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The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation.

In this paper, the (2+1)-dimensional generalized fifth-order KdV equation and the extended (3+1)-dimensional Jimbo-Miwa equation were transformed into the Hirota bilinear forms with Hirota direct method. In this process, the Hirota bilinear operator played a significant role. Based on the Hirota bilinear forms, the single soliton solutions and the single periodic wave solutions of these two types of equations were obtained respectively. Meanwhile, the figures of the single soliton solutions and the single periodic wave solutions were plotted. Furthermore, the results shed light on that when the amplitude of water wave approaches 0, the single periodic wave solutions tend to the single soliton solutions. The conclusion has been generalized from (2+1)-dimensional equations to (3+1)-dimensional equations.

In this study, the propagation of internal solitary waves in oceans at great depths was analyzed. Using multi-scale analysis and perturbation expansion, the basic equation is simplified to the classical Benjamin–Ono equation with variable coefficients. To better describe the propagation characteristics of solitary waves, we derived a higher-order variable-coefficient integral differential (Benjamin–Ono) equation. Subsequently, the bilinear form of the model was derived using Hirota’s bilinear method, and a multi-soliton solution was obtained. Based on the multi-soliton solution of the model, we further studied the interaction of the soliton, which led to the discovery of Mach reflection. Some conclusions were drawn, which are of potential value for further study of solitary waves in the ocean.

In this paper, the propagation of Rossby waves under barotropic-baroclinic interaction in polar co-ordinates is studied. By starting from the two-layer quasi-geotropic potential vorticity equation (of equal depth) with the β effect, the coupled KdV equations describing barotropic-baroclinic waves are derived using multi-scale analysis and the perturbation expansion method. Furthermore, in order to more accurately describe the propagation characteristics of barotropic-baroclinic waves, fifth-order coupled KdV-mKdV equations were obtained for the first time. On this basis, the Lie symmetry and conservation laws of the fifth-order coupled KdV-mKdV equations are analyzed in terms of their properties. Then, the elliptic function expansion method is applied to find the soliton solutions of the fifth-order coupled KdV-mKdV equations. Based on the solutions, we further simulate the evolution of Rossby wave amplitudes and investigate the influence of the high-order terms—time and wave number—on the propagation of barotropic waves and baroclinic waves. The results show that the appearance of the higher-order effect makes the amplitude of the wave lower, the width of the wave larger, and the whole wave flatter, which is obviously closer to actual Rossby wave propagation. The time and wave number will also influence wave amplitude and wave width.

The nonlinear Schrödinger (NLS) equation is an ideal model for describing optical soliton transmission. This paper first introduces an integer-order generalized coupled NLS equation describing optical solitons in birefringence fibers. Secondly, the semi-inverse and fractional variational method is used to extend the integer‐order model to the space–time fractional order. Moreover, various nonlinear forms of fractional NLS equations are discussed, including the Kerr, power, parabolic, dual-power, and log law. The exact soliton solutions, such as bright, dark, and singular solitons, are given. Finally, the behavior of the solution is shown by three-dimensional figures with different fractional orders, which reveals the propagation characteristics of optical solitons in birefringence fibers described by the generalized coupled space–time fractional NLS equation.

Gel’fand-Dikii (GD) formalism is an approach for generating integrable systems in terms of fractional powers of the δ differential operator. In this paper, it extends the GD formalism associated with the third-order δ differential operator L to the time scale. Then, the coupled Boussinesq equation on the time–space scale is given by taking special values, and it can be reduced on different time–space scales. Moreover, the exact solutions of the coupled Boussinesq equation on the time–space scale and the classical Boussinesq equation are constructed via employing the extensions of the Darboux theorem and Crum theorem on the time scale.

