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Superconvergence of Finite Element Approximations of the Two-Dimensional Cubic nonlinear schrodinger equation

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Advances in Applied Mathematics and Mechanics 2022年第3期14卷 652-665页

作者： Jianyun Wang Zhikun Tian School of Science Hunan University of TechnologyZhuzhouHunan 412007China School of Computational Science and Electronics Hunan Institute of EngineeringXiangtanHunan 411104China

The superconvergence of a two-dimensional time-independent nonlinear schrodinger equation are analyzed with the rectangular Lagrange typeﬁnite element of order k.Firstly,the error estimate and superclose property are given in H^(1)-norm with order O(h^(k+1))between theﬁnite element solution u_(h) and the interpolation func-tion uI by use of the elliptic projection operator.Then,the global superconvergence is obtained by the interpolation post-processing technique.In addition,some numerical examples with the order k=1 and k=2 are provided to demonstrate the theoretical analysis.

关键词： Superconvergence nonlinear schrodinger equation ﬁnite element method elliptic projection

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF A nonlinear schrodinger equation WITH WEAKLY DAMPED

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Journal of Computational Mathematics 2001年第4期19卷 393-406页

作者： Fa-yong Zhang Shu-juan Lu Department of Mathematics Heilongjiang UniversityHarbin 150080China Department of Mathematics Beijing University of Aeronautics&AstronauticsBeijing 100083China

A weakly demped schrodinger equation possessing a global attractor are considered. The dynamical properties of a class of finite difference scheme axe analysed. The exsitence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case.

关键词： global attractor nonlinear schrodinger equation finite difference method stibility and convergence

LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF THREE-DIMENSIONAL nonlinear schrodinger equation WITH WEAKLY DAMPED

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Journal of Computational Mathematics 2004年第4期22卷 593-604页

作者： Fa-yongZhang DepartmentofMathematics HeilongjiangUniversityHarbin150080China

The three-dimensional nonlinear SchrSdinger equation with weakly damped that possesses a global attractor are considered. The dynamical properties of the discrete dynamical system which generate by a class of finite d... 详细信息

关键词： nonlinear schrodinger equation Finite difference method Global attractor

Solving forward and inverse problems of the nonlinear schrodinger equation with the generalized PT-symmetric Scarf-Ⅱpotential via PINN deep learning

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Communications in Theoretical Physics 2021年第12期73卷 1-13页

作者： Jiaheng Li Biao Li School of Mathematics and Statistics Ningbo UniversityNingbo 315211China

In this paper,based on physics-informed neural networks(PINNs),a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations(PDEs)and other types of nonlinear physical models,we study the nonlinear schrodinger equation(NLSE)with the generalized PT-symmetric Scarf-Ⅱpotential,which is an important physical model in many fields of nonlinear physics.Firstly,we choose three different initial values and the same Dinchlet boundaiy conditions to solve the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential via the PINN deep learning method,and the obtained results are compared with ttose denved by the toditional numencal methods.Then,we mvestigate effect of two factors(optimization steps and activation functions)on the performance of the PINN deep learning method in the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential.Ultimately,the data-driven coefficient discovery of the generalized PT-symmetric Scarf-Ⅱpotential or the dispersion and nonlinear items of the NLSE with the generalized PT-symmetric Scarf-Ⅱpotential can be approximately ascertained by using the PINN deep learning method.Our results may be meaningful for further investigation of the nonlinear Schrodmger equation with the generalized PT-symmetric Scarf-Ⅱpotential in the deep learning.

关键词： nonlinear schrodinger equation generalized PT-symmetric scarf-Ⅱpotential physics-informed neural networks deep learning initial value and dirichlet boundary conditions data-driven coefficient discovery

nonlinear schrodinger equation with a Dirac delta potential:finite difference method

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Communications in Theoretical Physics 2020年第2期72卷 3-8页

作者：程彬陈亚铭徐传福李大力邓小刚 College of Computer National University of Defense Technology Changsha 410073China College of Aerospace Science and Engineering National University of Defense TechnologyChangsha 410073China

The nonlinear Schr?dinger equation with a Dirac delta potential is considered in this paper.It is noted that the equation can be transformed into an equation with a drift-admitting jump.Then following the procedure proposed in Chen and Deng(2018 Phys.Rev.E 98033302),a new second-order finite difference scheme is developed,which is justified by numerical examples.

