Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.     

Invariants of Multi-Component Ermakov Systems

It is shown that certain multi-component Ermakov systems admit Lewis-Ray-Reid invariants. This extends the result to the two-component Ermakov system.     

Quantum Systems Connected by a Time-Dependent Canonical Transformation

We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is timedependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.     

Normal coordinate in harmonic crystal obtained virtue of the classical correspondence of the invariant eigen-operator

Noticing that the equation with double-Poisson bracket, where On is normal coordinate, Hc is classical Hamiltonian, is the classical correspondence of the invariant eigen-operator equation （2004 Phys. Left. A. 321 75）, we can find normal coordinates in harmonic crystal by virtue of the invaxiant eigen-operator method.     

Energy Level of Three-Mode Harmonic Oscillator for Coordinate Operators Satisfying Cyclic Commutative Relations Obtained by IEO Method

Eigenvalue-solution to those Hamiltonians involving non-commutative coordinates is not easily obtained. In this paper we apply the invariant eigen-operator （IEO） method to solving the energy spectrmn of the three-mode harmonic oscillator in non-commutative space with the coordinate operators satisfying cyclic commutative relations, [X1, X2] = [X2, X3]=[X3, X1] = iθ, and this method seems effective and concise.     

Evolution and Generation of Dynamics of Two-Mode Squeezed States of Time-Dependent Coupled Boson System

The system of the Hamiltonian involving a driving part, a single mode part, and a two-mode squeezed one and a two-mode coupled one is discussed using the Lewis-Riesenfeld invariant theory. The time evolution operator is obtained. When the initial state is a coherent state, the quantum fluctuation of the system is calculated, and it is dependent on the squeezed part and the two-mode coupled part, but not dependent on the driving one.     

Kac-Moody-Virasoro Symmetry Algebra of （2＋1）-Dimensional Dispersive Long-Wave Equation with Arbitrary Order Invariant

By Lie symmetry method, the Lie point symmetries and its Kac Moody-Virasoro （KMV） symmetry algebra of （2＋1）-dimensional dispersive long-wave equation （DLWE） are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.     

Invariant expression for tunneling rate under canonical transformations

The original formula to calculate the tunneling rate through event horizons is apparently dependent on the type of coordinates used.In this paper,we propose an invariant expression under canonical transformations to study the tunneling effect.Moreover,the problem of factor 2 is solved naturally.As an application of this expression,we obtain the same tunneling rate both in the Schwarzschild and the Painleve′coordinates.It is shown that once the suitable formula to calculate tunneling rate is correctly identified,the tunneling method is manifestly covariant.     

Classification and Approximate Solutions to Perturbed Nonlinear Diffusion-Convection Equations

This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admit certain types of AGCSs is derived. Some approximate invariant solutions to the resulting equations can also be obtained.     

Symmetry Analysis of Two Types of （2＋1）-Dimensional Nonlinear Klein-Gorden Equation

By means of the classical symmetry method, we investigate two types of the （2＋1）-dimensional nonlinear Klein-Gorden equation. For the wave equation, we give out its symmetry group analysis in detail. For the second type of the （2＋1）-dimensional nonlinear Klein-Gorden equation, an optimal system of its one-dimensional subalgebras is constructed and some corresponding two-dimensional symmetry reductions are obtained.     

Analytic Study of a Bose-Einstein Condensate in Waveguide with an Obstacle Potential

We investigate the exact solutions of one-dimensional （1D） time-independent Gross-Pitaevskii equation （GPE）, which governs a Bose-Einstein condensate （BEC） in the magnetic waveguide with a square-Sech potential. Both the bound state and transmission state are found and the corresponding spatial configurations and transport properties of BEC are analyzed. It is shown that the well-known absolute transmission of the linear system can occur in the considered nonlinear system.     