The stochastic inertial bidirectional associative memory neural networks (SIBAMNNs) on time scales are considered in this paper, which can unify and generalize both continuous and discrete systems. It is of primary importance to derive the criteria for the existence and uniqueness of both periodic and almost periodic solutions of SIBAMNNs on time scales. Based on that, the criteria for their exponential stability on time scales are studied. Meanwhile, the effectiveness of all proposed criteria is demonstrated by numerical simulation. The above study proposes a new way to unify and generalize both continuous and discrete SIBAMNNs systems, and is applicable to some other practical neural network systems on time scales.

In this paper, the load time-series measured from 2016 to 2018 in Qingdao is investigated in order to make a prediction more accurately by using an artificial neural network model combined with some regular and irregular features. The results of spectral analysis show that several periodic variations in diurnal, semidiurnal and weekly frequencies are prominent, and considered as critical parts of predictior variables of the training and test sets in this forecasting model. However, a significant decline of load happens during the national statutory festivals in China, and is more obvious with the longer holidays, which should be taken into account as abnormal conditions. Moreover, both temperatuer and hummity filtered out by mutual information method are also added as two essential weather factors to improve the prediction accuracy, especially in the hot summer. Finally, the comparative results of five different experiments in term of mean absolute percent error show that the forecasting model combined with all periodic and non-periodic factors outperforms the others with one single or a few of factors. This multifactorial model, taking fully account of internal charateristics of load data and external important influences, is proved more suitable to predict the load trend in Qingdao.

Accurate and reliable short-term electric load forecasting (STLF) plays a critical role in power system to enhance its routine management efficiency and reduce operational costs. However, most of the existing STLF methods suffer from lack of appropriate feature selection procedure. In this paper, a multifactorial framework (MF) possessing the potential to contribute more satisfactory forecasting results and computational speed is proposed. Moreover, a graphical tool for easy and accurate computation of day-ahead load forecast is implemented via MATLAB App Designer. Firstly, we choose the candidate feature set by analyzing the raw electricity consumption data. Then, partial mutual information is adopted as criterion to eliminate these irrelevant and redundant ones among candidate features for the purpose of reducing the input subset and retaining these most relevant. At last, the selected features are used as the input of the well-established artificial neural network (ANN) model optimized by genetic algorithm and cross validation to implement prediction. The MF is applied for the load data measured from 2016 to 2018 in Jinan, and then some competitive experiments and extensive simulations are carried out and results indicates that the ANN-based model with selected features significantly outperforms other alternative models with single features or a few of features regarding mean absolute percent error. In addition, the parallel structure of ANN and the lower dimension of the input space enable the model to achieve faster calculation speed.

It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures.

Abstract Rossby waves, one of significant waves in the solitary wave, have important theoretical meaning in the atmosphere and ocean. However, the previous studies on Rossby waves commonly were carried out in the zonal area and could not be applied directly to the spherical earth. In order to overcome the problem, the research on ( 3+1 $3+1$)-dimensional Rossby waves in the paper is placed into the spherical area, and some new analytical solutions of ( 3+1 $3+1$)-dimensional Rossby waves are given through the classic Lie group method. Finally, the dissipation effect is analyzed in the sense of the above mentioned new analytical solutions. The new solutions on ( 3+1 $3+1$)-dimensional Rossby waves have important value for understanding the propagation of Rossby waves in the rotating earth with the influence of dissipation.

The coupled cubic-quintic nonlinear Schrödinger (CQNLS) equation is a universal mathematical model describing many physical situations, such as nonlinear optics and Bose–Einstein condensate. In this paper, in order to simplify the process of similar analysis with different forms of the coupled CQNLS equation, this dynamic system is extended to a time-space scale based on the Lax pair and zero curvature equation. Furthermore, Darboux transformation of the coupled CQNLS dynamic system on a time-space scale is constructed, and the N-soliton solution is obtained. These results effectively combine the theory of differential equations with difference equations and become a bridge connecting continuous and discrete analysis.