关键词： nonlinear schrodinger equation delta potential finite difference method

Explicit Symplectic Methods for the nonlinear schrodinger equation

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Communications in Computational Physics 2009年第8期6卷 639-654页

作者： Hua Guan Yandong Jiao Ju Liu Yifa Tang LSEC ICMSECAcademy of Mathematics&Systems ScienceChinese Academy of SciencesBeijing 100190China Department of Scientific Computation and Applied Software School of ScienceXi’an Jiaotong UniversityXi’an 710049China

By performing a particular spatial discretization to the nonlinear schrodinger equation(NLSE),we obtain a non-integrable Hamiltonian system which can be decomposed into three integrable parts(L-L-N splitting).We integrate each part by calculating its phase flow,and develop explicit symplectic integrators of different orders for the original Hamiltonian by composing the phase flows.A 2nd-order reversible constructed symplectic scheme is employed to simulate solitons motion and invariants behavior of the NLSE.The simulation results are compared with a 3rd-order non-symplectic implicit Runge-Kutta method,and the convergence of the formal energy of this symplectic integrator is also verified.The numerical results indicate that the explicit symplectic scheme obtained via L-L-N splitting is an effective numerical tool for solving the NLSE.

关键词： Explicit symplectic method L-L-N splitting nonlinear schrodinger equation

Solitary Waves of 1-nonlinear schrodinger equation in the Composite Right-and Left-Handed Metamaterial

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Journal of Partial Differential equations 2019年第4期32卷 293-303页

作者： HOUWE A. KLOFAI Yerima JUSTIN Mibaile GAMBO Betchewe DOKA Serge Y. Department of Marine engineering Limbe Nautical Arts and Fisheries InstituteP.O.Box.485 LimbeCameroon Department of Physics Higher Teachers Training CollegeUniversity of MarouaP.O.Box55MarouaCameroon Department of Physics Faculty of ScienceUniversity of MarouaP.O.Box814MarouaCameroon Department of Physics Faculty of ScienceUniversity of NgaoundereP.O.Box454NgaoundreCameroon

In this article,we analyze solitary waves in nonlinear left-handed transmission line with nonlinear diodes(Schottkys)which is an important issue,especially for soliton devices.By applying the Kirchhoffs laws and reductive direct method,the voltage in the spectral domain was obtained.Considering the Taylor series around a certain modulation frequency,we obtained one dimensional nonlinear schrodinger equation(NSE),which support envelops soliton,and bright soliton solutions.Using sine-cosine mathematical method,soliton solutions of the standard nonlinear Schrod--inger equation are obtained.The method used is straightforward and concise and can be applied to solve further of nonlinear PDEs in mathematical physics.

关键词： CRLH transmission line nonlinear schrodinger equation Sine-Cosine method solitary waves

A Family of Exact Solutions for the nonlinear schrodinger equation

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Journal of Shanghai University(English Edition) 2001年第4期5卷 273-275页

作者： HUANG De bin, LIU Zeng rong Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200436, China Department of Mathematics College of Sciences Shanghai University Shanghai China

In this paper, the nonlinear schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these stationary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.

关键词： nonlinear schrodinger equation stationary solutions exact solutions

Rogue wave solutions and rogue-breather solutions to the focusing nonlinear Schr?dinger equation

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Communications in Theoretical Physics 2024年第3期76卷 33-41页

作者： Si-Jia Chen Xing Lü College of Science Tianjin University of TechnologyTianjin 300384People's Republic of China Department of Mathematics Beijing Jiaotong UniversityBeijing 100044People's Republic of China Beijing Laboratory of National Economic Security Early-warning Engineering Beijing Jiaotong UniversityBeijing 100044People's Republic of China

Based on the long wave limit method,the general form of the second-order and third-order rogue wave solutions to the focusing nonlinear Schr?dinger equation are given by introducing some arbitrary parameters.The interaction solutions between the first-order rogue wave and one-breather wave are constructed by taking a long wave limit on the two-breather solutions.By applying the same method to the three-breather solutions,two types of interaction solutions are obtained,namely the first-order rogue wave and two breather waves,the second-order rogue wave and one-breather wave,respectively.The influence of the parameters related to the phase on the interaction phenomena is graphically demonstrated.Collisions occur among the rogue waves and breather waves.After the collisions,the shape of them remains unchanged.The abundant interaction phenomena in this paper will contribute to a better understanding of the propagation and control of nonlinear waves.

关键词： nonlinear schrodinger equation rogue wave solutions interaction solutions