In this paper,a self-adaptive method for the Maxwell’s Equations Derived Optimization(MEDO)is proposed.It is implemented by applying the Sequential Model-Based Optimization(SMBO)algorithm to the iterations of the MED...In this paper,a self-adaptive method for the Maxwell’s Equations Derived Optimization(MEDO)is proposed.It is implemented by applying the Sequential Model-Based Optimization(SMBO)algorithm to the iterations of the MEDO,and achieves the automatic adjustment of the parameters.The proposed method is named as adaptive Maxwell’s equations derived optimization(AMEDO).In order to evaluate the performance of AMEDO,eight benchmarks are used and the results are compared with the original MEDO method.The results show that AMEDO can greatly reduce the workload of manual adjustment of parameters,and at the same time can keep the accuracy and stability.Moreover,the convergence of the optimization can be accelerated due to the dynamical adjustment of the parameters.In the end,the proposed AMEDO is applied to the side lobe level suppression and array failure correction of a linear antenna array,and shows great potential in antenna array synthesis.展开更多
A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid-liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced....A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid-liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie-Ericksen theory is described by the first Rivlin-Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion-extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extru- date of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a seriesof new anisotropic non-Newtonian fluid problems can be addressed.展开更多
This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouvill...This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.展开更多
In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT pre...In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.展开更多
In this paper we prove a global attractivity result for the unique positive equilibrium point of a difference equation,which improves and generalizes some known ones in the existing literature.Especially,our results c...In this paper we prove a global attractivity result for the unique positive equilibrium point of a difference equation,which improves and generalizes some known ones in the existing literature.Especially,our results completely solve an open problem and some conjectures proposed in[1,2,3,4].展开更多
It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment...It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.展开更多
The mixed solutions of the derivative nonlinear Schrödinger equation from the trivial seed (zero solution) are derived by using the determinant representation. By adjusting the interaction and degeneracy of m...The mixed solutions of the derivative nonlinear Schrödinger equation from the trivial seed (zero solution) are derived by using the determinant representation. By adjusting the interaction and degeneracy of mixed solutions, it is possible to obtain different types of solutions: phase solutions, breather solutions, phase-breather solutions and rogue waves.展开更多
We study the Peregrine rogue waves within the framework of Derivative Nonlinear Schrödinger equation, which is used to describe the propagation of Alfven waves in plasma physics and sub-picosecond or femtosecond ...We study the Peregrine rogue waves within the framework of Derivative Nonlinear Schrödinger equation, which is used to describe the propagation of Alfven waves in plasma physics and sub-picosecond or femtosecond pulses in nonlinear optics. The interaction and degeneration of two soliton-like solutions and its relations for the breather solution have been analyzed. The Peregrine rogue waves have been considered from the two kinds of formation processes: it can be generated through the limitation of the infinitely large period of the breather solutions, and it can be interpreted as the soliton-like solutions with different polarities. As a special example, a special Peregrine rogue wave is generated by a breather solution and phase solution, which is given by the trivial seed (zero solution).展开更多
In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of...In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.展开更多
Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,...Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.展开更多
In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are sit...In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are situated in the solution function. As a result, various exact analytical solutions consisting of single and combined Jacobi elliptic functions solutions are obtained.展开更多
Based on the method of Hirota's bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The oneand two-soliton solutions are given as t...Based on the method of Hirota's bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The oneand two-soliton solutions are given as two typical examples in the illustration of the general procedures and the concrete cut-off technique of the series-form solution, and the n-soliton solution is also attained by induction method. Our study shows their equivalence to the existing soliton solutions by a simple parameter transformation. The methodological importance of bilinear derivative transform in dealing with an integrable nonlinear equation has also been emphasized. The evolution of one and two-soliton solution with respect to time and space has been discussed in detail. The collision among the solitons has been manifested through an example of two-soliton case, revealing the elastic essence of the collision and the invariance of the soliton form and characteristics.展开更多
In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obta...In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved.展开更多
A newly revised inverse scattering transform(IST) for the derivative nonlinear Schrdinger(DNLS+) equation with non-vanishing boundary condition(NVBC) and normal group velocity dispersion is proposed by introduc...A newly revised inverse scattering transform(IST) for the derivative nonlinear Schrdinger(DNLS+) equation with non-vanishing boundary condition(NVBC) and normal group velocity dispersion is proposed by introducing a suitable affine parameter in Zakharov-Shabat integral kern.The explicit breather-type one-soliton solution,which can reproduce one pure soliton at the de-generate case and one bright soliton solution at the limit of van-ishing boundary,has been derived to verify the validity of the revised IST.展开更多