In this paper, the time-fractional Fujimoto–Watanabe equation is investigated using the Riemann–Liouville fractional derivative. Symmetry groups and similarity reductions are obtained by virtue of the Lie symmetry analysis approach. Meanwhile, the time-fractional Fujimoto–Watanabe equation is transformed into three kinds of reduced equations and the third of which is based on Erdélyi–Kober fractional integro-differential operators. Furthermore, the conservation laws are also acquired by Ibragimov’s theory.

The Gerdjikov–Ivanov (GI) equation is one type of derivative nonlinear Schrödinger equation used widely in quantum field theory, nonlinear optics, weakly nonlinear dispersion water waves and other fields. In this paper, the coupled GI equation on a time–space scale is deduced from Lax pairs and the zero curvature equation on a time–space scale, which can be reduced to the classical and the semi-discrete GI equation by considering different time–space scales. Furthermore, the Darboux transformation (DT) of the GI equation on a time–space scale is constructed via a gauge transformation. Finally, N-soliton solutions of the GI equation are given through applying its DT, which are expressed by the Cayley exponential function. At the same time, one-solition solutions are obtained on three different time–space scales ( X = R, X = C and X = Kp ).

In this paper, by means of the Hirota bilinear method, a dimensionally reduced nonlinear evolution equation is investigated. Through its bilinear form, lump solutions are obtained. We construct interaction solutions between lump solutions and one soliton solution by choosing quadratic functions and exponential function. Interaction solutions with the combinations of exponential functions and sine function are also given. Meanwhile, the figures of these solutions are plotted. The dynamical characteristics and properties of obtained solutions are discussed, respectively. The results show that the corresponding physical quantities and properties of nonlinear waves are associated with the values of the parameters.

The general shallow wave equation can be used to describe the fluid motion which own uniform density and nearly uniform speed. In fact, these problems we face in nature do not satisfy these conditions, for example, the motion system of nonlinear rotation fluid governed by the frictional dissipation as well as the atmosphere system with greater viscosity. In these conditions, we need to amend the general shallow wave equation and consider the dissipative and viscous effect. In this article, starting from the shallow wave equation with dissipative and viscous effects in the horizontal direction, by virtue of β plane approximation and quasi-geostrophic approximation of large-scale motion, we derive the dissipative Petviashvili equation to describe the two-dimensional Rossby waves. Based on the ansatz function method, we obtain the exact analytical solutions of dissipative Petviashvili equation and discuss the influence of dissipation on the two-dimensional Rossby waves.

With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.

This article studies the generalized Mittag−Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

In this paper, a kind of nonlinear Schrödinger (NLS) equation, called an NLS-like equation, is Riemann−Hilbert investigated. We construct a 2 × 2 Lax pair associated with the NLS equation and combine the spectral analysis to formulate the Riemann−Hilbert (R−H) problem. Then, we mainly use the symmetry relationship of potential matrix Q to analyze the zeros of det P + and det P − ; the N-soliton solutions of the NLS-like equation are expressed explicitly by a particular R−H problem with an unit jump matrix. In addition, the single-soliton solution and collisions of two solitons are analyzed, and the dynamic behaviors of the single-soliton solution and two-soliton solutions are shown graphically. Furthermore, on the basis of the R−H problem, the evolution equation of the R−H data with the perturbation is derived.

In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential q ( x , t ) can be represented by the solution of this Riemann-Hilbert problem.

A generalized (2+1)-dimensional variable-coefficient KdV equation is introduced, which can describe the interaction between a water wave and gravity-capillary waves better than the (1+1)-dimensional KdV equation. The N-soliton solutions of the (2+1)-dimensional variable-coefficient fifth-order KdV equation are obtained via the Bell-polynomial method. Then the soliton fusion, fission, and the pursuing collision are analyzed depending on the influence of the coefficient eAij; when eAij=0, the soliton fusion and fission will happen; when eAij≠0, the pursuing collision will occur. Moreover, the Bäcklund transformation of the equation is gotten according to the binary Bell-polynomial and the period wave solutions are given by applying the Riemann theta function method.

