Approximate Analytic Solution of Solitary Wave for a Class of Nonlinear Disturbed Long-Wave System

In this paper, the approximate expressions of the solitary wave solutions for a class of nonlinear disturbed long-wave system are constructed using the homotopie mapping method.     

Approximate solution for the Klein-Gordon-SchrSdinger equation by the homotopy analysis method

The Homotopy analysis method is applied to obtain the approximate solution of the Klein-Gordon Schrodinger equation. The Homotopy analysis solutions of the Klein-Gordon Schrodinger equation contain an auxiliary parameter which provides a convenient way to control the convergence region and rate of the series solutions. Through errors analysis and numerical simulation, we can see the approximate solution is very close to the exact solution.     

一个燃烧模型的近似解

运用同伦理论探讨了一类非线性问题的近似解.然后,将其应用于一个燃烧模型,得到了该问题的近似解.关键词：非线性方程同伦变分迭代法燃烧模型近似解     

Approximate Solutions of Perturbed Nonlinear Schroedinger Equations

By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schroedinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.     

Approximate Solutions of Perturbed Nonlinear Schr(o)dinger Equations

By applying Lou's direct perturbation method to perturbed nonlinear Schr(o)dinger equation and the critical nonlinear Schr(o)dinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schr(o)dinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.     

Approximate solution for the Klein Gordon-Schrdinger equation by the homotopy analysis method

The Homotopy analysis method is applied to obtain the approximate solution of the Klein--Gordon--Schr?dinger equation. The Homotopy analysis solutions of the Klein--Gordon--Schr?dinger equation contain an auxiliary parameter which provides a convenient way to control the convergence region and rate of the series solutions. Through errors analysis and numerical simulation, we can see the approximate solution is very close to the exact solution.     

Application of the homotopy analysis method for the Gross-Pitaevskii equation with a harmonic trap

<正>The Homotopy analysis method(HAM) is adopted to find the approximate analytical solutions of the GrossPitaevskii equation,a nonlinear Schrodinger equation is used in simulation of Bose-Einstein condensates trapped in a harmonic potential.Comparisons between the analytical solutions and the numerical solutions have been made.The results indicate that they fit very well with each other when the atomic interaction is weak.     

具有非线性两模场和原子与场非线性耦合的Jaynes-Cummings模型的解

利用Lewis和Riesenfeld不变量方法求解了具有任意非线性场和依赖任意强度耦合的多光子双模场与二能级原子相互作用的JC模型,得到原子一光场的态矢、时间演化算符、原子布居和双模场的光子数算符的演化式.     

Classification and Approximate Solutions to a Class of Perturbed Nonlinear Wave Equations

A complete approximate symmetry classification of a class of perturbed nonlinear wave equations is performed using the method originated from Fushchich and Shtelen. Moreover, large classes of approximate invariant solutions of the equations based on the Lie group method are constructed.     

Approximate solution of the flow over a nonlinear magneto-hydrodynamic stretching sheet

The approximate solution of the magneto-hydrodynamic(MHD) boundary layer flow over a nonlinear stretching sheet is obtained by combining the Lie symmetry method with the homotopy perturbation method.The approximate solution is tabulated,plotted for the values of various parameters and compared with the known solutions.It is found that the approximate solution agrees very well with the known numerical solutions,showing the reliability and validity of the present work.     

A Bilinear Baecklund Transformation and N-Soliton-Like Solution of Three Coupled Higher-Order Nonlinear Schroedinger Equations with Symbolic Computation

A bilinear Baecklund transformation is presented for the three coupled higher-order nonlinear Schroedinger equations with the inclusion of the group velocity dispersion, third-order dispersion and Kerr-law nonlinearity, which can describe the dynamics of alpha helical proteins in living systems as well as the propagation of ultrashort pulses in wavelength-division multiplexed system. Starting from the Baecklund transformation, the analytical soliton solution is obtained from a trivial solution. Simultaneously, the N-soliton-like solution in double Wronskian form is constructed, and the corresponding proof is also given via the Wronskian technique. The results obtained from this paper might be valuable in studying the transfer of energy in biophysics and the transmission of light pulses in optical communication systems.     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.``     

利用线化和校正法求非线性单摆运动的周期

应用线化和校正方法,研究了单摆的非线性振动,作出了周期比和相对误差随摆角的变化曲线.将所得近似解与精确解比较可知,该方法具有简单实用,精度高,相对误差低等优点,对于求解非线性振动问题具有一定的实用价值.     

Three-Step Predictor-Corrector of Exponential Fitting Method for Nonlinear Schroedinger Equations

We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.     

A short comment on the effect of a shear layer on nonlinear water waves

In this paper we study nonlinear periodic deep water waves propagating on a background shear current,which decays exponentially with depth.We extend the study of Cheng,Cang and Liao(2009) by introducing a second parameter which measures the depth of the shear current.A high-order convergent analytical series solution is obtained by the homotopy analysis method(HAM).A detailed analysis of the impact of the depth parameter is given.We find that increasing this parameter so that the shear current is thinner re...     

Asymptotic Study of a Boundary Value Problem Governed by the Elasticity Operator with Nonlinear Term

In this paper, a nonlinear boundary value problem in a three dimensional thin domain with Tresca's friction law is considered. The small change of variable z = x₃/ε transforms the initial problem posed in the domain Ω^εinto a new problem posed on a fixed domain Ω independent of the parameter ε. As a main result, we obtain some estimates independent of the small parameter. The passage to the limit on ε, permits to prove the results concerning the limit of the weak problem and its uniqueness.     

Numerical Solutions of a Class of Nonlinear Evolution Equations with Nonlinear Term of Any Order

In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions： numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.     

Applications of Elliptic Equation to Nonlinear Coupled Systems

The elliptic equation is taken as a transformation and applied to solve nonlinear coupled systems. Itis shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wavesolutions, periodic wave solutions and so on, so this method can be taken as a unified method in solving nonlinear coupled systems.     

Approximate Solutions of Nonlinear Fractional Kolmogorov-Petrovskii-Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method

By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.