MULTIPLE SOLUTIONS TO CRITICAL MAGNETIC SCHRODINGER EQUATIONS

In this paper,we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation(-i▽-a(x))^(2)u+⒂λV(x)u=p|u|^(p-2)u+(∫R(n)|u(y)|^(2)_(a)^(*)/|x-y|^(a)dy)|u|2_(a)^(*)-2_(u)in R^(N),(0.1)where N≥4,2≤p<2^(*),2_α^(*)=(2N-α)/(N-2)with 0<α<4,λ>0,μ∈R,A(x)=(A_(1)(x),A_(2)(x),…,A_(N)(x))is a real local Hölder continuous vector function,i is the imaginary unit,and V(x)is a real valued potential function on R^(N).Supposing thatΩ=int V^(-1)(0)■R^(N)is bounded,we show that problem(0.1)possesses at least cat_(Ω)(Ω)nontrivial solutions ifλis large.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

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.

Exact Solutions of Forced Schrödinger Equation and How to Choose the External Forces

Schrödinger equations are very common equations in physics and mathematics for nonlinear physics to model the dynamics of wave propagation in waveguides such as power lines, atomic chains, optical fibers, and even in quantum mechanics. But all these equations are most often studied without worrying about what would happen if this equation were maintained, that is to say, had a second member synonymous with an external force. It is true that on a physical level, such equations can be considered as describing the generation of waves on a waveguide using an external force. However, the in-depth analysis of this aspect is not at the center of our reflection in this article, but for us, it is a question of proposing exact solutions to this type of equation and above all proposing the general form of the external force so that the obtaining exact solutions is possible.

Existence of a Sigh-Changing Solution Result for Logarithmic Schrödinger Equations with Weight Function

This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the corresponding functional I belongs to C1(HV1(ℝN),ℝ). Furthermore, by using the variational method, we prove the existence of a sigh-changing solution to problem (1).

Revision of Stationary Schrödinger Equation

Presence of centripetal force field in space shall cause time dilation of any clock at rest therein. Therefore, duration of unit of time determined by any clock in such field is not constant but varies with location of the clock in the field. This means that speed of light in vacuo in centripetal force field is not and cannot be a true physical constant but a function of location in such field because definition of c involves a unit of time and duration of that time unit varies with location in such field. However, classical Schrödinger equation assumes a prior the constancy of c in field, even though this may not be the case. Therefore, it is necessary to revise the classical equation in order to comply with the law of mass-energy equivalence of Einstein hence time dilation in centripetal force field.

Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

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 L2-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.

Stability of Standing Waves for the Nonlinear Schrödinger Equation with Mixed Power-Type and Hartree-Type Nonlinearities

This paper studies the existence of stable standing waves for the nonlinear Schrödinger equation with Hartree-type nonlinearity i∂tψ+Δψ+| ψ |pψ+(| x |−γ∗| ψ |2)ψ=0, (t,x)∈[ 0,T )×ℝN.Where ψ=ψ(t,x)is a complex valued function of (t,x)∈ℝ+×ℝN. The parameters N≥3, 0p4Nand 0γmin{ 4,N }. By using the variational methods and concentration compactness principle, we prove the orbital stability of standing waves.

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.

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.

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.

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.

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic-quintic nonlinear Schrdinger （CCQNLS） equations through the generalized Darboux transformation （DT）. A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higher-order localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed：（i） the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions; （ii） hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons; （iii） hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α. These results further uncover some striking dynamic structures in the CCQNLS system.

Solving unsteady Schr?dinger equation using the improved element-free Galerkin method

By employing the improved moving least-square （IMLS） approximation, the improved element-free Galerkin （IEFG） method is presented for the unsteady Schrodinger equation. In the IEFG method, the two-dimensional （2D） trial function is approximated by the IMLS approximation, the variation method is used to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. Because the number of coefficients in the IMLS approximation is less than in the moving least-square （MLS） approximation, fewer nodes are needed in the entire domain when the IMLS approximation is used than when the MLS approximation is adopted. Then the IEFG method has high computational efficiency and accuracy. Several numerical examples are given to verify the accuracy and efficiency of the IEFG method in this paper.

Superconvergence analysis of bi-k-degree rectangular elements for two-dimensional time-dependent Schrodinger equation

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1)in the H1norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1+ τ2) in the H1norm can be obtained in the Crank-Nicolson fully discrete scheme.

A new finite difference scheme for a dissipative cubic nonlinear Schrdinger equation

This paper considers the one-dimensional dissipative cubic nonlinear SchrSdinger equation with zero Dirichlet boundary conditions on a bounded domain. The equation is discretized in time by a linear implicit three-level central difference scheme, which has analogous discrete conservation laws of charge and energy. The convergence with two orders and the stability of the scheme are analysed using a priori estimates. Numerical tests show that the three-level scheme is more efficient.

Bound State Solutions of Schrodinger Equation for Generalized Morse Potential with Position-Dependent Mass

The effective mass one-dimensional Schroedinger equation for the generalized Morse potential is solved by using Nikiforov-Uvarov method. Energy eigenvalues and corresponding eigenfunctions are computed analytically. The results are also reduced to the constant mass case. Energy eigenvalues are computed numerically for some diatomic molecules. They are in agreement with the ones obtained before.