A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the full...A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the fully-discrete scheme are proved.Such error estimates are obtained by combining a new discrete fractional Gr¨onwall inequality,the corresponding Sobolev embedding theorems and some inverse inequalities.While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches.Numerical examples are presented to confirm the theoretical results.展开更多
In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical expo...In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.展开更多
It is an important issue to numerically solve the time fractional Schrödinger equation on unbounded domains, which models the dynamics of optical solitons propagating via optical fibers. The perfectly matched lay...It is an important issue to numerically solve the time fractional Schrödinger equation on unbounded domains, which models the dynamics of optical solitons propagating via optical fibers. The perfectly matched layer approach is applied to truncate the unbounded physical domain, and obtain an initial boundary value problem on a bounded computational domain, which can be efficiently solved by the finite difference method. The stability of the reduced initial boundary value problem is rigorously analyzed. Some numerical results are presented to illustrate the accuracy and feasibility of the perfectly matched layer approach. According to these examples, the absorption parameters and the width of the absorption layer will affect the absorption effect. The larger the absorption width, the better the absorption effect. There is an optimal absorption parameter, the absorption effect is the best.展开更多
In this paper,with the relative Morse index,we will study the existence of solutions of(1.1)under the assumptions that V satisfies some weaker conditions than those in[2].
In this paper,we study a class of the fractional Schrodinger equations involving logarithmic and critical nonlinearities.By using the Nehari manifold method and the concentration compactness principle,we show that the...In this paper,we study a class of the fractional Schrodinger equations involving logarithmic and critical nonlinearities.By using the Nehari manifold method and the concentration compactness principle,we show that the above problem admits at least one ground state solution and one ground state sign-changing solution.Moreover,by using variational methods,we prove that how the coefficient function of the critical nonlinearity affects the number of positive solutions.The main feature which distinguishes this paper from other related works lies in the fact that it is the first attempt to study the existence and multiplicity for the above problem involving both logarithmic and critical nonlinearities.展开更多
We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional...We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument.The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions.Numerical experiments are done to verify the theoretical results.展开更多
We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,...We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.展开更多
This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the n...This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.展开更多
We introduce a new integral scheme namely improved Kudryashov method for solving any nonlinear fractional differential model.Specifically,we apply the approach to the nonlinear space-time fractional model leading the ...We introduce a new integral scheme namely improved Kudryashov method for solving any nonlinear fractional differential model.Specifically,we apply the approach to the nonlinear space-time fractional model leading the wave to spread in electrical transmission lines(s-tfETL),the time fractional complex Schrödinger(tfcS),and the space-time M-fractional Schrödinger-Hirota(s-tM-fSH)models to verify the effectiveness of the proposed approach.The implementing of the introduced new technique based on the models provides us with periodic envelope,exponentially changeable soliton envelope,rational rogue wave,periodic rogue wave,combo periodic-soliton,and combo rational-soliton solutions,which are much interesting phenomena in nonlinear sciences.Thus the results disclose that the proposed technique is very effective and straight-forward,and such solutions of the models are much more fruitful than those from the generalized Kudryashov and the modified Kudryashov methods.展开更多
In this paper, we establish exact solutions for five complex nonlinear Schr¨odinger equations. The semiinverse variational principle(SVP) is used to construct exact soliton solutions of five complex nonlinear Sch...In this paper, we establish exact solutions for five complex nonlinear Schr¨odinger equations. The semiinverse variational principle(SVP) is used to construct exact soliton solutions of five complex nonlinear Schr¨odinger equations. Many new families of exact soliton solutions of five complex nonlinear Schr¨odinger equations are successfully obtained.展开更多
We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^(1+1). We ...We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^(1+1). We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H^(s_1) and wave data in H^(s_2) × H^(s_2-1)for 3/4- α < s_1≤0 and-1/2 < s_2 < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 < s_2 < 1/2 by using the conservation law for the L^2 norm of u.展开更多
