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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.

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.

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 Schrodinger (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.

Until now, the soliton solutions of nonlocal equations have also been put forward for the corresponding local equations. However, a natural motivation of our work is to find some soliton solutions to the mixed local–nonlocal equations that have not been reported for the associated nonlocal-type equations. A novel mixed local–nonlocal nonlinear Schrodinger (MLN-NLS) equation is presented, then we investigate the Hirota’s bilinearization transformation and obtain the one- and two-soliton solutions of the MLN-NLS equation. Based on the obtained solutions, the propagation and interaction structures are shown, the broken and unbroken soliton solutions are derived with Hirota’s transformation. We find that the MLN-NLS equation has some novel results, which is unlike the local or nonlocal cases. And these results seem to be the characteristic features of the mixed local–nonlocal nonlinear system in the present work.

We consider the nonautonomous discrete vector bright–dark solutions and their controllable behaviors in the coupled Ablowitz–Ladik equation with variable coefficients, which possesses complicated wave propagation in time. Based on the differential–difference symmetry transformation and the Lame polynomial solutions, we use the Jacobi elliptic functions $$\hbox {sn}(n, m), \hbox {cn}(n, m)$$ , $$\hbox {dn}(n, m)$$ and present the nonautonomous discrete vector bright–dark solutions, which are localized in space and keep the localization longer in time. Moreover, we also exhibit the wave propagation of nonautonomous Lame polynomial solutions of higher order and their dynamics for some chosen parameters and functions. And the managements and dynamic behaviors of these solutions are investigated analytically.

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.

The Hirota bilinear method is studied in a lot of local equations, but there are few work to solve nonlocal equations with external potential by Hirota bilinear method. In this letter, we succeed to bilinearize the generalized nonlocal Gross–Pitaevskii (NGP) equation with an arbitrary time-dependent linear potential through a nonstandard procedure and present more general bright soliton solutions, which describes the dynamics of soliton solutions in quasi-one-dimensional Bose–Einstein condensations. Under some reasonable assumptions, one-bright-soliton and two-bright-soliton solutions are constructed analytically by the improved Hirota method. From the gauge equivalence, we can see the difference between the solutions of the nonlocal GP equation and the solutions of the local GP equation.

We consider nonautonomous rogue wave solutions of a $$(3+1)$$ -dimensional (3D) nonlinear Schrodinger equation (NLSE) with time–space modulation terms, the dispersion and the nonlinear coefficients engendering temporal dependency. Similarity transformation is used to convert the nonautonomous equation into autonomous NLSE; we obtain the multi-rogue wave solutions employing the generalized Darboux transformation. Particularly, the rogue wave solutions possess several free parameters. Then, the first-order and second-order nonautonomous rogue wave solutions are considered for the 3D NLSE with variable coefficients. At last, the numerical simulations on the evolution and collision of rogue wave solutions are performed to verify the prediction of the analytical formulations. The obtained nonautonomous rogue wave solutions can be used to describe the dynamics waves in the nonlinear optical fibers and Bose–Einstein condensates, respectively.

We develop a nonstandard bilinearization procedure to generate more general bright soliton, breather soliton and quasi-rogue wave solutions for the PT-symmetric coupled nonlocal nonlinear Schrodinger (CNNLS) equations. We achieve some novel solutions by bilinearizing both the CNNLS equations and their associated auxiliary equations in a novel way. The soliton equations are written into bilinear operator forms by means of auxiliary equations, then the 1-soliton solution and 2-soliton solution of PT-symmetric CNNLS equations are constructed. Some novel interaction properties of the PT-symmetric two-soliton solutions are derived, which can present the potential applications to the soliton wave phenomena in nonlocal wave models.

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.

In this work, we study exact solutions of a (3+1)-dimensional higher-order coupled Schrodinger equation with higher-order terms, the dispersion and the nonlinear coefficients engendering temporal dependency. Similarity transformations are used to convert the nonautonomous equations into autonomous coupled Hirota equations, then we present the multi-rogue wave solutions with arbitrary constants employing the Darboux transformations. At last, the first order and second order rogue wave solutions with arbitrary constants are considered for the 3DHCNLSE with variable coefficients by using the similarity transformation and Darboux transformation. The obtained (3+1)-dimensional multi-rogue wave solutions can be used to describe the dynamics waves in the deep ocean and nonlinear optical fibers.

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 Schrodinger 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.

In this paper, we present an integrable coupling of lattice hierarchy and its continuous limits by using Lie algebra sl(4). 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.

A three-dimensional Gross-Pitaevskii (3D-GP) equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrodinger (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.

We present a new method to construct two (2 + 1)-dimensional lattice hierarchies. By considering a kind of spectral problems with two shift operators E ˜ and E ˆ , a new (2 + 1)-dimensional discrete soliton hierarchy and a new 3 × 3 lattice hierarchy are constructed by using 3 × 3 matrix spectral problem.

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.

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.

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.

A kind of non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with [email protected]?(4) [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 [email protected]?(4).

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.

Algebra of pseudodifferential operators for constructing the multicomponent Davey–Stewardson (DS) hierarchy of evolution equation is presented, then, we consider the Hamiltonian structure of the (1+2)-dimensional multicomponent DS equation hierarchy. Finally, the integrable coupling system of (1+2)-dimensional multicomponent DS hierarchy is obtained through enlarging matrix spectral problem.

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.

A set of new matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A ∼ 2 M . 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.

A kind of the integrable couplings of soliton equations hierarchy with self-consistent sources associated with sl ˜ ( 4 ) 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 sl ˜ ( 4 ) .

