Variable Separated Solutions and Four-Dromion Excitations for (2+1)-Dimensional Nizhnik-Novikov-Veselov Equation

Using the mapping approach via the projective Riccati equations, several types of variable separated solutions of the （2＋1）-dimensional Nizhnik-Novikov-Veselov equation are obtained, including multiple-soliton solutions, periodic-soliton solutions, and Weierstrass function solutions. Based on a periodic-soliton solution, a new type of localized excitation, i.e., the four-dromion soliton, is constructed and some evolutional properties of this localized structure are briefly discussed.

NUMERICAL SOLUTION OF NAVIER-STOKES EQUATION OF UNSTEADY SEPARATED FLOWS DUE TO A SPOILER'S OSCILLATION

The finite difference method (FDM) is applied in the present paper to solve the unsteady NHS equations for incompressible fluids. ADI and SLOR methods are served for the vorticity equation and the Poisson equation for ψ respectively. The upwind scheme is used for the convective terms. The moving boundary conditions are specially treated, and the effects of outlet conditions on the flow field are abo examined. Numerical results obtained show that the spoiler's oscillation induces forming, growing and shedding of the vortices. The shedding frequency of vortices is equal to that of the spoiler's oscillation. The forced unsteady separated flows under the present investigation depend mainly on the reduced frequency. At low reduced frequency, the vortices shed from the spoiler interact weakly with each other, and move downstream at an almost uniform speed of 038 V∞. At high reduced frequency, the interaction between the adjacent vortices strengthens. They close up to and rotate around each other, and eventually, merge into one vortex.

Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized Nonlinear Wave Equations

Using the generalized conditional symmetry approach, we obtain a number of new generalized (1+1)-dimensional nonlinear wave equations that admit derivative-dependent functional separable solutions.

Extension of Variable Separable Solutions for Nonlinear Evolution Equations

We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.

Classification and Functional Separable Solutions to Extended Nonlinear Wave Equations

The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.

The projective Riccati equation expansion method and variable separation solutions for the nonlinear physical differential equation in physics

Using the projective Riccati equation expansion （PREE） method, new families of variable separation solutions （including solitary wave solutions, periodic wave solutions and rational function solutions） with arbitrary functions for two nonlinear physical models are obtained. Based on one of the variable separation solutions and by choosing appropriate functions, new types of interactions between the multi-valued and single-valued solitons, such as a peakon-like semi-foldon and a peakon, a compacton-like semi-foldon and a compacton, are investigated.

New variable separation solutions for the generalized nonlinear diffusion equations

The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A（u, Ux）Uxx ＋ B（u, ux） is studied by using the conditional Lie-Blicklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie--Biicklund symmetries, are characterized. To construct functionally gener- alized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.

STUDIES ON REVERSE OSMOSIS SEPARATION OF AQUEOUS ORGANIC SOLUTIONS BY PAA/PSF COMPOSITE MEMBRANE

The reverse osmosis (RO) separation of aqueous organic solutions, such as alcohols, amines, aldehydes, acids, ketones, and esters etc., by PAA (polyacrylic acid)/PSF (polysulfone) composite membrane has been studied. It was found that the separation results for aliphatic alcohols, amines and aldehydes are satisfactory, the solute rejection (R-a) and the volume fluxes of solutions (J(V)) for 1000 ppm ethanol, ethylamine and ethyl aldehyde are 66.2%, 61.0%, 84.0% and 0.90 x 10(-6), 0.35 x 10(-6), 0.40 X 10(-6) m(3)/m(2) . s, respectively, at 5.0 MPa and 30 degrees C. R-a increased with increasing molecular weights of alcohols, amines and aldehydes, and the R-a for n-amyl alcohol, n-butylamine and n-butyl aldehyde reached 94.3%, 88.6% and 96.0%, respectively. Satisfactory separation results (R-a > 70%) for ketones, esters, phenols and polyols have been obtained with the PAA/PSF composite membrane. The effect of operating pressure on the properties of reverse osmosis has also been investigated. Analysis of experimental data with Spiegler-Kedem's transport model has been carried out and the membrane constants such as reflection coefficient sigma, solute and hydraulic permeabilities omega and L-p for several organic solutes have been obtained.

New Exact Solutions of (1+1)-Dimensional Coupled Integrable Dispersionless System

This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more general variable separation solutions with two arbitrary functions of （1＋1）-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p（x） and q（t） in our solutions, the annihilation phenomena of the fiat-basin soliton, arch-basin soliton, and fiat-top soliton are discussed.