In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation.Combining the idea of two-grid discretization,the ...In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation.Combining the idea of two-grid discretization,the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space,which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid.We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator.Finally,numerical simulations are provided to verify the correctness of the theoretical analysis.展开更多
In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrdinger equation(FSE) with this form of ...In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrdinger equation(FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrdinger equation to the fractional one. When applying this form of Schrdinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well,the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrdinger equation.展开更多
In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth{(−∆)su + u = u^2∗s−1 + λ(f(x, u) + h(x)), x ∈ R^N ,u ∈ Hs(R^N ), u(x) > 0, x ∈ RN ,where s∈(0,1),N>4...In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth{(−∆)su + u = u^2∗s−1 + λ(f(x, u) + h(x)), x ∈ R^N ,u ∈ Hs(R^N ), u(x) > 0, x ∈ RN ,where s∈(0,1),N>4 s,andλ>0 is a parameter,2s*=2 N/N-2 s is the fractional critical Sobolev exponent,f and h are some given functions.We show that there exists 0<λ*<+∞such that the problem has exactly two positive solutions ifλ∈(0,λ*),no positive solutions forλ>λ*,a unique solution(λ*,uλ*)ifλ=λ*,which shows that(λ*,uλ*)is a turning point in Hs(RN)for the problem.Our proofs are based on the variational methods and the principle of concentration-compactness.展开更多
We propose improved ring shaped like potential of the form,2V(r,θ)=V(r)+(2/2M r)[(βsin2θ+γcos2θ+2λ)/sinθcosθ]and its exact solutions are presented via the Nikiforov–Uvarov method.The angle dependent part V(θ...We propose improved ring shaped like potential of the form,2V(r,θ)=V(r)+(2/2M r)[(βsin2θ+γcos2θ+2λ)/sinθcosθ]and its exact solutions are presented via the Nikiforov–Uvarov method.The angle dependent part V(θ)=(2/22M r)[(βsin22θ+γcos2θ+λ)/sinθcosθ],which is reported for the first time embodied the novel angle dependent(NAD)potential and harmonic novel angle dependent potential(HNAD)as special cases.We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.展开更多
Combining symplectic algorithm,splitting technique and compact method,a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schr¨odinger equation with cubic-quintic nonlinear term...Combining symplectic algorithm,splitting technique and compact method,a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schr¨odinger equation with cubic-quintic nonlinear term.The scheme has fourth-order accuracy in space and second-order accuracy in time.The discrete charge conservation law and stability of the scheme are analyzed.Numerical examples are given to confirm the theoretical results.展开更多
In this paper,symplectic schemes and symmetric schemes are presented to simulate Nonlinear Schrodinger Equation(NLSE)in case of dark soliton motion.Firstly,by Ablowitz–Ladik model(A–L model),the NLSE is discretized...In this paper,symplectic schemes and symmetric schemes are presented to simulate Nonlinear Schrodinger Equation(NLSE)in case of dark soliton motion.Firstly,by Ablowitz–Ladik model(A–L model),the NLSE is discretized into a non-canonical Hamiltonian system.Then,different kinds of coordinate transformations can be used to standardize the non-canonical Hamiltonian system.Therefore,the symplectic schemes and symmetric schemes can be employed to simulate the solitons motion and test the preservation of the invariants of the A–L model and the conserved quantities approximations of the original NLSE.The numerical experiments show that symplectic schemes and symmetric schemes have similar simulation effect,and own significant superiority over non-symplectic and non-symmetric schemes in long-term tracking the motion of solitons,preserving the invariants and the approximations of conserved quantities.Moreover,it is obvious that coordinate transformations with more symmetry have a better simulation effect.展开更多
In this paper we study numerical issues related to the Schr ¨odinger equationwith sinusoidal potentials at infinity. An exact absorbing boundary condition in a formof Dirichlet-to-Neumann mapping is derived. This...In this paper we study numerical issues related to the Schr ¨odinger equationwith sinusoidal potentials at infinity. An exact absorbing boundary condition in a formof Dirichlet-to-Neumann mapping is derived. This boundary condition is based on ananalytical expression of the logarithmic derivative of the Floquet solution toMathieu’sequation, which is completely new to the author’s knowledge. The implementationof this exact boundary condition is discussed, and a fast evaluation method is used toreduce the computation burden arising from the involved half-order derivative operator.Some numerical tests are given to showthe performance of the proposed absorbingboundary conditions.展开更多
基金supported by the National Natural Science Foundation of China under grants No.11971010,11771162,12231003.
文摘A linearized transformed L1 Galerkin finite element method(FEM)is presented for numerically solving the multi-dimensional time fractional Schr¨odinger equations.Unconditionally optimal error estimates of the fully-discrete scheme are proved.Such error estimates are obtained by combining a new discrete fractional Gr¨onwall inequality,the corresponding Sobolev embedding theorems and some inverse inequalities.While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches.Numerical examples are presented to confirm the theoretical results.
