which is a new integrable hierarchy and possesses Hamiltonian structure. [Copyright &y& Elsevier]

Abstract: Based on the Kupershmidt deformation, we propose the generalized Kupershmidt deformation (GKD) to construct new systems from integrable bi-Hamiltonian system. As applications, the generalized Kupershmidt deformation of the fifth-order coupled KdV equations hierarchy with self-consistent sources and its Lax representation are presented. [Copyright &y& Elsevier]

Abstract: We present a kind of integrable couplings of soliton equations hierarchy with self-consistent sources by using the loop algebra . As an application, a new integrable couplings of the generalized Wadati–Konono–Ichikawa (G-WKI) equations hierarchy with self-consistent sources is derived through the enlarged loop algebra . [Copyright &y& Elsevier]

A generalized fractional trace identity and Hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified Riemann-Liouville derivative and exterior derivatives of fractional orders. Example of the fractional Kaup Newell (KN) soliton equation hierarchy and Hamiltonian system are considered, which is a new integrable hierarchy and possesses Hamiltonian structure. [ABSTRACT FROM AUTHOR]

Abstract: Some integrable coupling systems of existing papers are linear integrable couplings. In the Letter, beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing real nonlinear integrable couplings of continuous soliton hierarchy. A direct application to the AKNS spectral problem leads to a novel nonlinear integrable couplings, then we consider the Hamiltonian structures of nonlinear integrable couplings of AKNS hierarchy with the component-trace identity. [Copyright &y& Elsevier]

Abstract: A noncommutative version of the TC soliton equation hierarchy is presented, which possesses the zero curvature representation. Then, we show that noncommutative (NC) TC equation can be derived from the noncommutative (anti-)self-dual Yang–Mills equation by reduction. Finally, an integrable coupling system of the NC TC equation hierarchy is constructed by using of the enlarged Lax pairs. [Copyright &y& Elsevier]

Abstract: In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained. [Copyright &y& Elsevier]

Abstract: In this paper we present a soliton hierarchy of vector KN (Kaup–Newell) equations by using an arbitrary order matrix spectral problem, along with its bi-Hamiltonian formulation. Then, adjoint symmetry constraints are presented for manipulating binary nonlinearization for the associated arbitrary order matrix spectral problem. The resulting spatial and temporal constrained flows are shown to provide integrable decompositions of the vector KN equations. Finally, a class of integrable coupling systems of the vector KN soliton equations is obtained through enlarging the spectral problem. [Copyright &y& Elsevier]

Abstract: A kind of non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with [Y.F. Zhang, E.G. Fan, Characteristic Numbers of Matrix Lie Algebras, Commun. Theor. Phys (China) 49 (2008) 845] is presented. As an application example, the hierarchy of non-isospectral cubic Volterra lattice hierarchy with self-consistent sources is derived. Furthermore, we construct a non-isospectral integrable couplings of cubic Volterra lattice hierarchy with self-consistent sources by using the loop algebra . [Copyright &y& Elsevier]

Abstract: In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra . By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit. [Copyright &y& Elsevier]

Abstract: We present a kind of integrable couplings of soliton equations hierarchy with self-consistent sources by using of loop algebra . As an application, a hierarchy of C-KdV equations with self-consistent sources is derived through loop algebra . Furthermore, we construct a new integrable couplings of the C-KdV soliton hierarchy with self-consistent sources by using of the enlarged loop algebra . [Copyright &y& Elsevier]

Abstract: In [E.G. Fan, Phys. Lett. A 372 (2008) 6368], Fan present a lattice hierarchy and its continuous limits. In this Letter, we extend this method, by introducing a complex discrete spectral problem, a coupling lattice hierarchy is derived. It is shown that a new sequence of combinations of complex lattice spectral problem converges to the integrable coupling couplings of soliton equation hierarchy, which has the integrable coupling system of AKNS hierarchy as a continuous limit. [Copyright &y& Elsevier]

In this paper, we will extend Ma's method to construct the integrable couplings of soliton equation hierarchy with the Kronecker product and two-nilpotent matrix. A direct application to the MKdV spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is shown that the study of integrable couplings using the Kronecker product is an efficient and straightforward method. [ABSTRACT FROM AUTHOR]

Abstract: A set of new matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra . It follows that an isospectral problem is established. By use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component Yang hierarchy is given. Finally, the multi-component super-integrable coupling system of Yang hierarchy is presented by enlarging matrix spectral problem. [Copyright &y& Elsevier]

Abstract: A kind of the integrable couplings of soliton equations hierarchy with self-consistent sources associated with is presented. As an application example, we construct a new integrable couplings of the Dirac soliton hierarchy with self-consistent sources by using of loop algebra . [Copyright &y& Elsevier]

Abstract: A kind of new non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with is presented. As an application example, the integrable coupling hierarchy of non-isospectral Ablowitz–Ladik with self-consistent sources is derived by using of the loop algebra . [Copyright &y& Elsevier]

Abstract: A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with is presented. As an application example, the hierarchy of Yang equations with self-consistent sources is derived. Furthermore, we construct an integrable couplings of the Yang soliton hierarchy with self-consistent sources by using of the loop algebra . [Copyright &y& Elsevier]

Abstract: A new subalgebra of loop algebra is first constructed. It follows that an isospectral problem is established. Using Tu-pattern gives rise to a new integrable hierarchy, which possesses bi-Hamiltonian structure. Furthermore, by making use of bi-symmetry constraints, the generalized Hamiltonian regular representations for the hierarchy are obtained. Finally, we obtain an expanding integrable system of this hierarchy by applying a scalar transformation between two isospectral problems and constructing a five-dimensional loop algebra . [Copyright &y& Elsevier]

Abstract: The generalized zero curvature equation is presented, which is an important tool to construct the (2+1)-dimensional integrable soliton equation hierarchy. A few of new (2+1)-dimensional integrable soliton equation hierarchies are obtained through the generalized zero curvature equation. [Copyright &y& Elsevier]

Abstract: In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity. [Copyright &y& Elsevier]

Abstract: It is shown that the Kronecker product can be applied to construct a new integrable coupling system of discrete soliton equation hierarchy in this Letter. A direct application to the generalized Toda lattice spectral problem leads to a novel integrable coupling system. It is also indicated that the study of integrable couplings by using of the Kronecker product is an efficient and straightforward method. [Copyright &y& Elsevier]

Abstract: In this paper, we first present a fractional Hamiltonian function. Then, based on the dissipative Hamiltonian realization, using the fractional Hamiltonian function method, as a static measurable feedback, a criterion for the fractional nonlinear differential algebraic systems (FNDAS) is proposed to stabilize the multi-machine power system. Finally, FNDAS in presence of disturbance, a fractional H ∞ control strategy is constructed. The proposed stabilization and robust controller can effectively take advantage of the structural characteristics of FNDAS and is simple in form. [Copyright &y& Elsevier]

Abstract: A scheme for deducing Hamiltonian structures of the integrable couplings for the multicomponent hierarchies of evolution equations is presented, which is devoting to obtaining the Hamiltonian structures of the integrable couplings of the multicomponent Dirac hierarchy. [Copyright &y& Elsevier]

Abstract: This Letter reports on a study of the multicomponent discrete integrable equation hierarchy with variable spectral parameters. An instance of new multicomponent lattice equation is obtained. Correspondingly, a feasible way to construct the integrable couplings is presented. It indicates that the study of integrable couplings using the block matrix-form of Lie algebra is an important and effective method. [Copyright &y& Elsevier]