Perturbation to Noether-Lie Symmetry and Adiabatic Invariants for Mechanical Systems in Phase Space

Based on the concept of adiabatic invariant, the perturbation to Noether-Lie symmetry and adiabatic invariants for mechanical systems in phase space are studied. The criterion of the Noether Lie symmetry for the perturbed system is given, and the definition of the perturbation to Noether-Lie symmetry for the system under the action of small disturbance is presented. Meanwhile, the Noether adiabatic invariants and the generalized Hojman adiabatic invariants of the perturbed system are obtained.     

Representations and Classification of Traveling Wave Solutions to sinh-GSrdon Equation

Two concepts named atom solution and combinatory solution are defined. The classification of all single traveling wave atom solutions to sinh-Gordon equation is obtained, and qualitative properties of solutions are discussed. In particular, we point out that some qualitative properties derived intuitively from dynamic system method are not true. Finally, we prove that our solutions to sinh-Gordon equation include all solutions obtained in the paper [Z.T. Fu, et al., Commun. Theor. Phys. （Beijing, China） 45 （2006） 55]. Through an example, we show how to give some new identities on Jacobian elliptic functions.     

A New Route to the Interpretation of Hopf Invariant

We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the Ф-mapping topological current theory. The main purpose of this paper is to present a new theoretical framework, which can directly give the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclidean space R^2n-1. For the sake of this purpose we introduce a topological tensor current, which can naturally deduce the （n- 1）-dimensional topological defect in R^2n-1 space. If these （n- 1）-dimensional topological defects are closed oriented submanifolds of R^2n-1, they are just the （n - 1）-dimensional knots. The linking number of these knots is well defined. Using the inner structure of the topological tensor current, the relationship between Hopf invariant and the linking numbers of the higher-dimensional knots can be constructed.     

Symmetry, Reductions and New Exact Solutions of ANNV Equation Through Lax Pair

In this paper, the direct symmetry method is extended to the Lax pair of the ANNV equation. As a result, symmetries of the Lax pair and the ANNV equation are obtained at the same time. Applying the obtained symmetry, the （2＋1）-dimensional Lax pair is reduced to （1＋1）-dimensional Lax pair, whose compatibility yields the reduction of the ANNV equation. Based on the obtained reductions of the ANNV equation, a lot of new exact solutions for the ANNV equation are found. This shows that for an integrable system, both the symmetry and the reductions can be obtained through its corresponding Lax pair.     

Analytical Solitary Wave Solutions to a （3＋1）-Dimensional Gross-Pitaevskii Equation with Variable Coefficients

A （3＋1）-dimensional Gross-Pitaevskii （GP） equation with time variable coefficients is considered, and is transformed into a standard nonlinear Schrodinger （NLS） equation. Exact solutions of the （3＋1）D GP equation are constructed via those of the NLS equation. By applying specific time-modulated nonlinearities, dispersions, and potentials, the dynamics of the solutions can be controlled. Solitary and periodic wave solutions with snaking and breathing behavior are reported.     

Fusion,fission, and annihilation of complex waves for the （2＋l）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system

With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Objective Reduction Solutions to Higher-Order Boussinesq System in （2＋1）-Dimensions

With the help of an objective reduction approach （ORA）, abundant exact solutions of （2＋1）-dimensional higher-order Boussinesq system （including some hyperboloid function solutions, trigonometric function solutions, and a rational function solution） are obtained. It is shown that some novel soliton structures, like single linearity soliton structure, breath soliton structure, single linearity y-periodic solitary wave structure, libration dromion structure, and kink-like multisoliton structure with actual physical meaning exist in the （2＋1）-dimensional higher-order Boussinesq system.     

Perturbation to Lie-Mei Symmetry and Adiabatic Invariants for Birkhoffian Systems

Based on the concept of adiabatic invariant, the perturbation to Lie-Mei symmetry and adiabatic invariants for Birkhoffian systems are studied. The definition of the perturbation to Lie-Mei symmetry for the system is presented, and the criterion of the perturbation to Lie-Mei symmetry is given. Meanwhile, the Hojman adiabatic invariants and the Mei adiabatic invariants for the perturbed system are obtained.