基金supported by the NNSF of China(12171014, 12271539, 12171326)the Beijing Municipal Commission of Education (KZ202010028048)the Research Foundation for Advanced Talents of Beijing Technology and Business University (19008022326)。
文摘In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.
文摘It is an important issue to numerically solve the time fractional Schrödinger equation on unbounded domains, which models the dynamics of optical solitons propagating via optical fibers. The perfectly matched layer approach is applied to truncate the unbounded physical domain, and obtain an initial boundary value problem on a bounded computational domain, which can be efficiently solved by the finite difference method. The stability of the reduced initial boundary value problem is rigorously analyzed. Some numerical results are presented to illustrate the accuracy and feasibility of the perfectly matched layer approach. According to these examples, the absorption parameters and the width of the absorption layer will affect the absorption effect. The larger the absorption width, the better the absorption effect. There is an optimal absorption parameter, the absorption effect is the best.
基金Supported by DEU of Henan(Grant No.19A110011)and PSF of China(Grant No.188576).
文摘In this paper,with the relative Morse index,we will study the existence of solutions of(1.1)under the assumptions that V satisfies some weaker conditions than those in[2].
基金The first author is supported by the National Natural Science Foundation of China(Grant No.12101599)the China Postdoctoral Science Foundation(Grant No.2021M703506)+2 种基金the second author is supported by National Natural Science Foundation of China(Grant Nos.11871199 and 12171152)Shandong Provincial Natural Science Foundation,P.R.China(Grant No.ZR2020MA006)Cultivation Project of Young and Innovative Talents in Universities of Shandong Province。
文摘In this paper,we study a class of the fractional Schrodinger equations involving logarithmic and critical nonlinearities.By using the Nehari manifold method and the concentration compactness principle,we show that the above problem admits at least one ground state solution and one ground state sign-changing solution.Moreover,by using variational methods,we prove that how the coefficient function of the critical nonlinearity affects the number of positive solutions.The main feature which distinguishes this paper from other related works lies in the fact that it is the first attempt to study the existence and multiplicity for the above problem involving both logarithmic and critical nonlinearities.
基金supported by the NSFC (No.12001067)by the Natural Science Foundation of Chongqing,China (No.cstc2019jcyj-bshX0038)by the China Postdoctoral Science Foundation funded Project No.2019M653333.
文摘We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument.The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions.Numerical experiments are done to verify the theoretical results.
基金Project supported by the Global Change Research Program of China(Grant No.2015CB953904)the National Natural Science Foundation of China(Grant Nos.11275072 and 11435005)+2 种基金the Doctoral Program of Higher Education of China(Grant No.20120076110024)the Network Information Physics Calculation of Basic Research Innovation Research Group of China(Grant No.61321064)Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things,China(Grant No.ZF1213)
文摘We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.
基金Jialing Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11801277)Tingchun Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11571181)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20171454)Qing Lan Project.Yushun Wang’s work is supported by the National Natural Science Foundation of China(Grant Nos.11771213 and 12171245).
文摘This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.
文摘We introduce a new integral scheme namely improved Kudryashov method for solving any nonlinear fractional differential model.Specifically,we apply the approach to the nonlinear space-time fractional model leading the wave to spread in electrical transmission lines(s-tfETL),the time fractional complex Schrödinger(tfcS),and the space-time M-fractional Schrödinger-Hirota(s-tM-fSH)models to verify the effectiveness of the proposed approach.The implementing of the introduced new technique based on the models provides us with periodic envelope,exponentially changeable soliton envelope,rational rogue wave,periodic rogue wave,combo periodic-soliton,and combo rational-soliton solutions,which are much interesting phenomena in nonlinear sciences.Thus the results disclose that the proposed technique is very effective and straight-forward,and such solutions of the models are much more fruitful than those from the generalized Kudryashov and the modified Kudryashov methods.
文摘In this paper, we establish exact solutions for five complex nonlinear Schr¨odinger equations. The semiinverse variational principle(SVP) is used to construct exact soliton solutions of five complex nonlinear Schr¨odinger equations. Many new families of exact soliton solutions of five complex nonlinear Schr¨odinger equations are successfully obtained.
基金supported by National Natural Science Foundation of China (Grant No. 11201498)
文摘We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^(1+1). We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H^(s_1) and wave data in H^(s_2) × H^(s_2-1)for 3/4- α < s_1≤0 and-1/2 < s_2 < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 < s_2 < 1/2 by using the conservation law for the L^2 norm of u.
