Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation

Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

Solitons for the cubic—quintic nonlinear Schrödinger equation with varying coefficients

In this paper, by means of similarity transfomations, we obtain explicit solutions to the cubic--quintic nonlinear Schrödinger equation with varying coefficients, which involve four free functions of space. Four types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.     

Solitons for the cubic-quintic nonlinear Schrdinger equation with varying coefficients

In this paper,by means of similarity transfomations,we obtain explicit solutions to the cubic-quintic nonlinear Schrdinger equation with varying coefficients,which involve four free functions of space.Four types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.     

Analytical solutions and rogue waves in （3＋1）-dimensional nonlinear SchrSdinger equation

Analytical solutions in terms of rational-like functions are presented for a(3+1)-dimensional nonlinear Schrdinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz.Several free functions of time t are involved to generate abundant wave structures.Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

An exact solution to the relativistic equation of motion of acharged particle driven by a linearly polarized electromagneticwave

An exact analytic solution is found for a basic electromagnetic wave-charged particle interaction by solving the nonlinear equations of motion. The particle position, velocity, and corresponding time are found to be explicit functions of the total phase of the wave. Particle position and velocity are thus implicit functions of time. Applications include describing the motion of a free electron driven by an intense laser beam     

非线性LC电路方程的显式精确行波解

理论上考察了具有耗散的非线性LC电路中的行波. 借助于作者最近发展的精确求解非线性偏微分方程的扩展的双曲函数方法解析地研究了模拟非线性电路中冲击波的四阶耗散非线性波动方程. 一致地获得了丰富的显式精确解析行波解, 包括精确冲击波解和奇异的行波解, 和三角函数有理形式的周期波解. 关键词： LC电路')" href="#">非线性LC电路 非线性耗散波动方程 冲击波 周期波     

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

In this paper Green’s functions for the Boltzmann equation around a global Maxwellian are used to construct the non-characteristic nonlinear Knudsen layers as well as their time-asymptotic stability. Furthermore, the detailed pointwise structures, nonlinear wave couplings, and wave interactions with boundary are studied.     

Nonlinear periodic waves on the charged free surface of a perfect fluid

Analytical expressions for the profile of a nonlinear wave and for a nonlinear correction to its frequency are derived in the fourth-order approximation in amplitude of a periodic traveling wave on a uniformly charged free surface of an infinitely deep perfect incompressible fluid. It is found that corrections to the amplitude and frequency of the nonlinear wave are absent if the problem is solved under the initial condition that provides the constancy of the first-order amplitude and wavelength in time. Nonlinear analysis of conditions for instability of the fluid free surface against the surface charge shows that the critical charge density and wave-number of the least stable wave are not constant (as in the linear theory) and decrease with growing amplitude of the wave.     

Exact wave functions for atomic electron interacting with photon fields

Many nonlinear quantum optical physics phenomena need more accurate wave functions and corresponding energy or quasienergy levels to account for. An analytic expression of wave functions with corresponding energy levels for an atomic electron interacting with a photon field is presented as an exact solution to the Schrödinger-like equation involved with both atomic Coulomb interaction and electron-photon interaction. The solution is a natural generalization of the quantum-field Volkov states for an otherwise free electron interacting with a photon field. The solution shows that an Nlevel atom in light form stationary states without extra energy splitting in addition to the Floquet mechanism. The treatment developed here with computing codes can be conveniently transferred to quantum optics in classical-field version as research tools to benefit the whole physics community.     

各向异性海森伯自旋链中的超椭圆函数波解

在霍尔斯坦-普里马科夫表象中研究了各向异性海森伯自旋链模型.在半经典近似条件下,考虑高阶非线性项和周期性边界条件,应用相干态求出了用雅可比椭圆函数的反函数的组合表示的超椭圆函数波解,并讨论了解的物理意义.     

The extended (G′/G)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law non-linearity by the extended (G′/G)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.     

Six‐wave mixing induced by free‐carrier plasma in silicon nanowire waveguides

Nonlinear wave mixing in mesoscopic silicon structures is a fundamental nonlinear process with broad impact and applications. Silicon nanowire waveguides, in particular, have large third‐order Kerr nonlinearity, enabling salient and abundant four‐wave‐mixing dynamics and functionalities. Besides the Kerr effect, in silicon waveguides two‐photon absorption generates high free‐carrier densities, with corresponding fifth‐order nonlinearity in the forms of free‐carrier dispersion and free‐carrier absorption. However, whether these fifth‐order free‐carrier nonlinear effects can lead to six‐wave‐mixing dynamics still remains an open question until now. Here we report the demonstration of free‐carrier‐induced six‐wave mixing in silicon nanowires. Unique features, including inverse detuning dependence of six‐wave‐mixing efficiency and its higher sensitivity to pump power, are originally observed and verified by analytical prediction and numerical modeling. Additionally, asymmetric sideband generation is observed for different laser detunings, resulting from the phase‐sensitive interactions between free‐carrier six‐wave‐mixing and Kerr four‐wave‐mixing dynamics. These discoveries provide a new path for nonlinear multi‐wave interactions in nanoscale platforms.

     

Accurate analytic presentation of solution for the spiked harmonic oscillator problem

High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 10⁻³ and 10⁻⁷ for the energies and, correspondingly, 10⁻² and 10⁻⁷ for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems.     

Application of the (G^{\prime}/G)-expansion method for the Burgers, Burgers–Huxley and modified Burgers–KdV equations

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.     

原子相对论波函数的平均自洽场计算

利用Dirac-Slater相对论自洽场理论研究了原子中电子的相对论波函数,说明了其主要特征:电子相对论波函数大小分量的变化趋势不完全"同步",即核外电子的分布没有严格的零几率点;通常电子相对论波函数的大小分量相差非常悬殊,但当束缚电子波函数的空间压缩较为严重、或自由电子的动能较大时,其相对论波函数的小分量的相对值就会增大,并在此基础上分析了束缚电子与自由电子波函数的主要异同点.     

一类非线性演化方程的新多级准确解

在Lamé方程和新的Lamé函数的基础上，应用小扰动方法和Jacobi椭圆函数展开法求解一类非线性演化方程（如mKdV方程，非线性Klein-Gordon方程Ⅱ等），获得多种新的多级准确解 .这些多级准确解对应着不同形式的周期波解.这些解在极限条件下可以退化为多种形式的孤 立波解，如带状孤立子、钟形孤立子等. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

Periodic travelling and non-travelling wave solutions of the nonlinear Klein-Gordon equation with imaginary mass

Exact solutions, including the periodic travelling and non-travelling wave solutions, are presented for the nonlinear Klein-Gordon equation with imaginary mass. Some arbitrary functions are permitted in the periodic non-travelling wave solutions, which contribute to various high dimensional nonlinear structures.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10⁻⁴ and 10⁻⁸ for the energies and, correspondingly, 10⁻² and 10⁻⁴ for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.     

Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.     

Localized nonlinear waves in combined time-dependent magnetic–optical potentials with spatiotemporally modulated nonlinearities

We construct explicit novel solutions of the nonlinear Schrödinger equation with spatiotemporal modulation of the nonlinearities and potentials. By using a modified similarity transformation we explore some localized nonlinearities and combined time-dependent magnetic–optical potentials in the form of linear-lattice ones and harmonic-lattice ones. Several families of exact localized nonlinear wave solutions in terms of Mathieu and elliptic functions corresponding to these potentials are then studied, such as snakelike solitons and breathing solitons. The stability of the obtained localized nonlinear wave solutions is investigated numerically such that some stable solutions are found.