We introduce and study the relation between Pesin-Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measure for nonautonomous dynamical systems,which is an extensi...We introduce and study the relation between Pesin-Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measure for nonautonomous dynamical systems,which is an extension of the classical definition of Bowen topological entropy.We show that the Pesin-Pitskel topological pressure can be determined by the local pressures of measures in nonautonomous case and establish a variational principle for Pesin-Pitskel topological pressure on compact subsets in the context of nonautonomous dynamical systems.展开更多
We analytically present a family of nonautonomous dark solitons and rogue waves in a planar graded-index grating waveguide with an additional long-period grating.The dark solitons whose dynamics described by the expli...We analytically present a family of nonautonomous dark solitons and rogue waves in a planar graded-index grating waveguide with an additional long-period grating.The dark solitons whose dynamics described by the explicit expressions such as the valley,background and wave central position are investigated.We find that dark soliton's depth and the long-period grating have effects on soliton's wave central position;the gain or loss term affects directly both the background and valley of the soliton.For rogue waves,it is reported that one can modulate the distribution of the light intensity by adjusting the parameters of the long-period grating.Additionally,more rogue waves with different evolution behaviors in this special waveguide are demonstrated clearly.展开更多
A delayed n-species nonautonomous Lotka-Volterra type competitive system without dominating instantaneous negative feedback is investigated. By means of a suitable Lyapunov functional, sufficient conditions are derive...A delayed n-species nonautonomous Lotka-Volterra type competitive system without dominating instantaneous negative feedback is investigated. By means of a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive solutions of the system. As a corollary, it is shown that the global asymptotic stability of the positive solution is maintained provided that the delayed negative feedbacks dominate other interspecific interaction effects with delays and the delays are sufficiently small.展开更多
In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.
Based on the three-dimensional Liu chaotic system, this paper appends a feedback variable to construct a novel hyperchaotic Liu system. Then, a control signal is further added to construct a novel nonautonomous hyperc...Based on the three-dimensional Liu chaotic system, this paper appends a feedback variable to construct a novel hyperchaotic Liu system. Then, a control signal is further added to construct a novel nonautonomous hyperchaotic Liu system. Through adjusting the frequency of the control signal, the chaotic property of the system can be controlled to show some different dynamic behaviors such as periodic, quasi-periodic, chaotic and hyperchaotic dynamic behaviours. By numerical simulations, the Lyapunov exponent spectrums, bifurcation diagrams and phase diagrams of the two new systems are studied, respectively. Furthermore, the synchronizing circuits of the nonautonomous hyperchaotic Liu system are designed via the synchronization control method of single variable coupling feedback. Finally, the hardware circuits are implemented, and the corresponding waves of chaos are observed by an oscillograph.展开更多
By utilizing a fixed point theorem on cone, some new results on the existence ofpositive periodic solutions for nonautonomous differential equations with delay are derived.
To study controlled evolution of nonautonomous matter-wave breathers and rogue waves in spinor Bose–Einstein condensates with spatiotemporal modulation,we focus on a system of three coupled Gross–Pitaevskii equation...To study controlled evolution of nonautonomous matter-wave breathers and rogue waves in spinor Bose–Einstein condensates with spatiotemporal modulation,we focus on a system of three coupled Gross–Pitaevskii equations with spacetime-dependent external potentials and temporally modulated gain-loss distributions.With different external potentials and gain-loss distributions,various solutions for controlled nonautonomous matterwave breathers and rogue waves are derived by the Darboux transformation method,such as breathers and rogue waves on arched and constant backgrounds which have the periodic and parabolic trajectories.Effects of the gain-loss distribution and linear potential on the breathers and rogue waves are studied.Nonautonomous two-breathers on the arched and constant backgrounds are also derived.展开更多
Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the v...Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.展开更多
This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary condition.The authors prove the existence of compact ...This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary condition.The authors prove the existence of compact kernel sections for the associated process by using a suitable decomposition of the equations.展开更多
In this paper, for controlling the spread of plant diseases, a nonautonomous SEIS (Susceptible → Exposed → Infectious → Susceptible) epidemic model with a general nonlinear incidence rate and time-varying impulsive...In this paper, for controlling the spread of plant diseases, a nonautonomous SEIS (Susceptible → Exposed → Infectious → Susceptible) epidemic model with a general nonlinear incidence rate and time-varying impulsive control strategy is proposed and investigated. This novel model could result in an objective criterion on how to control plant disease transmission by replanting of healthy plants and removal of infected plants. Using the method of small amplitude perturbation, the sufficient conditions under which guarantee the globally attractive of the disease-free periodic solution and the permanence of the disease are obtained, that is, the disease dies out if R12>1.展开更多
