Modulation space estimates for damped fractional wave equation

Instead of the L~p estimates,we study the modulation space estimates for the solution to the damped wave equation.Decay properties for both the linear and semilinear equations are obtained.The estimates in modulation space differ in many aspects from those in L~p space.     

SCALE-INVARIANT SOLUTION FOR FRACTIONAL ANOMALOUS DIFFUSION EOUATION

Fractional diffusion equation for transport phenomena in fractal media is an integro-partial differential equation. For solving the problem of this kind of equation the invariants of the group of scaling transformations are given and then used for deriving the integro-ordinary differential equation for the scale-invariant solution. With the help of Mellin transform the scale-invariant solution is obtained in terms of Fox functions. And the series and asymptotic representations for the solution are considered.     

Lump wave and hybrid solutions of a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.     

PLANE ELASTIC PROBLEMS OF DIFFERENT MEDIA WITH CRACKS ON THE INTERFACE

The equllibrium problem for the infinite elastic plane consisting of two different media with many cracks on the interface is discussed.It is transferred to a boundary value problem for analytic functions and then further reduced to a singular integral equation,the unique solvability and an effective method of solution for which are eatablished.A practical example in applications is illustrated,the solution of which is obtained in closed form.     

On a Linear Equation Arising in Isometric Embedding of Torus-like Surface

The solvability of a linear equation and the regularity of the solution are discussed. The equation is arising in a geometric problem which is concerned with the realization of Alexandroff＇s positive annul in R^3.     

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.     

THE INITIAL BOUNDARY VALUE PROBLEM FOR QUASI-LINEAR SCHROEDINGER-POISSON EQUATIONS

In this article, the author studies the iuitial （Dirichlet.） boundary problem for a high field version of the Schroedinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schroedinger equations on the unit cube. h global existence and uniqueness is established for a solution to this problem.     

SOLITON SOLUTIONS TO AN EQUATION IN 2+1 DIMENSIONS

This paper considers a 2+1 dimensional equation. A bilinear form for the equation and its 3-soliton solutions are obtained by the Hirota method. The N-soliton solution is given in the form of pfaffian. At the same time the proof of the solutions are given.     

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions.In the first part of this article,we are concerned with the decay rate of solutions of one dimension convection diffusion equation.On the other hand,in the second part of this article,we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.     

Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

The present work aims to determine the solution of trigonometric functional equation f with involution from group to field by using the properties of involution function,and the solution and Ulam-Hyers stability of the trigonometric functional equation are also discussed.Furthermore,this method generalizes the main theorem and gives the supplement in some reference.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

再论弹性大挠度问题von Kármán方程与量子本征值问题Schrǒdinger方程的关系

本文是文[1]的继续和改善。利用本文的结果,还可以改善文[2～3]中有关弹性大挠度问题的讨论。在本文中,我们再次对弹性大挠度问题的von Kármán方程进行简化,使它最终成为非线性Schr?dinger方程。其次,在本文中我们对AKNS方程在多维条件下进行了更为对称的拓展。由于非线性Schr?dinger方程与AKNS方程即Dirac方程的可积性条件相联系,因此,弹性大挠度问题可以用逆散射方法求得其精确解,也就是说,它完全成了量子本征值问题。对于正交各向异性大挠度问题,本文也作了推论。     

New analysis and application of fractional order Schrödinger equation using with Atangana–Batogna numerical scheme

In this work, an analytical approximation to the solution of Schrodinger equation has been provided. The fractional derivative used in this equation is the Caputo derivative. The existence and uniqueness conditions of solutions for the proposed model are derived based on the power law. While solving the fractional order Schrodinger equation, Atangana–Batogna numerical method is presented for fractional order equation. We obtain an efficient recurrence relation for solving these kinds of equations. To illustrate the usefulness of the numerical scheme, the numerical simulations are presented. The results show that the numerical scheme is very effective and simple.     

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.     

二维空间中具有积分型非线性Schrodinger方程组的初值问题

具有积分型非线性ｓｃｈｒｏｄｉｎｇｅｒ方程是在研究非线性Ｌａｎｇｍｕｉｒ波时考虑到离子惯性作用而导出的．本文讨论了二维空间中具有积分型非线性ｓｃｈｒｏｄｉｎｇｅｒ方程组的初值问题，用积分估计方法证明了整体解的存在唯一性．     

A geometric characterization of the nonlinear Schrodinger equation and its applications

We prove that the nonlinear Schrodinger equation of attractive type (NLS+ describes just spher-ical surfaces (SS) and the nonlinear Schrodinger equation of repulsive type (NLS-) determines only pseudo-spherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equationsthere exists a local gauge transformation (despite of changing the independent variables or not) which trans-forms a solution of one into any solution of the other, it is impossible to have such a gauge transformationbetween the NLS+ and the NLS-.     

Stochastic Schrodinger Equation for a Quantum Oscillator with Dissipation

In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.     

The quasihomogeneous function method and Fock's problem

The diffraction of a high-frequency wave by a smooth convex body near the tangency point of the limiting ray to the surface is restated as the scattering problem for the Schrodinger equation with a linear potential on a half-axis. Various prior estimates for the scattering problem are used in order to prove existence, uniqueness, and smoothness theorems. The corresponding solution satisfies the principle of limiting absorption. The formal solution of the corresponding Schrodinger equation in the form of quaslhomogeneous functions is essentially used in our constructions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 144–151, 1985.     

A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH

In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in R~N which includes the so-called modified nonlinear Schrodinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.     

一类带调和势的耗散非线性Schrdinger方程的全局和爆破性质(英文)

针对一类带调和势的耗散非线性schrodinger方程,本文运用一些不等式和先验估计方法研究了其解的行为特征．