Asymptotical solutions of coupled nonlinear Schrodinger equations with perturbations

In this paper Lou＇s direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear SchrSdinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.

A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrodinger Equation

In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

Convective Schrodinger Equation:Insights on the Potential Energy’s Role to Wave Particle Decay

In this paper, we coupled the Quantum Mechanics conventional Schrödinger’s equation, for the particles, with the Maxwell’s wave equation, in order to study the potential’s role on the conversion of the electromagnetic field energy to mass and vice versa. We show that the dissipation (“conductivity”) factor and the particle implicit proper frequency are both related to the potential energy. We have also derived a new expression for the Schrödinger’s Equation considering the potential energy into this equation not as an ad hoc term, but also as an operator (Hermitian), which has the scalar potential energy as a natural eigenvalue of this operator.

Some Comments on the Real Meaning of Schrodinger’s Equation Revealed by the Fact that Electron Is Always a Particle

We all physicist have long been believed that an elementary particle is a wave as well as a particle, but we discuss in this paper that an electron (probably all fermions) is always a particle. Author claim that quantum mechanics (QM) is not such mysterious as Bohr stated that the wave turn to the particle by observation. We can understand QM by natural human sense. The wave nature of electrons is only an appearance or a phenomena but not intrinsic or substantial. An electron is an individual body, which interferes with other individual electrons. Interference is the key word instead of the wave to understand the quantum mechanics. Interference produces the wave nature and the uncertainty. When we determine that an electron is nothing but a particle, we will see the true meaning of wave function and the Schr?dinger’s equation.

Inverse Scattering Problem for the Schrodinger’s Equation

The analytic properties of the scattering amplitude are discussed. And, the representation of the potential by the scattering amplitude is obtained.

The Estimates L_(1)-L_(∞) for the Reduced Radial Equation of Schrodinger

Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.

Mixture of a New Integral Transform and Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations

In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].

He’s Homotopy Perturbation Method and Fractional Complex Transform for Analysis Time Fractional Fornberg-Whitham Equation

In this article,time fractional Fornberg-Whitham equation of He’s fractional derivative is studied.To transform the fractional model into its equivalent differential equation,the fractional complex transform is used and He’s homotopy perturbation method is implemented to get the approximate analytical solutions of the fractional-order problems.The graphs are plotted to analysis the fractional-order mathematical modeling.

SERIES PERTURBATIONS APPROXIMATE SOLUTIONS TO N-S EQUATIONS AND MODIFICATION TO ASYMPTOTIC EXPANSION MATCHED METHOD

A method that series perturbations approximate solutions to N-S equations with boundary conditions was discussed and adopted. Then the method was proved in which the asymptotic solutions of viscous fluid flow past a sphere were deducted. By the ameliorative asymptotic expansion matched method, the matched functions, are determined easily and the ameliorative curve of drag coefficient is coincident well with measured data in the case that Reynolds number is less than or equal to 40 000.

Circular Scale of Time Applied in Classifying the Quantum-Mechanical Energy Terms Entering the Framework of the Schrodinger Perturbation Theory

The paper applies a one-to-one correspondence which exists between individual Schr?dinger perturbation terms and the diagrams obtained on a circular scale of time to whole sets of the Schr?dinger terms belonging to a definite perturbation order. In effect the diagram properties allowed us to derive the recurrence formulae giving the number of higher perturbative terms from the number of lower order terms. This recurrence formalism is based on a complementary property that any perturbation order N can be composed of two positive integer components Na , Nb combined into N in all possible ways. Another result concerns the degeneracy of the perturbative terms. This degeneracy is shown to be only twofold and the terms having it are easily detectable on the basis of a circular scale. An analysis of this type demonstrates that the degeneracy of the perturbative terms does not exist for very low perturbative orders. But when the perturbative order exceeds five, the number of degenerate terms predominates heavily over that of nondegenerate terms.

Circular Scale of Time as a Guide for the Schrodinger Perturbation Process of a Quantum-Mechanical System

We point out that a suitable scale of time for the Schrödinger perturbation process is a closed line having rather a circular and not a conventional straight-linear character. A circular nature of the scale concerns especially the time associated with a particular order N of the perturbation energy which provides us with a full number of the perturbation terms predicted by Huby and Tong. On the other hand, a change of the order N—connected with an increased number of the special time points considered on the scale—requires a progressive character of time. A classification of the perturbation terms is done with the aid of the time-point contractions present on a scale characteristic for each N. This selection of terms can be simplified by a partition procedure of the integer numbers representing N-1. The detailed calculations are performed for the perturbation energy of orders N=7 and N=8 .

非线性扰动Schrodinger方程的低正则性问题

利用文[2]中的Fourier限制模方法和压缩映射原理,研究带有Kerr law非线性项的非线性扰动Schrodinger方程在低正则性空间H^s(s>-1/4)的局部适定性.

在无穷远点震荡的Schrodinger-Poisson方程的无穷多个解

研究了具有在无穷远点震荡的非线性项和一个连续的非线性扰动项的Schr?dinger-Poisson方程.采用变分方法和Szulkin类泛函对称临界点原理,得到一个初始问题的重要结论.然后构造一个特殊函数,并应用这一重要结论最后证得非扰动Schr?dinger-Poisson方程有无穷多个不同解,且扰动问题不同解的数量比非扰动问题的更多.

Exact invariants and adiabatic invariants of Raitzin＇s canonical equations of motion for a nonlinear nonholonomic mechanical system

The exact invariants and the adiabatic invariants of Raitzin＇s canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher-order adiabatic invariant of a mechanical system under the action of a small perturbation, the forms of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results.

Generalized Extended tanh-function Method for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher＇s equation, the nonlinear schr＆#168;odinger equation to illustrate the validity and ad-vantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.

广义拟线性Schrodinger方程正解的存在性

本文利用扰动法和变分法研究一类拟线性Schrodinger方程正解的存在性.本文的新颖性在于对权函数在远离原点处的光滑性不作要求.

Navier-Stokes Equations—Millennium Prize Problems

In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.

含混合项的拟线性Schrodinger方程的正规化基态解

该文研究了一类含混合项拟线性Schrodinger方程正规化基态解的存在性.推广了文献[1-2]中的结果,与他们研究的情形相比,该文中方程对应能量泛函的结构更加复杂.

Global Estimation of the Cauchy Problem Solutions’the Navier-Stokes Equation

The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided.

The Cosmological Evolution of Baryonic Matter’S Density Perturbations Under Influence of the Quintessence

For deeper understanding the process of baryonic matter evolution in the expanding Universe it is necessary to know the physical property of concrete field that represents the background of substrate type of dark energy. Beside, it is necessary to explore in details the influence of such field on the continuous medium of baryonic matter. These statements were realized for the quintessence field that describes by two gravitating scalar fields. They give own contributions at the total pressure and at the total mass density of baryonic matter. It allowed show that evolution of baryonic matter’s density perturbations obeys the equation of forced oscillations and admits the resonance case, when amplitude of baryonic matter’s density perturbations gets the strong short-time splash. This splash interprets as a new macroscopic mechanism of the initial matter density perturbations appearance.