The Existence of Solutions for a Class of Schr¨odinger Equations via Morse Index Theory

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].

A New Framework of Convergence Analysis for Solving the General Nonlinear Schrodinger Equation using the Fourier Pseudo-Spectral Method in Two Dimensions

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.

Linearized Transformed L1 Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schr¨odinger Equations

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.

Exact Solutions of Five Complex Nonlinear Schr¨odinger Equations by Semi-Inverse Variational Principle

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.

Two-Grid Crank-Nicolson FiniteVolume Element Method for the Time-Dependent Schrodinger Equation

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.

Symplectic schemes and symmetric schemes for nonlinear Schr¨odinger equation in the case of dark solitons motion

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.

Compact splitting symplectic scheme for the fourth-order dispersive Schrodinger 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.

A New Explicit Symplectic Fourier Pseudospectral Method for Klein-Gordon-Schrodinger Equation

In this paper,we propose an explicit symplectic Fourier pseudospectral method for solving the Klein-Gordon-Schr odinger equation.The key idea is to rewrite the equation as an infinite-dimensional Hamiltonian system and discrete the system by using Fourier pseudospectral method in space and symplectic Euler method in time.After composing two different symplectic Euler methods for the ODEs resulted from semi-discretization in space,we get a new explicit scheme for the target equation which is of second order in space and spectral accuracy in time.The canonical Hamiltonian form of the resulted ODEs is presented and the new derived scheme is proved strictly to be symplectic.The new scheme is totally explicitwhereas symplectic scheme are generally implicit or semi-implicit.Linear stability analysis is carried and a necessary Courant-Friedrichs-Lewy condition is given.The numerical results are reported to test the accuracy and efficiency of the proposed method in long-term computing.

Novel Conformal Structure-Preserving Algorithms for Coupled Damped Nonlinear Schr odinger System

This paper introduces two novel conformal structure-preserving algorithms for solving the coupled damped nonlinear Schr¨odinger(CDNLS)system,which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws.The proposed algorithms can preserve corresponding conformal multi-symplectic conservation lawand conformalmomentum conservation lawin any local time-space region,respectively.Moreover,it is further shown that the algorithms admit the conformal charge conservation law,and exactly preserve the dissipation rate of charge under appropriate boundary conditions.Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.

An Exact Absorbing Boundary Condition for the Schr¨odinger Equation With Sinusoidal Potentials at Infinity

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.

Quantum speed limit of a single atom in a squeezed optical cavity mode

We theoretically study the quantum speed limit of a single atom trapped in a Fabry-Perot microresonator.The cavity mode will be squeezed when a driving laser is applied to the second-order nonlinear medium,and the effective Hamiltonian can be obtained under the Bogoliubov squeezing transformation.The analytical expression of the evolved atom state can be obtained by using the non-Hermitian Schr¨odinger equation for the initial excited state,and the quantum speed limit time coincides very well for both the analytical expression and the master equation method.From the perspective of quantum speed limit,it is more conducive to accelerate the evolution of the quantum state for the large detuning,strong driving,and coupling strength.For the case of the initial superposition state,the form of the initial state has more influence on the evolution speed.The quantum speed limit time is not only dependent on the system parameters but also determined by the initial state.

类氦离子^(1,3)P激发态Schrdinger方程的直接解

利用相关函数（CF）－超球谐（HH）－广义Laguerre（GLF）方法直接求解类氦离子n1，3P（n＝1，2，3）低躺激发态的Schrodinger方程，得氦原子的本征能量分别为－2．13317Eh（l3P），－2．12383Eh（11P），－2．05810Eh（23P），－2．05516Eh（21P），－2．03235Eh（33P）和－2．03109Eh（31P），它们与文献值在第6位上很好地吻合．这说明CFHHGLF方法也适用于类氦离子1，3P激发态．还给出了总角动量L＝1的对称超球谐基函数和有关矩阵元的解析式．

脉冲Schrdinger方程和最优控制

本文用Fourier变换,得到有界区域上Schrdinger算iΔ所生成的半群的表达式。并用它引进了受控脉冲型Schrdinger方程的PCl-温和解,证明PCl-温和解的存在唯一性。讨论相应的最优控制问题,证明最优控制的存在性,导出最优化的必要条件。

Localized waves in three-component coupled nonlinear Schrdinger equation

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.