In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive relea...In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.展开更多
With the help of self-similarity transformation, we construct and study the nonautonomous vortices with different topological charges inside a planar graded-index nonlinear waveguide, analytically, and numerically. Al...With the help of self-similarity transformation, we construct and study the nonautonomous vortices with different topological charges inside a planar graded-index nonlinear waveguide, analytically, and numerically. Although these vortices are approximate, they can reflect the real properties of self-similar optical beam during a short-term propagation. Existence of these autonomous vortices require delicate balances between the system parameters such as diffraction, nonlinearity, gain, and external potential. We are concerned with some special but interesting situations, and discussing the changes of the height, width, energy, and central position of the vortices as the increase of propagation distance. Moreover, we are also interested in the azimuthal modulational instability of the system, and comparing our prediction for the modulational instability growth rates to numerical results.展开更多
The necessary and sufficient condition of the stability of linear nonautonomous system under the frequently-acting perturbation has been given and proved on the basis of[1] and [2], and the theorem of the equivalence ...The necessary and sufficient condition of the stability of linear nonautonomous system under the frequently-acting perturbation has been given and proved on the basis of[1] and [2], and the theorem of the equivalence on the uniform and asymptotical stability in the sense of Liapunov ami the stability under the frequently-acting perturbation of linear nonautonomous system has been given in this paper. Besides, the analysis of the dynamic stabilitv of robot has been presented by applying the theorem in this paper, which is closer to reality展开更多
We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potenti...We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain （loss） coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.展开更多
In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore,...In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov's indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.展开更多
Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-...Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.展开更多
A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed an...A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed and verified with the Wronskian technique. Collisions among the three solitons are discussed and illustrated, and effects of the coefficientsσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) on the collisions are graphically analyzed, whereσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) are the first-, second-, third-order dispersion parameters and an inhomogeneous parameter related to the phase modulation and gain(loss), respectively. The head-on collisions among the three solitons are observed, where the collisions are elastc. Whenσ1(x, t) is chosen as the function of x, amplitudes of the solitons do not alter, but the speed of one of the solitons changes.σ2(x, t) is found to affect the amplitudes and speeds of the two of the solitons. It reveals that the collision features of the solitons alter withσ3(x, t)=-1.8x. Additionally, traveling directions of the three solitons are observed to be parallel when we change the value of v(x, t).展开更多
We report the analytical nonantonomous soliton solutions （NSSs） for two-component Bose-Einstein condensates with the presence of a time-dependent potential. These solutions show that the time-dependent potential can...We report the analytical nonantonomous soliton solutions （NSSs） for two-component Bose-Einstein condensates with the presence of a time-dependent potential. These solutions show that the time-dependent potential can affect the velocity of NSS. The velocity shows the characteristic of both increasing and oscillation with time. A detailed analysis for the asymptotic behavior of NSSs demonstrates that the collision of two NSSs of each component is elastic.展开更多
We prove the existence and the orbital stability of standing waves for the nonau- tonomous Schrodinger system……under suitable conditions on the coefficient functions a and b. We follow the idea of analyzing the comp...We prove the existence and the orbital stability of standing waves for the nonau- tonomous Schrodinger system……under suitable conditions on the coefficient functions a and b. We follow the idea of analyzing the compactness of minimizing sequence of the constrained minimization problems.展开更多
A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarit...A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.展开更多
基金Supported by NSFC(Nos.11971236,11901419)the Foundation in Higher Education Institutions of Henan Province(No.23A110020)。
文摘We introduce and study the relation between Pesin-Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measure for nonautonomous dynamical systems,which is an extension of the classical definition of Bowen topological entropy.We show that the Pesin-Pitskel topological pressure can be determined by the local pressures of measures in nonautonomous case and establish a variational principle for Pesin-Pitskel topological pressure on compact subsets in the context of nonautonomous dynamical systems.