基金the helpful suggestions.This work was supported by National Natural Science Foundation of China(Nos.11771375,11971152)Shandong Province Natural Science Foundation(No.ZR2018MA008)NSF of Henan Province(No.202300410167).
文摘In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation.Combining the idea of two-grid discretization,the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space,which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid.We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator.Finally,numerical simulations are provided to verify the correctness of the theoretical analysis.
基金Supported by National Natural Science Foundation of China under Grant Nos.11472247 and 11872335
文摘In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrdinger equation(FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrdinger equation to the fractional one. When applying this form of Schrdinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well,the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrdinger equation.
基金supported by National Natural Science Foundation of China(Grant Nos.11771468 and 11971027)supported by National Natural Science Foundation of China(Grant Nos.11771234 and 11926323)。
文摘In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth{(−∆)su + u = u^2∗s−1 + λ(f(x, u) + h(x)), x ∈ R^N ,u ∈ Hs(R^N ), u(x) > 0, x ∈ RN ,where s∈(0,1),N>4 s,andλ>0 is a parameter,2s*=2 N/N-2 s is the fractional critical Sobolev exponent,f and h are some given functions.We show that there exists 0<λ*<+∞such that the problem has exactly two positive solutions ifλ∈(0,λ*),no positive solutions forλ>λ*,a unique solution(λ*,uλ*)ifλ=λ*,which shows that(λ*,uλ*)is a turning point in Hs(RN)for the problem.Our proofs are based on the variational methods and the principle of concentration-compactness.
文摘We propose improved ring shaped like potential of the form,2V(r,θ)=V(r)+(2/2M r)[(βsin2θ+γcos2θ+2λ)/sinθcosθ]and its exact solutions are presented via the Nikiforov–Uvarov method.The angle dependent part V(θ)=(2/22M r)[(βsin22θ+γcos2θ+λ)/sinθcosθ],which is reported for the first time embodied the novel angle dependent(NAD)potential and harmonic novel angle dependent potential(HNAD)as special cases.We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.
基金This work is partially supported by grants from the National Natural Science Foun-dation of China(Nos.11701196,11701197 and 11501224)the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(No.ZQN-YX502)the Key Laboratory of Applied Mathematics of Fujian Province University(Putian University)(No.SX201802).
文摘Combining symplectic algorithm,splitting technique and compact method,a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schr¨odinger equation with cubic-quintic nonlinear term.The scheme has fourth-order accuracy in space and second-order accuracy in time.The discrete charge conservation law and stability of the scheme are analyzed.Numerical examples are given to confirm the theoretical results.
基金This work was supported by the Fundamental Research Funds for the Central Universities(Nos.2018ZY14,2019ZY20 and 2015ZCQ-LY-01)Beijing Higher Education Young Elite Teacher Project(YETP0769)the National Natural Science Foundation of China(Grant Nos.61571002,61179034 and 61370193).
文摘In this paper,symplectic schemes and symmetric schemes are presented to simulate Nonlinear Schrodinger Equation(NLSE)in case of dark soliton motion.Firstly,by Ablowitz–Ladik model(A–L model),the NLSE is discretized into a non-canonical Hamiltonian system.Then,different kinds of coordinate transformations can be used to standardize the non-canonical Hamiltonian system.Therefore,the symplectic schemes and symmetric schemes can be employed to simulate the solitons motion and test the preservation of the invariants of the A–L model and the conserved quantities approximations of the original NLSE.The numerical experiments show that symplectic schemes and symmetric schemes have similar simulation effect,and own significant superiority over non-symplectic and non-symmetric schemes in long-term tracking the motion of solitons,preserving the invariants and the approximations of conserved quantities.Moreover,it is obvious that coordinate transformations with more symmetry have a better simulation effect.
基金the National Natural Science Foundation of China underGrant No. 10401020.
文摘In this paper we study numerical issues related to the Schr ¨odinger equationwith sinusoidal potentials at infinity. An exact absorbing boundary condition in a formof Dirichlet-to-Neumann mapping is derived. This boundary condition is based on ananalytical expression of the logarithmic derivative of the Floquet solution toMathieu’sequation, which is completely new to the author’s knowledge. The implementationof this exact boundary condition is discussed, and a fast evaluation method is used toreduce the computation burden arising from the involved half-order derivative operator.Some numerical tests are given to showthe performance of the proposed absorbingboundary conditions.