Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities.Moiréphotonic and optical lattices—two-dimen...Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities.Moiréphotonic and optical lattices—two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones—are a new emerging investigative tool for studying nonlinear localized waves of diverse types.Herein,a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction(linear nonlocality)and moiréoptical lattices is investigated.Specifically,the flat-band feature is well preserved in shallow moiréoptical lattices which,interact with the defocusing nonlinearity of the media,can support fundamental gap solitons,bound states composed of several fundamental solitons,and topological states(gap vortices)with vortex charge s=1 and 2,all populated inside the finite gaps of the linear Bloch-wave spectrum.Employing the linear-stability analysis and direct perturbed simulations,the stability and instability properties of all the localized gap modes are surveyed,highlighting a wide stability region within the first gap and a limited one(to the central part)for the third gap.The findings enable insightful studies of highly localized gap modes in linear nonlocality(fractional)physical systems with shallow moirépatterns that exhibit extremely flat bands.展开更多
This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study’s core obj...This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study’s core objectives include comparing software performance using standardized benchmarks, employing key performance metrics for quantitative assessment, and examining the influence of varying hardware specifications on software efficiency across HP ProBook, HP EliteBook, Dell Inspiron, and Dell Latitude laptops. Results from this investigation reveal insights into the capabilities of these software tools in diverse computing environments. On the HP ProBook, Python consistently outperforms MATLAB in terms of computational time. Python also exhibits a lower robustness index for problems 3 and 5 but matches or surpasses MATLAB for problem 1, for some initial guess values. In contrast, on the HP EliteBook, MATLAB consistently exhibits shorter computational times than Python across all benchmark problems. However, Python maintains a lower robustness index for most problems, except for problem 3, where MATLAB performs better. A notable challenge is Python’s failure to converge for problem 4 with certain initial guess values, while MATLAB succeeds in producing results. Analysis on the Dell Inspiron reveals a split in strengths. Python demonstrates superior computational efficiency for some problems, while MATLAB excels in handling others. This pattern extends to the robustness index, with Python showing lower values for some problems, and MATLAB achieving the lowest indices for other problems. In conclusion, this research offers valuable insights into the comparative performance of Python, MATLAB, and Scilab in solving nonlinear systems of equations. It underscores the importance of considering both software and hardware specifications in real-world applications. The choice between Python and MATLAB can yield distinct advantages depending on the specific problem and computational environment, providing guidance for researchers and practitioners in selecting tools for their unique challenges.展开更多
Safety critical control is often trained in a simulated environment to mitigate risk.Subsequent migration of the biased controller requires further adjustments.In this paper,an experience inference human-behavior lear...Safety critical control is often trained in a simulated environment to mitigate risk.Subsequent migration of the biased controller requires further adjustments.In this paper,an experience inference human-behavior learning is proposed to solve the migration problem of optimal controllers applied to real-world nonlinear systems.The approach is inspired in the complementary properties that exhibits the hippocampus,the neocortex,and the striatum learning systems located in the brain.The hippocampus defines a physics informed reference model of the realworld nonlinear system for experience inference and the neocortex is the adaptive dynamic programming(ADP)or reinforcement learning(RL)algorithm that ensures optimal performance of the reference model.This optimal performance is inferred to the real-world nonlinear system by means of an adaptive neocortex/striatum control policy that forces the nonlinear system to behave as the reference model.Stability and convergence of the proposed approach is analyzed using Lyapunov stability theory.Simulation studies are carried out to verify the approach.展开更多
The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obta...The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.展开更多
A block diagonal form about a nonlinear system is defined. Based on the de finition, a design method ca1led block diagonal controller (BDC) is proPOsed bo feedbacklinearization. Thus, a linearization design of a class...A block diagonal form about a nonlinear system is defined. Based on the de finition, a design method ca1led block diagonal controller (BDC) is proPOsed bo feedbacklinearization. Thus, a linearization design of a class of nonlinear system can be simply re-alized. The result of design has been proved by mathematical simulation of a certain anti-ship missile.展开更多
It is shown that there exists Λ>0 such that, for every λ∈(0,Λ), the semilinear elliptic system: - Δ u=λu|u| q-1 +u|u| p-1 -v inΩ, - Δ v=δu-γv in Ω, u=v=0 on Ω, where Ω∈R N(N≥2) is ...It is shown that there exists Λ>0 such that, for every λ∈(0,Λ), the semilinear elliptic system: - Δ u=λu|u| q-1 +u|u| p-1 -v inΩ, - Δ v=δu-γv in Ω, u=v=0 on Ω, where Ω∈R N(N≥2) is a bounded domain with smooth boundary and 0<q<1<p,has a minimal positive solution (u λ,v λ). Moreover: u λ and v λ are strictly increasing with respect to λ.展开更多
A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable syste...A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.展开更多
A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations u' = nu, epsilon nu' + f(x, u, u')nu' - g(x, ...A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations u' = nu, epsilon nu' + f(x, u, u')nu' - g(x, u, u') nu = 0 (0 < epsilon much less than 1). The asymptotic expansions of solutions are constructed, the remainders are estimated. The former works are improved and generalized.展开更多
The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a syst...The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.展开更多
