能量临界分数阶非线性Schrodinger方程的整体弱解

利用紧性方法给出能量临界分数阶非线性Schr9dinger方程Cauchy问题解的存在性,并证明Cauchy问题存在整体解.通过构造逼近方程,对满足逼近方程的解序列取极限,得到的极限函数即为能量临界分数阶非线性Schr9dinger方程的整体弱解,并证明该弱解满足能量不等式和质量守恒性质.

Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schr?dinger equation

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.

Soliton propagation for a coupled Schrödinger equation describing Rossby waves

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.

Breather and its interaction with rogue wave of the coupled modified nonlinear Schrodinger equation

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.

POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRODINGER EQUATIONS

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.

Nondegenerate solitons of the(2+1)-dimensional coupled nonlinear Schrodinger equations with variable coefficients in nonlinear optical fibers

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.

Coupled-generalized nonlinear Schr¨odinger equations solved by adaptive step-size methods in interaction picture

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.

Semigroup of Weakly Continuous Operators Associated to a Generalized Schrödinger Equation

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.

Stability of Perfectly Matched Layers for Time Fractional Schrödinger Equation

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.

Exact solitary wave solutions of a nonlinear Schrdinger equation model with saturable-like nonlinearities governing modulated waves in a discrete electrical lattice

In this paper, we introduce and propose exact and explicit analytical solutions to a novel model of the nonlinear Schr¨odinger(NLS) equation. This model is derived as the equation governing the dynamics of modulated cutoff waves in a discrete nonlinear electrical lattice. It is characterized by the addition of two terms that involve time derivatives to the classical equation. Through those terms, our model is also tantamount to a generalized NLS equation with saturable;which suggests that the discrete electrical transmission lines can potentially be used to experimentally investigate wave propagation in media that are modeled by such type of nonlinearity. We demonstrate that the new terms can enlarge considerably the forms of the solutions as compared to similar NLS-type equations. Sine–Gordon expansion-method is used to derive numerous kink, antikink, dark, and bright soliton solutions.

基于变限积分法的非线性Schr dinger方程的数值格式

首先,利用变限积分法和四阶Runge-Kutta法分别离散含五次项的非线性Schr dinger方程的空间和时间变量,并构造初边值问题的全离散格式;其次,在理论上证明其数值解的有界性、存在唯一性以及收敛阶;最后,用数值模拟验证理论分析的有效性.

一类非线性Schrodinger方程变号解的存在性

研究了一类非线性Schr dinger方程:-Δu+V x u=f x,u,x∈RN,其中f的原函数满足的超二次条件比(AR)条件更弱.利用下降流不变集方法,证明了该方程存在变号解.

非线性Schrödinger方程的算子分裂配点格式

采用算子分裂格式结合重心Lagrange插值配点法求解非线性Schrödinger方程.首先,将非线性Schrödinger方程分解为线性部分和非线性部分,线性部分在空间方向上采用重心Lagrange插值配点格式进行离散,在时间方向上采用Crank-Nicolson格式进行离散,非线性部分采用解析求解;然后,对线性子问题空间半离散技术进行相容性分析.最后,采用数值算例进行验证.结果表明:算子分裂配点格式具有高精度,且满足离散的质量守恒和能量守恒.

一类Kirchhoff-Schr dinger-Poisson方程的非平凡解

-a+b∫R 3 u 2 d xΔu+u+λ∅u=a(x)u p-2 u,u∈H(R 3)-Δ∅=u 2,∅∈D 1,2(R 3)为一类非自治的Kirchhoff-Schr dinger-Poisson方程,其中,a>0,λ>0,b≥0,4<p<6,a(x)∈C(R 3)且a(x)为变号。对a(x)赋予适当的条件,通过变分法、山路定理和一些分析技巧,给出了该方程的非平凡解的存在性定理,并利用强极值原理得到了它的正解。

Error estimates of H^1-Galerkin mixed finite element method for Schrdinger equation

An H1-Galerkin mixed finite element method is discussed for a class of second order Schrdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

In this paper,we study the following generalized quasilinear Schrdinger equations with critical or supercritical growths-div(g^2(u)▽u) + g(u)g′(u)|▽u|~2+ V(x)u = f(x,u) + λ|u|^(p-2)u,x∈R^N,where λ>0,N≥3,g:R → R^+ is a C^1 even function,g(0) = 1,g′(s) ≥ 0 for all s ≥ 0,lim_(|s|→+∞)g(s)/|s|^(α-1):= β > 0 for some α≥ 1 and(α-1)g(s) > g′(s)s for all s > 0 and p ≥α2*.Under some suitable conditions,we prove that the equation has a nontrivial solution for smallλ > 0 using a change of variables and variational method.

The Factorization Method to Solve a Class of Inverse Potential Scattering Problems for Schrdinger Equations

This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of (A A) 1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.

NODAL BOUND STATES WITH CLUSTERED SPIKES FOR NONLINEAR SCHRDINGER EQUATIONS

We consider the following nonlinear Schr¨odinger equations -ε2△u + u = Q(x)|u|p-2u in RN, u ∈ H1(RN),where ε is a small positive parameter, N ≥ 2, 2 < p < ∞ for N = 2 and 2 < p <2N/N-2 for N ≥ 3. We prove that this problem has sign-changing(nodal) semi-classical bound states with clustered spikes for sufficiently small ε under some additional conditions on Q(x).Moreover, the number of this type of solutions will go to infinity as ε→ 0+.

Life Is Quantum Biology Effects Explained from the Schrödinger Equation, Serious or Life-Threatening Conditions or Diseases, and COVID-19 Results

Background: A plant based approach was undertaken to ascertain the efficacy of the Technology/Innovation of a new medical Technology based on similar activity to the circulatory system of plants. Basic Technology formulations were introduced into plant structures to evaluate the responses for cell division and plant growth which ultimately lead to the final product. This Technology/Innovation transforms and enhances common botanical/herb products such that they can be used to treat COVID-19 infection and Serious or Life-threatening conditions or diseases. They can restore the diseased cell’s abnormal metabolism to homeostasis and return it to normal function by slowing or reversing the cell’s malfunctioning metabolism, or in other words, slowing or reversing the disease process. Methods: A systematic treatment for COVID-19 infection and Serious or Life-threatening illness in patients is based on theories of traditional Chinese medicine. The Technology/Innovation is developed as dietary supplement products for treatment of Serious or Life-threatening conditions or diseases. Results: The results consist of over one hundred Serious or Life-threatening patient medical records. The U.S. Food and Drug Administration (FDA) reviewed/evaluated/accepted the Technology/Innovation patient medical record results, which included fifty-four COVID-19 patients with Prioritized Review Expedited No.: 2020-2867 CR2020-2596 information on September 16, 2020 and twenty-seven cancer patients Cancer Survival Rate Results in (Table 1) with FDA Establishment Identification No.: 3012327979 information on March 25, 2019 as FDA-Regulated medical products and meet the requirement under FDA Significant Scientific Agreement standards.

The exact solutions to (2+1)-dimensional nonlinear Schrdinger equation

By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrdinger equation are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.