基金Supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 10975180,11047025,and 11075126the "Applied nonlinear Science and Technology" from the most important among all the top priority disciplines of Zhejiang Province
文摘We analytically present a family of nonautonomous dark solitons and rogue waves in a planar graded-index grating waveguide with an additional long-period grating.The dark solitons whose dynamics described by the explicit expressions such as the valley,background and wave central position are investigated.We find that dark soliton's depth and the long-period grating have effects on soliton's wave central position;the gain or loss term affects directly both the background and valley of the soliton.For rogue waves,it is reported that one can modulate the distribution of the light intensity by adjusting the parameters of the long-period grating.Additionally,more rogue waves with different evolution behaviors in this special waveguide are demonstrated clearly.
文摘A delayed n-species nonautonomous Lotka-Volterra type competitive system without dominating instantaneous negative feedback is investigated. By means of a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive solutions of the system. As a corollary, it is shown that the global asymptotic stability of the positive solution is maintained provided that the delayed negative feedbacks dominate other interspecific interaction effects with delays and the delays are sufficiently small.
基金Supported by the National Natural Science Foundation of China (10701032)Natural Science Foundation of Hebei Province (A2008000132)the Doctoral Foundation of Hebei Normal University (L2005B02)
文摘In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.
基金Project supported by the National Natural Science Foundation of China (Grant No 60572089)the Natural Science Foundation of Chongqing (Grant No CSTC,2008BB2087)
文摘Based on the three-dimensional Liu chaotic system, this paper appends a feedback variable to construct a novel hyperchaotic Liu system. Then, a control signal is further added to construct a novel nonautonomous hyperchaotic Liu system. Through adjusting the frequency of the control signal, the chaotic property of the system can be controlled to show some different dynamic behaviors such as periodic, quasi-periodic, chaotic and hyperchaotic dynamic behaviours. By numerical simulations, the Lyapunov exponent spectrums, bifurcation diagrams and phase diagrams of the two new systems are studied, respectively. Furthermore, the synchronizing circuits of the nonautonomous hyperchaotic Liu system are designed via the synchronization control method of single variable coupling feedback. Finally, the hardware circuits are implemented, and the corresponding waves of chaos are observed by an oscillograph.
基金Supported by the Natural Science Foundation of Guangdong Province（032469）
文摘By utilizing a fixed point theorem on cone, some new results on the existence ofpositive periodic solutions for nonautonomous differential equations with delay are derived.
基金supported by the National Natural Science Foundation of China (Grant Nos.11975172 and 12261131495)。
文摘To study controlled evolution of nonautonomous matter-wave breathers and rogue waves in spinor Bose–Einstein condensates with spatiotemporal modulation,we focus on a system of three coupled Gross–Pitaevskii equations with spacetime-dependent external potentials and temporally modulated gain-loss distributions.With different external potentials and gain-loss distributions,various solutions for controlled nonautonomous matterwave breathers and rogue waves are derived by the Darboux transformation method,such as breathers and rogue waves on arched and constant backgrounds which have the periodic and parabolic trajectories.Effects of the gain-loss distribution and linear potential on the breathers and rogue waves are studied.Nonautonomous two-breathers on the arched and constant backgrounds are also derived.
基金the National Natural Science Foundation of China (11871188, 12031019)。
文摘Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.
基金the NNSFC(10771139 and 10771074)NSF of Wenzhou University(2007L024)NSF of Guangdong Province(004020077)
文摘This note discusses the long time behavior of solutions for nonautonomous weakly dissipative Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary condition.The authors prove the existence of compact kernel sections for the associated process by using a suitable decomposition of the equations.
文摘In this paper, for controlling the spread of plant diseases, a nonautonomous SEIS (Susceptible → Exposed → Infectious → Susceptible) epidemic model with a general nonlinear incidence rate and time-varying impulsive control strategy is proposed and investigated. This novel model could result in an objective criterion on how to control plant disease transmission by replanting of healthy plants and removal of infected plants. Using the method of small amplitude perturbation, the sufficient conditions under which guarantee the globally attractive of the disease-free periodic solution and the permanence of the disease are obtained, that is, the disease dies out if R12>1.