The truncation equation for the derivative nonlinear Schrodinger equation has been dis- cussed in this paper. The existence of a special heteroclinic orbit has been found by using geometrical singular perturbation the...The truncation equation for the derivative nonlinear Schrodinger equation has been dis- cussed in this paper. The existence of a special heteroclinic orbit has been found by using geometrical singular perturbation theory together with Melnikov's technique.展开更多
By means of some algebraic techniques,especially the Binet-Cauchy formula,an explicit multi-soliton solution of the derivative nonlinear Schrdinger equation with vanishing boundary condition is attained based on a pur...By means of some algebraic techniques,especially the Binet-Cauchy formula,an explicit multi-soliton solution of the derivative nonlinear Schrdinger equation with vanishing boundary condition is attained based on a pure Marchenko formalism without needing the usual scattering data except for given N simple poles. The one-and two-soliton solutions are given as two special examples in illustration of the general formula of multi-soliton solution. Their effectiveness and equivalence to other approaches are also demonstrated. Meanwhile,the asymptotic behavior of the multi-soliton solution is discussed in detail. It is shown that the N-soliton solution can be viewed as a summation of N one-soliton solutions with a definite displacement and phase shift of each soliton in the whole process(from t →∞ to t → +∞ ) of the elastic collisions.展开更多
In this paper,we consider the initial value problem of a class of fractional differential equations.Firstly,we obtain the existence and uniqueness of the solutions by using Picard’s method of successive approximation...In this paper,we consider the initial value problem of a class of fractional differential equations.Firstly,we obtain the existence and uniqueness of the solutions by using Picard’s method of successive approximation.Then we discuss the dependence of the solutions on the initial value.展开更多
In this work,a fourth-order numerical scheme in space and two second-order numerical schemes in both time and space are proposed for the derivative nonlinear Schrodinger equation.We verify the mass conservation for th...In this work,a fourth-order numerical scheme in space and two second-order numerical schemes in both time and space are proposed for the derivative nonlinear Schrodinger equation.We verify the mass conservation for the two-level implicit scheme.The influence on the soliton solution by adding a small random perturbation to the initial condition is discussed.The numerical experiments are given to test the accuracy order for different schemes,respectively.We also test the conservative property of mass and Hamiltonian for these schemes from the numerical point of view.展开更多
The derivative nonlinear Schrodinger equation, which is extensively applied in plasma physics and nonlinear optics, is analytically studied by Hirota method. Space periodic solutions are determined by means of Hirota...The derivative nonlinear Schrodinger equation, which is extensively applied in plasma physics and nonlinear optics, is analytically studied by Hirota method. Space periodic solutions are determined by means of Hirota's bilinear formalism, and the rogue wave solution is derived as a long-wave limit of the space periodic solution.展开更多
This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equati...This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].展开更多
基金the National Nature Science Foundation of China(No.61427803).
文摘In this paper,a self-adaptive method for the Maxwell’s Equations Derived Optimization(MEDO)is proposed.It is implemented by applying the Sequential Model-Based Optimization(SMBO)algorithm to the iterations of the MEDO,and achieves the automatic adjustment of the parameters.The proposed method is named as adaptive Maxwell’s equations derived optimization(AMEDO).In order to evaluate the performance of AMEDO,eight benchmarks are used and the results are compared with the original MEDO method.The results show that AMEDO can greatly reduce the workload of manual adjustment of parameters,and at the same time can keep the accuracy and stability.Moreover,the convergence of the optimization can be accelerated due to the dynamical adjustment of the parameters.In the end,the proposed AMEDO is applied to the side lobe level suppression and array failure correction of a linear antenna array,and shows great potential in antenna array synthesis.
基金the National Natural Science Foundation of China(10372100,19832050)(Key project).
文摘A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid-liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie-Ericksen theory is described by the first Rivlin-Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion-extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extru- date of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a seriesof new anisotropic non-Newtonian fluid problems can be addressed.
基金partially supportedby Ministerio de Ciencia e Innovacion-SPAINFEDER,project MTM2010-15314supported by the Ministry of Science and Education of the Republic of Kazakhstan through the Project No.0713 GF
文摘This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.
文摘In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.
基金the National Natural Science Foundation of China(61473340)the Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province+1 种基金the National Natural Science Foundation of Zhejiang Province(LQ13A010019)the National Natural Science Foundation of Zhejiang University of Science and Technology(F701108G14).
文摘In this paper we prove a global attractivity result for the unique positive equilibrium point of a difference equation,which improves and generalizes some known ones in the existing literature.Especially,our results completely solve an open problem and some conjectures proposed in[1,2,3,4].
文摘It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.
基金supported by the National Natural Science Foundation of China under Grant No.11601187 and Major SRT Project of Jiaxing University.
文摘The mixed solutions of the derivative nonlinear Schrödinger equation from the trivial seed (zero solution) are derived by using the determinant representation. By adjusting the interaction and degeneracy of mixed solutions, it is possible to obtain different types of solutions: phase solutions, breather solutions, phase-breather solutions and rogue waves.