This paper presents a new decomposition method for solving large-scale systems of nonlinear equations. The new method is of superlinear convergence speed and has rather less computa tional complexity than the Newton-t...This paper presents a new decomposition method for solving large-scale systems of nonlinear equations. The new method is of superlinear convergence speed and has rather less computa tional complexity than the Newton-type decomposition method as well as other known numerical methods, Primal numerical experiments show the superiority of the new method to the others.展开更多
In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximat...In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.展开更多
In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method...In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.展开更多
The quaternion approach to solve the coupled nonlinear Schrodinger equations (CNSEs) in fibers is proposed, converting the CNSEs to a single variable equation by using a conception of eigen-quaternion of coupled qua...The quaternion approach to solve the coupled nonlinear Schrodinger equations (CNSEs) in fibers is proposed, converting the CNSEs to a single variable equation by using a conception of eigen-quaternion of coupled quater- nion. The crosstalk of quarter-phase-shift-key signals caused by fiber nonlinearity in polarization multiplexing systems with 100 Cbps bit-rate is investigated and simulated. The results demonstrate that the crosstalk is like a rotated ghosting of input constellation. For the 50 km conventional fiber link, when the total power is less than 4roW, the crosstalk effect can be neglected; when the power is larger than 20roW, the crosstalk is very obvious. In addition, the crosstalk can not be detected according to the output eye diagram and state of polarization in Poincare sphere in the trunk fiber, making it difficult for the monitoring of optical trunk link.展开更多
On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparis...On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C, is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.展开更多
Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-soliton solutions are actually visual artifacts. The so-called loopsoliton solution consists of three solutions, and is ...Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-soliton solutions are actually visual artifacts. The so-called loopsoliton solution consists of three solutions, and is not a real solution. This paper answers the question as to whether or not nonlinear wave equations exist for which a "real" loopsolution exists, and if so, what are the precise parametric representations of these loop traveling wave solutions.展开更多
According to the improved sine-cosine method and Wu-elimination method, a new algorithm to construct solitary wave solutions for systems of nonlinear evolution equations is put forward. The algorithm has some conclusi...According to the improved sine-cosine method and Wu-elimination method, a new algorithm to construct solitary wave solutions for systems of nonlinear evolution equations is put forward. The algorithm has some conclusions which are better than what the hyperbolic function method known does and simpler in use. With the aid of MATHEMATICA, the algorithm can be carried out in computer.展开更多
In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for ...In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.展开更多
基金This work was supported by the National Natural Science Foundation of China(NSFC)(No.12074423)Young Scholar of Chinese Academy of Sciences in Western China(No.XAB2021YN18)China Postdoctoral Science Foundation(No.2023M733722).
文摘Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities.Moiréphotonic and optical lattices—two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones—are a new emerging investigative tool for studying nonlinear localized waves of diverse types.Herein,a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction(linear nonlocality)and moiréoptical lattices is investigated.Specifically,the flat-band feature is well preserved in shallow moiréoptical lattices which,interact with the defocusing nonlinearity of the media,can support fundamental gap solitons,bound states composed of several fundamental solitons,and topological states(gap vortices)with vortex charge s=1 and 2,all populated inside the finite gaps of the linear Bloch-wave spectrum.Employing the linear-stability analysis and direct perturbed simulations,the stability and instability properties of all the localized gap modes are surveyed,highlighting a wide stability region within the first gap and a limited one(to the central part)for the third gap.The findings enable insightful studies of highly localized gap modes in linear nonlocality(fractional)physical systems with shallow moirépatterns that exhibit extremely flat bands.
文摘This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study’s core objectives include comparing software performance using standardized benchmarks, employing key performance metrics for quantitative assessment, and examining the influence of varying hardware specifications on software efficiency across HP ProBook, HP EliteBook, Dell Inspiron, and Dell Latitude laptops. Results from this investigation reveal insights into the capabilities of these software tools in diverse computing environments. On the HP ProBook, Python consistently outperforms MATLAB in terms of computational time. Python also exhibits a lower robustness index for problems 3 and 5 but matches or surpasses MATLAB for problem 1, for some initial guess values. In contrast, on the HP EliteBook, MATLAB consistently exhibits shorter computational times than Python across all benchmark problems. However, Python maintains a lower robustness index for most problems, except for problem 3, where MATLAB performs better. A notable challenge is Python’s failure to converge for problem 4 with certain initial guess values, while MATLAB succeeds in producing results. Analysis on the Dell Inspiron reveals a split in strengths. Python demonstrates superior computational efficiency for some problems, while MATLAB excels in handling others. This pattern extends to the robustness index, with Python showing lower values for some problems, and MATLAB achieving the lowest indices for other problems. In conclusion, this research offers valuable insights into the comparative performance of Python, MATLAB, and Scilab in solving nonlinear systems of equations. It underscores the importance of considering both software and hardware specifications in real-world applications. The choice between Python and MATLAB can yield distinct advantages depending on the specific problem and computational environment, providing guidance for researchers and practitioners in selecting tools for their unique challenges.