We introduce how to obtain the bilinear form and the exact periodic wave solutions of a class of (2+1)-dimensional nonlinear integrable differential equations directly and quickly with the help of the generalized Dp-operators, binary Bell polynomials, and a general Riemann theta function in terms of the Hirota method. As applications, we solve the periodic wave solution of BLMP equation and it can be reduced to soliton solution via asymptotic analysis when the value of p is 5.

Two high-dimensional Lie algebras are presented for which four (1+1)-dimensional expanding integrable couplings of the D-AKNS hierarchy are obtained by using the Tu scheme; one of them is a united integrable coupling model of the D-AKNS hierarchy and the AKNS hierarchy. Then (2+1)-dimensional DS hierarchy is derived by using the TAH scheme; in particular, the integrable couplings of the DS hierarchy are obtained.

Integrable coupling system of a lattice soliton equation hierarchy is deduced. The Hamiltonian structure of the integrable coupling is constructed by using the discrete quadratic-form identity. The Liouville integrability of the integrable coupling is demonstrated. Finally, the discrete integrable coupling system with self-consistent sources is deduced.

From rotational potential vorticity-conserved equation with topography effect and dissipation effect, with the help of the multiple-scale method, a new integro-differential equation is constructed to describe the Rossby solitary waves in deep rotational fluids. By analyzing the equation, some conservation laws associated with Rossby solitary waves are derived. Finally, by seeking the numerical solutions of the equation with the pseudospectral method, by virtue of waterfall plots, the effect of detuning parameter and dissipation on Rossby solitary waves generated by topography are discussed, and the equation is compared with KdV equation and BO equation. The results show that the detuning parameter α plays an important role for the evolution features of solitary waves generated by topography, especially in the resonant case; a large amplitude nonstationary disturbance is generated in the forcing region. This condition may explain the blocking phenomenon which exists in the atmosphere and ocean and generated by topographic forcing.

Rossby solitary waves generated by a wavy bottom are studied in stratified fluids. From the quasigeostrophic vorticity equation including a wavy bottom and dissipation, by employing perturbation expansions and stretching transforms of time and space, a forced KdV-ILW-Burgers equation is derived through a new scale analysis, modelling the evolution of Rossby solitary waves. By analysis and calculation, based on the conservation relations of the KdV-ILW-Burgers equation, the conservation laws of Rossby solitary waves are obtained. Finally, the numerical solutions of the forced KdV-ILW-Burgers equation are given by using the pseudospectral method, and the evolutional feature of solitary waves generated by a wavy bottom is discussed. The results show that, besides the solitary waves, an additional harmonic wave appears in the wavy bottom forcing region, and they propagate independently and do not interfere with each other. Furthermore, the wavy bottom forcing can prevent wave breaking to some extent. Meanwhile, the effect of dissipation and detuning parameter on Rossby solitary waves is also studied. Research on the wavy bottom effect on the Rossby solitary waves dynamics is of interest in analytical geophysicalfluid dynamics.

Based on a well-known Lie algebra, the multicomponent Guo hierarchy with self-consistent sources is proposed. With the help of a set of non-semisimple Lie algebra, the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources are obtained. It enriches the content of the integrable couplings of hierarchies with self-consistent sources. Finally, the Hamiltonian structures are worked out by employing the variational identity.

We discuss the Lie point symmetries and discrete symmetries of the inviscid Burgers equation. By employing the Lie group method of infinitesimal transformations, symmetry reductions and similarity solutions of the governing equation are given. Based on discrete symmetries analysis, two groups of discrete symmetries are obtained, which lead to new exact solutions of the inviscid Burgers equation.

In this paper, a study of the coupled Schrödinger Maxwell-Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three-level atoms, is presented via the Riemann-Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann-Hilbert problem is established successfully through a standard dressing procedure via the Riemann-Hilbert approach, and then the potential function related to the solution of the Riemann-Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann-Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann-Hilbert problem without reflection for the coupled Schrödinger Maxwell-Bloch equation, and the multi-soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters. [ABSTRACT FROM AUTHOR]