Quantum enigma hidden in continuum mechanics

It is reported that there exist deformable media which display quantum effects just as quantum entities do. As such, each quantum entity usually treated as a point particle may be represented by a deformable medium, the dynamic behavior of which is prescribed by four dynamic state variables, including mass density, velocity, internal pressure, and intrinsic angular momentum. In conjunction with the finding of the characteristic equation characterizing the physical nature of such media, it is found that a complex field quantity may be introduced to uncover a perhaps unexpected correlation, i.e., the governing dynamic equations for such media may be exactly reduced to the SchrSdinger equation, from which the closed-form solutions for all the four dynamic state variables can be obtained. It turns out that this complex field quantity is just the wavefunction in the SchrSdinger equation. Moreover, the dynamic effects peculiar to spin are derivable as direct consequences. It appears that these results provide a missing link in quantum theory, in the sense of disclosing the physical origin and nature of both the wavefunction and the wave equation. Now, the inherent indeterminacy in quantum theory may be rendered irrelevant. The consequences are explained for certain long-standing fundamental issues.

Electron localization of linear symmetric molecular ion H3-(2＋)

Electron localization in the dissociation of the symmetric linear molecular ion H3-（2＋） is investigated. The numerical simulation shows that the electron localization distribution is dependent on the central frequency and peak electric field amplitude of the external ultrashort ultraviolet laser pulse. When the electrons of the ground state are excited onto the 2pσ-2Σu-＋ by a one-photon process, most electrons of the dissociation states are localized at the protons on both sides symmetrically. Almost no electron is stabilized at the middle proton due to the odd symmetry of the wave function. With the increase of the frequency of the external ultraviolet laser pulse, the electron localization ratio of the middle proton increases, for more electrons of the ground state are excited onto the higher 3pσ-2Σu-＋ ustate. 50.9% electrons of all the dissociation events can be captured by the middle Coulomb potential well through optimizing the central frequency and peak electric field amplitude of the ultraviolet laser pulse. Besides, a direct current（DC） electric field can be utilized to control the electron motions of the dissociation states after the excitation of an ultraviolet laser pulse, and 68.8% electrons of the dissociation states can be controlled into the middle proton.

Recent Progress in Symplectic Algorithms for Use in Quantum Systems

In this paper we survey recent progress in symplectic algorithms for use in quantum systems in the following topics:Symplectic schemes for solving Hamiltonian systems;Classical trajectories of diatomic systems,model molecule A2B,Hydrogen ion H+2 and elementary atmospheric reaction N(4S)+O2(X 3Σ−g)→NO(X 2Π)+O(3P)calculated by means of Runge-Kutta methods and symplectic methods;the classical dissociation of the HF molecule and classical dynamics of H+2 in an intense laser field;the symplectic form and symplectic-scheme shooting method for the time-independent Schr¨odinger equation;the computation of continuum eigenfunction of the Schr¨odinger equation;asymptotic boundary conditions for solving the time-dependent Schr¨odinger equation of an atom in an intense laser field;symplectic discretization based on asymptotic boundary condition and the numerical eigenfunction expansion;and applications in computing multi-photon ionization,above-threshold ionization,Rabbi oscillation and high-order harmonic generation of laser-atom interaction.

Exact Solutions of Schrdinger Equation for the Position-Dependent Effective Mass Harmonic Oscillator

A one-dimensional harmonic oscillator with position-dependent effective mass is studied. We quantize the oscillator to obtain a quantum Hamiltonian, which is manifestly Hermitian in configuration space, and the exact solutions to the corresponding Schr¨odinger equation are obtained analytically in terms of modified Hermite polynomials.It is shown that the obtained solutions reduce to those of simple harmonic oscillator as the position dependence of the mass vanishes.

Rogue Wave Solutions for Nonlinear Schrdinger Equation with Variable Coefficients in Nonlinear Optical Systems

In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrdinger(NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics.

Vector Breathers in an Averaged Dispersion-Managed Birefringent Fiber System

A variable-coefficient coupled nonlinear Schrodinger equation in an averaged dispersion-managed birefringent fiber is investigated. Based on the one-to-one correspondence between variable-coefficient and constant-coefficient equations, an analytical breather solution is derived. As an example to exhibit dynamical behaviors of solution, its controllable excitations including rear excitation, peak excitation and initial excitation are discussed.