General method for the reduction of the effective rotational hamiltonian of molecules. Rational approximants and Padé approximants

A general method for reducing the effective rotational Hamiltonian for moleculesH _rot to the empirically constructed form in the {J_±, J_z} representation (molecules of the asymmetric-top type), applicable to Hamiltonians having improved convergence with nonpolynomial dependence on the angular momentum J, is developed. Rational forms for the reducedHrot and reduced Padé approximants for the rotational energy operator of the molecules are proposed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 32–37, August, 1985.     

Extended Jacobi elliptic function method and its applications to （2＋l）-dimensional dispersive long-wave equation

An extended Jacobi elliptic function method is proposed for constructing the exact double periodic solutions of nonlinear partial differential equations (PDEs) in a unified way. It is shown that these solutions exactly degenerate to the many types of soliton solutions in a limited condition. The Wu－Zhang equation (which describes the (2+1)-dimensional dispersive long wave) is investigated by this means and more formal double periodic solutions are obtained.     

The periodic wave solutions for two systems of nonlinear wave equations

The periodic wave solutions for the Zakharov system of nonlinear wave equations and a long－short-wave interaction system are obtained by using the F-expansion method, which can be regarded as an overall generalization of Jacobi elliptic function expansion proposed recently. In the limit cases, the solitary wave solutions for the systems are also obtained.     

A method for constructing exact solutions and application to Benjamin Ono equation

By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Solitary Wave Solutions of Triple Sine-Gordon Equation

Through a mapping relation between the triple sine-Gordon and Ф5 model, we discuss the equation in detail. A series of solitary wave solutions for the triple sine-Gordon equation is obtained by the existing ones of the Ф5 model. Some detailed solitary wave solutions are given for some special coefficients which have significance in physics.     

Solution of ODE u＂ ＋ p（u）（u＇）^2＋ q（u） = 0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2 ＋ q（u） = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u＂＋p（u）（u＇）^2＋q（u） = 0 includes the equation （u＇）^2 = f（u） as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.     

Auto-Backlund Transformation and Analytic Solutions of （2＋1）-Dimensional Boussinesq Equation

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of （2＋1）-dimensional Boussinesq equation. An auto-Backlund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass ＆ function. Some of them are novel.     

New exact excitations and soliton fission and fusion for the （2＋1）-dimensional Broer-Kaup-Kupershmidt system

With the help of an extended mapping approach, a series of new types of exact excitations with two arbitrary functions of the (2 1)-dimensional Broer-Kaup-Kupershmidt (BKK) system is derived. Based on the derived solitary wave excitation, some specific soliton fission and fusion solutions of the higher-dimensional BKK system are also obtained.     

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.     

On High-Frequency Soliton Solutions to a （2＋1）-Dimensional Nonlinear Partial Differential Evolution Equation

A （2＋1）-dimensional nonlinear partial differential evolution （NLPDE） equation is presented as a model equation for relaxing high-rate processes in active barothropic media. With the aid of symbolic computation and Hirota＇s method, some typical solitary wave solutions to this （2＋1）-dimensional NLPDE equation are unearthed. As a result, depending on the dissipative parameter, single and multivalued solutions are depicted.     

Doubly Periodic Propagating Wave for （2d-1）-Dimensional Breaking Soliton Equation

Using the variable separation approach, we obtain a general exact solution with arbitrary variable separation functions for the （2＋1）-dimensional breaking soliton system. By introducing Jacobi elliptic functions in the seed solution, two families of doubly periodic propagating wave patterns are derived. We investigate these periodic wave solutions with different modulus m selections, many important and interesting properties are revealed. The interaction of Jabcobi elliptic function waves are graphically considered and found to be nonelastic.     

Double Elliptic Equation Expansion Approach and Novel Solutions of （2＋1）-Dimensional Break Soliton Equation

In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in （2＋1） dimensions.     

Darboux Transformation and Multi-Solitons for Complex mKdV Equation

An explicR N-fold Darboux transformation with multi-parameters for coupled mKdV equation is constructed with the help of a gauge transformation of the Ablowitz-Kaup-Newell-Segur （AKNS） system spectral problem. By using the Darboux transformation and the reduction technique, some multi-soliton solutions for the complex mKdV equation are obtained.     

Some New Exact Solutions to the Dispersive Long－Wave Equation in（2＋1）－Dimensional Spaces

In this paper, by using a further extended tanh method- and symbolic computation system, some new soliton-like and period form solutions of the dispersive long-wave equation in (2 l )-dimensional spaces are obtained.     

Exact Periodic-Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and（3＋1）-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

Searching for the （3＋1）—Dimensional Painleve Integrable Model and its Solitary Wave Solution

A （3 1)-dimensional integrable model constructed by conformal invariants is proven to be integrable.The solitary wave solution of the model is obtained by a simple algebraic transformation relation between the (3 1)-dimensional Harry-Dym equation and the cubic nonlinear Klein-Gordon equation.     

Non—equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method

In this paper, we propose a new approach to implementing boundary conditions in the lattice Boltzmann method (LBM). The basic idea is to decompose the distribution function at the boundary node into its equilibrium and non-equilibrium parts, and then to approximate the non-equilibrium part with a first-order extrapolation of the non-equilibrium part of the distribution at the neighbouring fluid node. Schemes for velocity and pressure boundary conditions are constructed based on this method. The resulting schemes are of second-order accuracy. Numerical tests show that the numerical solutions of the LBM together with the present boundary schemes are in excellent agreement with the analytical solutions. Second-order convergence is also verified from the results. It is also found that the numerical stability of the present schemes is much better than that of the original extrapolation schemes proposed by Chen et al. (1996 Phys. Fluids 8 2527).     

用推广的Jacobian椭圆函数方法解离散的mKdV格子

本文用推广的Jacobian椭圆函数方法求解了离散的mKdV格子，得到了Jacobian椭圆函数双周期解，当模取1时，可得到钟型孤波解和冲击型孤波解。     

Multi Dromion-Solitoff and Fractal Excitations for （2＋1）-Dimensional Boit i-Leon-Manna-Pempinelli System

In this short note, a new projective equation （Ф = σФ ＋ Ф^2） is used to obtain the variable separation solutions with two arbitrary functions of the （2＋1）-dimensional Boiti-Leon-Manna Pempinelli system （BLMP）. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as multi dromion-solitoffs and fractal-solitons are investigated.     

长短波相互作用方程的Jacobi椭圆函数求解

推广了Jacobi椭圆函数展开方法,研究了复非线性演化方程组的求解问题,得到了长短波相互作用方程的准确包络周期解.该结果在一定条件下包含了相应的孤波解. 关键词： Jacobi椭圆函数方法 长短波相互作用方程 孤波解