By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrdinger equation are derived. In the limit c...By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrdinger equation are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.展开更多
An exact closed form of solution to the hyperradial Schrdinger equation is constructed for any generalcase comprising any hypercentral power and inverse-power potential.The hypercentral potential depends only on thehy...An exact closed form of solution to the hyperradial Schrdinger equation is constructed for any generalcase comprising any hypercentral power and inverse-power potential.The hypercentral potential depends only on thehyperradius,which itself is a function of Jacobi relative coordinates that are functions of particle positions(r_1,r_2,…...,r_N).This article is mainly devoted to the dernonstrat of the fact that any ψ of the form ψ=power series×exp(polynomial)=[f(x)exp(g(x))]is potentially a solution of the Schrdinger equation,where the polynomial g(x)is an ansatz dependingon the interaction potential.展开更多
We consider the existence of infinitely many sign-changing solutions for the nonlinear time-independent schrodinger equations of the form where Vλ(x) =λa(x) + 1. This problem originates from various problems in phys...We consider the existence of infinitely many sign-changing solutions for the nonlinear time-independent schrodinger equations of the form where Vλ(x) =λa(x) + 1. This problem originates from various problems in physics and mathematical physics. In constructive field theory, (1.1) is called a nonlinear Euclidean scalar field equation. In chemical dynamics, a solution of (1.1) is a stationary state of the reaction diffusion展开更多
In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al...In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.展开更多
In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercriti...In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.展开更多
We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equ...We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equation,we obtain the Manakov model of all-focusing,all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions.Especially,we find that the all-defocusing type Manakov model admits bright-bright soliton solutions.Furthermore,we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions.Therefrom,we go over how the free parameters affect the Manakov model’s allfocusing type’s two-soliton and three-soliton solutions’collision locations,propagation directions,and wave amplitudes.These findings are useful for setting a simulation scene in Rossby waves research.The answers we have found are helpful for studying physical properties of the equation in Rossby waves.展开更多
We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NL...We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.展开更多
We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localiz...We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.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
We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota b...We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.展开更多
We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(R...We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(RK4IP-CQE), and the other is the local error adaptive step-control method(RK4IP-LEM). The methods are developed in the vector form of fourthorder Runge–Kutta iterative scheme in the interaction picture by converting a vector equation in frequency domain. By simulating the supercontinuum generated from the high birefringence photonic crystal fiber, the calculation accuracies and the efficiencies of the two adaptive step-size methods are discussed. The simulation results show that the two methods have the same global average error, while RK4IP-LEM spends more time than RK4IP-CQE. The decrease of huge calculation time is due to the differences in the convergences of the relative photon number error and the approximated local error between these two adaptive step-size algorithms.展开更多
In this work, we prove the existence and uniqueness of the solution of the generalized Schrödinger equation in the periodic distributional space P’. Furthermore, we prove that the solution depends continuously r...In this work, we prove the existence and uniqueness of the solution of the generalized Schrödinger equation in the periodic distributional space P’. Furthermore, we prove that the solution depends continuously respect to the initial data in P’. Introducing a family of weakly continuous operators, we prove that this family is a semigroup of operators in P’. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we provide some consequences of this study.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, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger...In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.展开更多
The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odin...The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.展开更多
This paper studies the stability of Schr¨odinger equation with boundary control matched disturbance. The time-varying gain extended state observer is utilized to estimate disturbance and state. Meanwhile, the aut...This paper studies the stability of Schr¨odinger equation with boundary control matched disturbance. The time-varying gain extended state observer is utilized to estimate disturbance and state. Meanwhile, the authors get a continuous controller by the active disturbance rejection control strategy, which shows that the closed-loop system of Schr¨odinger equation is asymptotically stable.These results are illustrated by simulation examples.展开更多
The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid...The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.展开更多
文摘By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrdinger equation are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.