文摘In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.
文摘With the help of self-similarity transformation, we construct and study the nonautonomous vortices with different topological charges inside a planar graded-index nonlinear waveguide, analytically, and numerically. Although these vortices are approximate, they can reflect the real properties of self-similar optical beam during a short-term propagation. Existence of these autonomous vortices require delicate balances between the system parameters such as diffraction, nonlinearity, gain, and external potential. We are concerned with some special but interesting situations, and discussing the changes of the height, width, energy, and central position of the vortices as the increase of propagation distance. Moreover, we are also interested in the azimuthal modulational instability of the system, and comparing our prediction for the modulational instability growth rates to numerical results.
文摘The necessary and sufficient condition of the stability of linear nonautonomous system under the frequently-acting perturbation has been given and proved on the basis of[1] and [2], and the theorem of the equivalence on the uniform and asymptotical stability in the sense of Liapunov ami the stability under the frequently-acting perturbation of linear nonautonomous system has been given in this paper. Besides, the analysis of the dynamic stabilitv of robot has been presented by applying the theorem in this paper, which is closer to reality
基金Supported by the National Natural Science Foundation of China under Grant Nos.10875106 and 11175158
文摘We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain （loss） coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.
文摘In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov's indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.
基金Project supported by the BUPT Excellent Ph.D.Students Foundation(Grant No.CX2019201)the National Natural Science Foundation of China(Grant Nos.11772017 and 11805020)+1 种基金the Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications),China(Grant No.IPOC:2017ZZ05)the Fundamental Research Funds for the Central Universities of China(Grant No.2011BUPTYB02)。
文摘Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.
基金Supported by the Fundamental Research Funds for the Central Universities under Grant No.2018MS132
文摘A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed and verified with the Wronskian technique. Collisions among the three solitons are discussed and illustrated, and effects of the coefficientsσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) on the collisions are graphically analyzed, whereσ1(x, t),σ2(x, t),σ3(x, t) and v(x, t) are the first-, second-, third-order dispersion parameters and an inhomogeneous parameter related to the phase modulation and gain(loss), respectively. The head-on collisions among the three solitons are observed, where the collisions are elastc. Whenσ1(x, t) is chosen as the function of x, amplitudes of the solitons do not alter, but the speed of one of the solitons changes.σ2(x, t) is found to affect the amplitudes and speeds of the two of the solitons. It reveals that the collision features of the solitons alter withσ3(x, t)=-1.8x. Additionally, traveling directions of the three solitons are observed to be parallel when we change the value of v(x, t).
基金supported by the Key Project of the Chinese Ministry of Education(Grant No.2011015)the Natural Science Foundation of Hebei Province of China(Grant No.A2012202023)
文摘We report the analytical nonantonomous soliton solutions （NSSs） for two-component Bose-Einstein condensates with the presence of a time-dependent potential. These solutions show that the time-dependent potential can affect the velocity of NSS. The velocity shows the characteristic of both increasing and oscillation with time. A detailed analysis for the asymptotic behavior of NSSs demonstrates that the collision of two NSSs of each component is elastic.
文摘We prove the existence and the orbital stability of standing waves for the nonau- tonomous Schrodinger system……under suitable conditions on the coefficient functions a and b. We follow the idea of analyzing the compactness of minimizing sequence of the constrained minimization problems.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175158)the Natural Science Foundation of Zhejiang Province of China(Grant No.LY12A04001)
文摘A class of analytical solitary-wave solutions to the generalized nonautonomous cubic–quintic nonlinear Schrdinger equation with time-and space-modulated coefficients and potentials are constructed using the similarity transformation technique. Constraints for the dispersion coefficient, the cubic and quintic nonlinearities, the external potential, and the gain （loss） coefficient are presented at the same time. Various shapes of analytical solitary-wave solutions which have important applications of physical interest are studied in detail, such as the solutions in Feshbach resonance management with harmonic potentials, Faraday-type waves in the optical lattice potentials, and localized solutions supported by the Gaussian-shaped nonlinearity. The stability analysis of the solutions is discussed numerically.