A new subalgebra of loop algebra A ∼ 2 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 G ∼ .

A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using of differential forms and exterior derivatives of fractional orders. Example of the fractional Hamiltonian system of the C-KdV soliton equation hierarchy is constructed, which is a new Hamiltonian structure.

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.

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.

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.

A fractional order soliton equation hierarchy is presented, which is a new Liouville integrable hierarchy and possesses bi-Hamiltonian structure. Finally, we obtain a integrable coupling system of this fractional order soliton equation hierarchy by applying a upper triangular matrix of Lie algebras.

In terms of the properties of the known loop algebra A ∼ 1 and difference operators, a new algebraic system GM is constructed, which is devoted to working out the multi-component Toda hierarchy. Then an extended algebraic system G ∼ M of GM is presented, from which the integrable coupling system of the multi-component Toda equations is obtained.

In terms of the properties of the known loop algebra A ∼ 1 and difference operators, a new algebraic system X is constructed, which is devote to working out the well-known generalized cubic Volterra lattice equations hierarchy. Then an extended algebraic system X ∼ of X is presented, from which the integrable coupling system of generalized cubic Volterra lattice equations are obtained.

A new simple loop algebra G ∼ M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component TD hierarchy is obtained. Furthermore, an expanding loop algebra F ∼ M of the loop algebra G ∼ M is presented. Based on F ∼ M , the multi-component integrable couplings system with five arbitrary functions of the multi-component TD hierarchy is worked out. The method can be applied to other nonlinear evolution equations hierarchies.

A new simple loop algebra G ∼ M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component WKI hierarchy is obtained. Furthermore, an expanding loop algebra F ∼ M of the loop algebra G ∼ M is presented. Based on F ∼ M , the multi-component integrable couplings system with two arbitrary functions of the multi-component WKI hierarchy is worked out. How to design the isospectral problem of the multi-component hierarchy of equations is a technique and interesting topic. The method can be applied to other nonlinear evolution equations hierarchy.

A new loop algebra G ˜ M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component coupled Burgers hierarchy is obtained. Furthermore, an expanding loop algebra F ˜ M of the loop algebra G ˜ M is presented. It follows that the multi-component integrable coupling system of the multi-component coupled Burgers hierarchy is worked out. In general, the method can be applied to other evolution equations hierarchies.

Starting from a discrete spectral problem, we derive a hierarchy of nonlinear discrete equations which include the Ablowitz-Ladik (AL) equation. We analytically study the discrete rogue-wave (DRW) solutions of AL equation with three free parameters. The trajectories of peaks and depressions of profiles for the first- and second-order DRWs are produced by means of analytical and numerical methods. In particular, we study the solutions with dispersion in parity-time ( PT) symmetric potential for Ablowitz-Musslimani equation. And we consider the non-autonomous DRW solutions, parameters controlling and their interactions with variable coefficients, and predict the long-living rogue wave solutions. Our results might provide useful information for potential applications of synthetic PT symmetric systems in nonlinear optics and condensed matter physics.

The passivity and feedback passification of T-S fuzzy systems with norm bounded uncertainties are considered. Attention is focused on the design of memoryless and delayed feedback controllers because the kinds of controllers have not been fully investigated. An appropriate selection of Lyapunov-Krasovskii function and introduction of free weighting matrices, which make the obtained condition less conservative, generalize sufficient conditions. In terms of a linear matrix inequality(LMI), we obtain delay-dependent sufficient conditions which guarantees the passivity based stabilization of the closed-loop system for all admissible uncertainties. A numerical example is given to show the correctness and effectiveness of our theoretical results.

The passivity and feedback passification of T-S fuzzy systems with time-varying norm bounded uncertainties are considered. Attention is focused on the design of static output feedback because the kinds of controllers have not been fully investigated, which is more complex and still unsolved. An appropriate selection of Lyapunov-Krasovskii function and introduction of free weighting matrices, which make the obtained condition less conservative, generalize sufficient conditions. In terms of linear matrix inequality(LMI), we obtain delay-dependent sufficient conditions which guarantees the passivity based stabilization of the closed-loop system for all admissible uncertainties. An numerical example is given to show the correctness and effectiveness of our theoretical results.

This paper investigates the robust delay-dependent stabilization problem for uncertain fuzzy systems with state and input delays which is more general fuzzy model. Taking Leibniz-Newton formula into account, sufficient matrix inequalities conditions for stabilization of such systems are derived via a novel Lyapunov- Krasovskii functional (LKF), which are dependent on the size of the delays and are based on the solutions of linear matrix inequalities (LMIs). Finally, numerical examples are included to show the effectiveness and improvement of the proposed method.

This paper proposes a novel control method for delay-dependent stability of a class of time-delay chaotic systems via standard feedback control (SFC) and delayed feedback control (DFC). Based on a novel Lyapunov-Krasovskii stability theory and linear matrix inequality (LMI) technique, the delay-dependent stability criteria via SFC and DFC are derived. And the method can be obtained to stabilize the system to an unstable fixed point. An example is included to show the advantage of the proposed method.

This paper deals with the controller design problem of a networked control system with multiple time-delays and the presence of network-induced delay. The Takagi-Sugeno (T-S) fuzzy model is adopted for representing networked control systems. Based on a novel Lyapunov-Krasovskii functional and LMI technique, some sufficient conditions for the NCS which is subject to the asymptotic stability are derived, and the method of the controller design is also given in terms of LMIs. Finally, an example shows the effectiveness of the proposed method.

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.

We present a kind of integrable couplings of soliton equations hierarchy with self-consistent sources by using the loop algebra sl̃(4). 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 sl̃(4).