文摘We study the Peregrine rogue waves within the framework of Derivative Nonlinear Schrödinger equation, which is used to describe the propagation of Alfven waves in plasma physics and sub-picosecond or femtosecond pulses in nonlinear optics. The interaction and degeneration of two soliton-like solutions and its relations for the breather solution have been analyzed. The Peregrine rogue waves have been considered from the two kinds of formation processes: it can be generated through the limitation of the infinitely large period of the breather solutions, and it can be interpreted as the soliton-like solutions with different polarities. As a special example, a special Peregrine rogue wave is generated by a breather solution and phase solution, which is given by the trivial seed (zero solution).
基金Foundation item:The NSF(11071097,11101217)of Chinathe NSF(BK20141476)of Jiangsu Province of China
文摘In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.
基金supported by the National Natural Science Foundation of China(Grants 11372354 and 10825207)
文摘Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.
文摘In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are situated in the solution function. As a result, various exact analytical solutions consisting of single and combined Jacobi elliptic functions solutions are obtained.
基金Supported by the National Natural Science Foundation of China (10775105)
文摘Based on the method of Hirota's bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The oneand two-soliton solutions are given as two typical examples in the illustration of the general procedures and the concrete cut-off technique of the series-form solution, and the n-soliton solution is also attained by induction method. Our study shows their equivalence to the existing soliton solutions by a simple parameter transformation. The methodological importance of bilinear derivative transform in dealing with an integrable nonlinear equation has also been emphasized. The evolution of one and two-soliton solution with respect to time and space has been discussed in detail. The collision among the solitons has been manifested through an example of two-soliton case, revealing the elastic essence of the collision and the invariance of the soliton form and characteristics.
基金the National Natural Science Foundation of China (Nos.10432010 and 10571010)
文摘In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved.
基金Supported by the National Natural Science Foundation of China(10775105)
文摘A newly revised inverse scattering transform(IST) for the derivative nonlinear Schrdinger(DNLS+) equation with non-vanishing boundary condition(NVBC) and normal group velocity dispersion is proposed by introducing a suitable affine parameter in Zakharov-Shabat integral kern.The explicit breather-type one-soliton solution,which can reproduce one pure soliton at the de-generate case and one bright soliton solution at the limit of van-ishing boundary,has been derived to verify the validity of the revised IST.
文摘The truncation equation for the derivative nonlinear Schrodinger equation has been dis- cussed in this paper. The existence of a special heteroclinic orbit has been found by using geometrical singular perturbation theory together with Melnikov's technique.
基金Supported by the National Natural Science Foundation of China (10775105)
文摘By means of some algebraic techniques,especially the Binet-Cauchy formula,an explicit multi-soliton solution of the derivative nonlinear Schrdinger equation with vanishing boundary condition is attained based on a pure Marchenko formalism without needing the usual scattering data except for given N simple poles. The one-and two-soliton solutions are given as two special examples in illustration of the general formula of multi-soliton solution. Their effectiveness and equivalence to other approaches are also demonstrated. Meanwhile,the asymptotic behavior of the multi-soliton solution is discussed in detail. It is shown that the N-soliton solution can be viewed as a summation of N one-soliton solutions with a definite displacement and phase shift of each soliton in the whole process(from t →∞ to t → +∞ ) of the elastic collisions.
基金by the National Natural Science Foundation of China(Nos.11571159,11531010)by the Natural Science Foundation of Fujian Province(Nos.2017J01562).
文摘In this paper,we consider the initial value problem of a class of fractional differential equations.Firstly,we obtain the existence and uniqueness of the solutions by using Picard’s method of successive approximation.Then we discuss the dependence of the solutions on the initial value.
基金National Natural Science Foundation of China(No.11671044)Beijing Municipal Education Commission under Grant(No.PXM2016014224000028)the Science Challenge project(No.JCKY2016212A501).
文摘In this work,a fourth-order numerical scheme in space and two second-order numerical schemes in both time and space are proposed for the derivative nonlinear Schrodinger equation.We verify the mass conservation for the two-level implicit scheme.The influence on the soliton solution by adding a small random perturbation to the initial condition is discussed.The numerical experiments are given to test the accuracy order for different schemes,respectively.We also test the conservative property of mass and Hamiltonian for these schemes from the numerical point of view.
基金Supported by the Teaching Steering Committee Research Project of Higher-Learning Institutions of Ministry of Education(JZW-16-DD-15)
文摘The derivative nonlinear Schrodinger equation, which is extensively applied in plasma physics and nonlinear optics, is analytically studied by Hirota method. Space periodic solutions are determined by means of Hirota's bilinear formalism, and the rogue wave solution is derived as a long-wave limit of the space periodic solution.
基金Supported by NSFC(Grant Nos.11601487 and 11526189)
文摘This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].