On a generalized Ablowitz–Kaup–Newell–Segur hierarchy in inhomogeneities of media: soliton solutions and wave propagation influenced from coefficient functions and scattering data

In this paper, a generalized Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy in inhomogeneities of media described by variable coefficients is investigated, which includes some important nonlinear evolution equations as special cases, for example, the celebrated Korteweg–de Vries equation modeling waves on shallow water surfaces. To be specific, the known AKNS spectral problem and its time evolution equation are first generalized by embedding a finite number of differentiable and time-dependent functions. Starting from the generalized AKNS spectral problem and its generalized time evolution equation, a generalized AKNS hierarchy with variable coefficients is then derived. Furthermore, based on a systematic analysis on the time dependence of related scattering data of the generalized AKNS spectral problem, exact solutions of the generalized AKNS hierarchy are formulated through the inverse scattering transform method. In the case of reflectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. It is graphically shown that the dynamical evolutions of such soliton solutions are influenced by not only the time-dependent coefficients but also the related scattering data in the process of propagations.     

Evolution of the exact spatiotemporal periodic wave and soliton solutions of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients

In this paper the spatiotemporal evolution of the periodic wave is investigated analytically when the laser passes through the inhomogeneous nonlinear medium. Firstly, the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients is solved analytically by an improved homogeneous balance principle and F-expansion technique. A number of exact periodic traveling wave and spatiotemporal soliton solutions are obtained. Then, their propagation characteristics are analyzed in detail. It is found that the evolutions of propagation of spatiotemporal soliton and periodic wave solutions are regular when the diffraction and dispersion coefficients are the identical distributed coefficients, but the evolutions of propagation of these solutions are irregular with other coefficients.     

第二类变系数KdV方程的新类型无穷序列精确解

为了构造变系数非线性发展方程的无穷序列新精确解, 发掘第一种椭圆辅助方程的构造性和机械化性特点, 获得了该方程的 新类型解和相应的 Bäcklund 变换. 在符号计算系统 Mathematica 的帮助下, 以第二类变系数 KdV 方程为应用实例, 构造了三种类型的无穷序列新精确解. 这里包括无穷序列光滑类孤子解、无穷序列尖峰孤立子解和无穷序列紧孤立子解. 这种方法也可以获得其他变系数非线性发展方程的无穷序列新精确解.     

Dynamics of matter-wave solitons in Bose-Einstein condensates with time-dependent scattering length and complex potentials

We investigate the dynamics of matter-wave solitons in the one-dimensional (1-D)Gross-Pitaevskii (GP) equation describing Bose-Einstein condensates (BECs) withtime-dependent scattering length in varying trapping potentials with feeding/loss term. Byperforming a modified lens-type transformation, we reduce the GP equation into a classicalnonlinear Schrödinger (NLS) equation with distributed coefficients and find its integrablecondition. Under the integrable condition, we apply the generalized Jacobian ellipticfunction method (GJEFM) and present exact analytical solutions which describe thepropagation of a bright and dark solitons in BECs. Their stability is examined usinganalytic method. The obtained exact solutions show that the amplitude of bright and darksolitons depends on the scattering length, while their motion and the total number of BECatoms depend on the external trapping potential. Our results also shown that the loss ofatoms can dominate the aggregation of atoms by the attractive interaction, and thus thepeak density can decrease in time despite that the strength of the attractive interactionis increased.     

Exact Solutions of the Schrödinger Equation with Irregular Singularity

The 1D nonrelativistic Schrödinger equation possessing an irregular singularpoint is investigated. We apply a general theorem about existence and structureof solutions of linear ordinary differential equations to the Schrödinger equationand obtain suitable ansatz functions and their asymptotic representations for alarge class of singular potentials. Using these ansatz functions, we work out allpotentials for which the irregular singularity can be removed and replaced by aregular one. We obtain exact solutions for these potentials and present sourcecode for the computer algebra system Mathematica to compute the solutions. Forall cases in which the singularity cannot be weakened, we calculate the mostgeneral potential for which the Schrödinger equation is solved by the ansatzfunctions obtained and develop a method for finding exact solutions.     

(2+1)-Dimensional Spatial Localized Modes in Cubic-Quintic Nonlinear Media with the PT-Symmetric Potentials

We obtain exact spatial localized mode solutions of a (2+1)-dimensional nonlinear Schrödinger equation with constant diffraction and cubic-quintic nonlinearity in PT-symmetric potential, and study the linear stability of these solutions. Based on these results, we further derive exact spatial localized mode solutions in a cubic-quintic medium with harmonic and PT-symmetric potentials. Moreover, the dynamical behaviors of spatial localized modes in the exponential diffraction decreasing waveguide and the periodic distributed amplification system are investigated.     

第二种椭圆方程构造变系数非线性发展方程的无穷序列新精确解

本文为了获得非线性发展方程的无穷序列新精确解,进一步研究获得了第二种椭圆方程的几类新型解和Bäcklund变换.在此基础上,借助符号计算系统Mathematica,用带强迫项变系数组合KdV方程、(2+1)维和(3+1)维变系数Zakharov-Kuznetsov 方程为应用实例,构造了无穷序列新精确解.这里包括无穷序列Jacobi 椭圆函数光滑孤立子解、无穷序列Jacobi椭圆函数紧孤立子解、无穷序列三角函数紧孤立子解和无穷序列尖峰孤立子解. 关键词： 第二种椭圆方程 Bä cklund 变换 变系数非线性发展方程 无穷序列新精确解     

Off-shell photon distribution amplitudes in the low-energy effective theory of QCD

The mechanism of the initial inflationary scenario of the Universe and of its late-time acceleration can be described by assuming the existence of some gravitationally coupled scalar fields $\phi $ , with the inflaton field generating inflation and the quintessence field being responsible for the late accelerated expansion. Various inflationary and late-time accelerated scenarios are distinguished by the choice of an effective self-interaction potential $V(\phi )$ , which simulates a temporarily non-vanishing cosmological term. In this work, we present a new formalism for the analysis of scalar fields in flat isotropic and homogeneous cosmological models. The basic evolution equation of the models can be reduced to a first-order non-linear differential equation. Approximate solutions of this equation can be constructed in the limiting cases of the scalar-field kinetic energy and potential energy dominance, respectively, as well as in the intermediate regime. Moreover, we present several new accelerating and decelerating exact cosmological solutions, based on the exact integration of the basic evolution equation for scalar-field cosmologies. More specifically, exact solutions are obtained for exponential, generalized cosine hyperbolic, and power-law potentials, respectively. Cosmological models with power-law scalar field potentials are also analyzed in detail.     

Poloidal–toroidal decomposition in a finite cylinder: II. Discretization, regularization and validation

The Navier–Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non-axisymmetric helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective.     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.