Fusion and fission solitons for the (2+1)-dimensional generalized Breor-Kaup system

With a projective equation and a linear variable separation method, this paper derives new families of variable separation solutions （including solitory wave solutions, periodic wave solutions, and rational function solutions） with arbitrary functions for （2＋1）-dimensional generalized Breor-Kaup （GBK） system. Based on the derived solitary wave excitation, it obtains fusion and fission solitons.

Optimization of air quantity regulation in mine ventilation networks using the improved differential evolution algorithm and critical path method

In mine ventilation networks, the reasonable airflow distribution is very important for the production safety and economy. Three basic problems of the natural, full-controlled and semi-controlled splitting were reviewed in the paper. Aiming at the high difficulty semi-controlled splitting problem, the general nonlinear multi-objectives optimization mathematical model with constraints was established based on the theory of mine ventilation networks. A new algorithm, which combined the improved differential evaluation and the critical path method （CPM） based on the multivariable separate solution strategy, was put forward to search for the global optimal solution more efficiently. In each step of evolution, the feasible solutions of air quantity distribution are firstly produced by the improved differential evolu- tion algorithm, and then the optimal solutions of regulator pressure drop are obtained by the CPM. Through finite steps iterations, the optimal solution can be given. In this new algorithm, the population of feasible solutions were sorted and grouped for enhancing the global search ability and the individuals in general group were randomly initialized for keeping diversity. Meanwhile, the individual neighbor- hood in the fine group which may be closely to the optimal solutions were searched locally and slightly for achieving a balance between global searching and local searching, thus improving the convergence rate. The computer program was developed based on this method. Finally, the two ventilation networks with single-fan and multi-fans were solved. The results show that this algorithm has advantages of high effectiveness, fast convergence, good robustness and flexibility. This computer program could be used to solve lar^e-scale ~eneralized ventilation networks o^timization problem in the future.

Variable Separation for(1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.

Evolutional Properties of Localized Excitations for Generalized Broer-Kaup System in （2＋1） Dimensions

Using a special Painleve-Baecklund transformation as well as the extended mapping approach and the linear superposition theorem, we obtain new families of variable separation solutions to the （2＋1）-dimensional generalized Broer-Kaup （GBK） system. Based on the derived exact solution, we reveal some novel evolutional behaviors of localized excitations, i.e. fission and fusion phenomena in the （2＋1）-dimensional GBK system.

A Note on "Doubly Periodic Propagating Wave for (2+1)-Dimensional Breaking Soliton Equation"

In a recent paper [Commun. Theor. Phys. （Beijing, China） 49 （2008） 268], Huang et al. gave a general variable separation solution to the （2＋1）-dimensional breaking soliton equation via a special Biicldund transformation and the variable separation approach. In terms of the derived variable separation solution and by introducing Jacobi elliptic functions, they claimed that nonelastic types of interaction between Jacobi elliptic function waves are investigated both analytically and graphically. We show that some inappropriateness or errors exist in their paper, and say nothing of the conclusion that some nonelastic types of interaction between Jacobi elliptic function waves in the （2＋1）-dimensional breaking soliton equation have been found.

Non-completely elastic interactions in a(2+1)-dimensional dispersive long wave equation

With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the （24-1）-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.

Conditional Lie-Bcklund Symmetry and New Variable Separation Solutions of the Third Order KdV-Type Equations

The functionally generalized variable separation solutions of a general KdV-type equations u_t=u_(xxx) +A(u, u_x)u_(xx) + B(u, u_x) are investigated by developing the conditional Lie-Backlund symmetry method. A complete classification of the considered equations, which admit multi-dimensional invariant subspaces governed by higher-order conditional Lie-B¨acklund symmetries, is presented. As a result, several concrete examples are provided to construct functionally generalized variable separation solutions of some resulting equations.

Invariant subspaces and conditional Lie-Bcklund symmetries of inhomogeneous nonlinear difusion equations

The inhomogeneous nonlinear diffusion equation is studied by invariant subspace and condi- tional Lie=Bgcklund symmetry methods. It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary differential equations, which is equivalent to a kind of higher=order conditional Lie-B^icklund symmetries of the equations. As a consequence, a number of new solutions to the inhomogeneous nonlinear diffusion equations are constructed explicitly or reduced to solving finite-dimensional dynamical sys- tems.

Interaction Behaviors Between Special Dromions in the(2+1)-Dimensional Broer-Kaup-Kupershmidt Equation

A modified mapping method is used to obtain variable separation solution with two arbitrary functions of the(2+1)-dimensional Broer-Kaup-Kupershmidt equation.Based on the variable separation solution and by selecting appropriate functions,we discuss the completely elastic head-on collision between two dromion-lattices,non-completely elastic "chase and collision" between two multi-dromion-pairs and completely non-elastic interaction phenomenon between anti-dromion and dromion-pair.