基金supported by the Royal Academy of Engineering and the Office of the Chie Science Adviser for National Security under the UK Intelligence Community Postdoctoral Research Fellowship programme。
文摘Safety critical control is often trained in a simulated environment to mitigate risk.Subsequent migration of the biased controller requires further adjustments.In this paper,an experience inference human-behavior learning is proposed to solve the migration problem of optimal controllers applied to real-world nonlinear systems.The approach is inspired in the complementary properties that exhibits the hippocampus,the neocortex,and the striatum learning systems located in the brain.The hippocampus defines a physics informed reference model of the realworld nonlinear system for experience inference and the neocortex is the adaptive dynamic programming(ADP)or reinforcement learning(RL)algorithm that ensures optimal performance of the reference model.This optimal performance is inferred to the real-world nonlinear system by means of an adaptive neocortex/striatum control policy that forces the nonlinear system to behave as the reference model.Stability and convergence of the proposed approach is analyzed using Lyapunov stability theory.Simulation studies are carried out to verify the approach.
基金supported by National Natural Science Foundation of China(61273016)The Natural Science Foundation of Zhejiang Province(Y6100016)The Public Welfare Technology Application Research Project of Zhejiang Province Science and Technology Department(2015C33088)
文摘The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.
文摘A block diagonal form about a nonlinear system is defined. Based on the de finition, a design method ca1led block diagonal controller (BDC) is proPOsed bo feedbacklinearization. Thus, a linearization design of a class of nonlinear system can be simply re-alized. The result of design has been proved by mathematical simulation of a certain anti-ship missile.
基金Supported by National Basic Research Program of China (973 Program) (2010CB731800), National Natural Science Foundation of China (60934003, 61074065), Key Project for Natural Science Research of Hebei Education Department (ZD200908), and the Doctor Foundation of Northeastern University at Qinhuangdao(XNB201507)
文摘It is shown that there exists Λ>0 such that, for every λ∈(0,Λ), the semilinear elliptic system: - Δ u=λu|u| q-1 +u|u| p-1 -v inΩ, - Δ v=δu-γv in Ω, u=v=0 on Ω, where Ω∈R N(N≥2) is a bounded domain with smooth boundary and 0<q<1<p,has a minimal positive solution (u λ,v λ). Moreover: u λ and v λ are strictly increasing with respect to λ.
文摘A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.
基金The project supported by National Natural Science Foundation of China under Grant No.10671156the Program for New CenturyExcellent Talents in Universities under Grant No.NCET-04-0968
文摘A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations u' = nu, epsilon nu' + f(x, u, u')nu' - g(x, u, u') nu = 0 (0 < epsilon much less than 1). The asymptotic expansions of solutions are constructed, the remainders are estimated. The former works are improved and generalized.
文摘The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.
文摘This paper presents a new decomposition method for solving large-scale systems of nonlinear equations. The new method is of superlinear convergence speed and has rather less computa tional complexity than the Newton-type decomposition method as well as other known numerical methods, Primal numerical experiments show the superiority of the new method to the others.
基金Supported by the Qianjiang Rencai Project Foundation of Zhejiang Province (J20070288)
文摘In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.
文摘In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.
基金Supported by the National Natural Science Foundation of China under Grant No 61275075the Beijing Natural Science Foundation under Grant Nos 4132035 and 4144080
文摘The quaternion approach to solve the coupled nonlinear Schrodinger equations (CNSEs) in fibers is proposed, converting the CNSEs to a single variable equation by using a conception of eigen-quaternion of coupled quater- nion. The crosstalk of quarter-phase-shift-key signals caused by fiber nonlinearity in polarization multiplexing systems with 100 Cbps bit-rate is investigated and simulated. The results demonstrate that the crosstalk is like a rotated ghosting of input constellation. For the 50 km conventional fiber link, when the total power is less than 4roW, the crosstalk effect can be neglected; when the power is larger than 20roW, the crosstalk is very obvious. In addition, the crosstalk can not be detected according to the output eye diagram and state of polarization in Poincare sphere in the trunk fiber, making it difficult for the monitoring of optical trunk link.
文摘On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C, is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.
基金supported by the National Natural Science Foundation of China (Nos.10671179 and 10831003)
文摘Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-soliton solutions are actually visual artifacts. The so-called loopsoliton solution consists of three solutions, and is not a real solution. This paper answers the question as to whether or not nonlinear wave equations exist for which a "real" loopsolution exists, and if so, what are the precise parametric representations of these loop traveling wave solutions.
文摘According to the improved sine-cosine method and Wu-elimination method, a new algorithm to construct solitary wave solutions for systems of nonlinear evolution equations is put forward. The algorithm has some conclusions which are better than what the hyperbolic function method known does and simpler in use. With the aid of MATHEMATICA, the algorithm can be carried out in computer.
文摘In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.