文摘An exact closed form of solution to the hyperradial Schrdinger equation is constructed for any generalcase comprising any hypercentral power and inverse-power potential.The hypercentral potential depends only on thehyperradius,which itself is a function of Jacobi relative coordinates that are functions of particle positions(r_1,r_2,…...,r_N).This article is mainly devoted to the dernonstrat of the fact that any ψ of the form ψ=power series×exp(polynomial)=[f(x)exp(g(x))]is potentially a solution of the Schrdinger equation,where the polynomial g(x)is an ansatz dependingon the interaction potential.
文摘We consider the existence of infinitely many sign-changing solutions for the nonlinear time-independent schrodinger equations of the form where Vλ(x) =λa(x) + 1. This problem originates from various problems in physics and mathematical physics. In constructive field theory, (1.1) is called a nonlinear Euclidean scalar field equation. In chemical dynamics, a solution of (1.1) is a stationary state of the reaction diffusion
基金supported by the National Science Foundation grant DMS-1818998.
文摘In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.
文摘In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.12102205 and 12161065)the Scientific Research Ability of Youth Teachers of Inner Mongolia Agricultural University(Grant Nos.JC2021001 and BR220126)+1 种基金the Natural Science Foundation of Inner Mongolia Autonomous Region of China(Grant No.2022QN01003)the Research Program of Inner Mongolia Autonomous Region Education Department(Grant Nos.NJYT23099 and NMGIRT2208).
文摘We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equation,we obtain the Manakov model of all-focusing,all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions.Especially,we find that the all-defocusing type Manakov model admits bright-bright soliton solutions.Furthermore,we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions.Therefrom,we go over how the free parameters affect the Manakov model’s allfocusing type’s two-soliton and three-soliton solutions’collision locations,propagation directions,and wave amplitudes.These findings are useful for setting a simulation scene in Rossby waves research.The answers we have found are helpful for studying physical properties of the equation in Rossby waves.
基金Project supported by the Natural Science Foundation of Beijing Municipality (Grant No.1212007)the National Natural Science Foundation of China (Grant No.11705284)the Open Project Program of State Key Laboratory of Petroleum Resources and Prospecting,China University of Petroleum (Grant No.PRP/DX-2211)。
文摘We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.
基金the National Natural Science Foundation of China(Grant Nos.11871232 and 12201578)Natural Science Foundation of Henan Province,China(Grant Nos.222300420377 and 212300410417)。
文摘We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.
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.
基金supported by the National Natural Science Foundation of China (Grant Nos.11975204 and 12075208)the Project of Zhoushan City Science and Technology Bureau (Grant No.2021C21015)the Training Program for Leading Talents in Universities of Zhejiang Province。
文摘We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.
基金supported by the National Key Research and Development Program of China (Grant Nos. 2021YFC2201803 and 2020YFC2200104)。
文摘We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(RK4IP-CQE), and the other is the local error adaptive step-control method(RK4IP-LEM). The methods are developed in the vector form of fourthorder Runge–Kutta iterative scheme in the interaction picture by converting a vector equation in frequency domain. By simulating the supercontinuum generated from the high birefringence photonic crystal fiber, the calculation accuracies and the efficiencies of the two adaptive step-size methods are discussed. The simulation results show that the two methods have the same global average error, while RK4IP-LEM spends more time than RK4IP-CQE. The decrease of huge calculation time is due to the differences in the convergences of the relative photon number error and the approximated local error between these two adaptive step-size algorithms.
文摘In this work, we prove the existence and uniqueness of the solution of the generalized Schrödinger equation in the periodic distributional space P’. Furthermore, we prove that the solution depends continuously respect to the initial data in P’. Introducing a family of weakly continuous operators, we prove that this family is a semigroup of operators in P’. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we provide some consequences of this study.
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.
文摘In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.
文摘The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.
基金supported by the National Nature Science Foundation of China under Grant Nos.11671240,61503230 and 61403239the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi under Grant No.2014101the Science Council of Shanxi Province under Grant No.2015021010
文摘This paper studies the stability of Schr¨odinger equation with boundary control matched disturbance. The time-varying gain extended state observer is utilized to estimate disturbance and state. Meanwhile, the authors get a continuous controller by the active disturbance rejection control strategy, which shows that the closed-loop system of Schr¨odinger equation is asymptotically stable.These results are illustrated by simulation examples.
文摘The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.