互反型高维可积Kaup-Newell系统

可积系统研究是物理和数学等学科的重要研究课题.然而,通常的可积系统研究往往被限制在(1+1)维和(2+1)维,其原因是高维可积系统极其稀少.最近,我们发现利用形变术可以从低维可积系统导出大量的高维可积系统.本文利用形变术,将(1+1)维的Kaup-Newell(KN)系统推广到(4+1)维系统.新系统除了包含原来的(1+1)维的KN系统外,还包含三种(1+1)维KN系统的互反形式.模型也包含了许多新的(D+1)维(D≤3)的互反型可积系统.(4+1)维互反型KN系统的Lax可积性和对称可积性也被证明.新的互反型高维KN系统的求解非常困难.本文仅研究(2+1)维互反型导数非线性薛定谔方程的行波解,并给出薛定谔方程孤子解的隐函数表达式.

Non-Lie Symmetry Groups of （2＋1）-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.

Exact Solutions for a Higher-Order Nonlinear Schrodinger Equation in Atmospheric Dynamics

By giving prior assumptions on the form of the solutions, we succeed to find several exact solutions for a higher-order nonlinear Schroetinger equation derived from one important model in the study of atmospheric and ocean dynamical systems. Our analytical solutions include bright and dark solitary waves, and periodical solutions, which can be used to explain atmospheric phenomena.

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.

A Maple Package on Symbolic Computation of Conserved Densities for （1＋l）-Dimensional Nonlinear Evolution Systems

A new algorithm for symbolic computation of polynomial-type conserved densities for nonlinear evolution systems is presented. The algorithm is implemented in Maple. The improved algorithm is more efficient not only in removing the redundant terms of the genera/form of the conserved densities but also in solving the conserved densities with the associated flux synchronously without using Euler operator. Furthermore, the program conslaw.mpl can be used to determine the preferences for a given parameterized nonlinear evolution systems. The code is tested on several well-known nonlinear evolution equations from the soliton theory.

Approximate Generalized Conditional Symmetries for Perturbed Evolution Equations

The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.

Interactions among Periodic Waves and Solitary Waves of the(2+1)-Dimensional Konopelchenko-Dubrovsky Equation

By using the truncated Painlevéanalysis and the generalized tanh function expansion approaches,many interaction solutions among solitons and other types of nonlinear excitations of the Konopelchenko–Dubrovsky(KD)equation can be obtained.Particularly,the soliton-cnoidal wave interaction solutions are studied by means of the Jacobi elliptic functions and the third type of incomplete elliptic integrals.

CTE Solvability and Exact Solution to the Broer-Kaup System

A consistent tanh expansion(CTE)is used to solve the Broer–Kaup(BK)system.It is proved that the BK system is CTE solvable.Some exact interaction solutions among different nonlinear excitations such as solitons,rational waves,periodic waves,error function waves and any Burgers waves are explicitly given.

Coupled Nonlinear Schrodinger Equation: Symmetries and Exact Solutions

The symmetries, symmetry reductions, and exact solutions of a coupled nonlinear Schrodinger （CNLS） equation derived from the governing system for atmospheric gravity waves are researched by means of classical Lie group approach in this paper. Calculation shows the CNLS equation is invariant under some Galilean transformations, scaling transformations, phase shifts, and space-time translations. Some ordinary differential equations are derived from the CNLS equation. Several exact solutions including envelope cnoidal waves, solitary waves and trigonometric function solutions for the CNLS equation are also obtained by making use of symmetries.

Approximate Symmetries and Infinite Series Symmetry Reduction Solutions to Perturbed Kuramoto-Sivashinsky Equation

Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.

Damping Law of Photocount Distribution in a Dissipative Channel

For a dissipative channel governed by the master equation of the density operator describing the photon loss,we find that the photocount distribution formula at time𝑢can be related to the initial photocount distribution by replacing the efficiency of the detector𝜊ζwith𝜊ζe^(−2kt)𝜆𝑢,as if the quantum efficiencyζ𝜊of the detector becomes𝜊ζe^(−2kt)𝜆𝑢.This law greatly simplifies the theoretical study of the photocount distribution for quantum optical fields.

Bäcklund Transformations and Interaction Solutions of the Burgers Equation

The Burgers equation is one of the most important prototypic models in nonlinear physics.Various exact solutions of the Burgers equation have been found by many methods.However,it is very difficult to find interactive solutions among different types of nonlinear excitations.We develop a generalized tanh function expansion approach,which can be considered as the Bäcklund transformation,to find interactive solutions between the soliton and other types of Burgers waves.

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.

Direct Perturbation Method: Nonlinear Schrodinger Equation with Loss

Using the nonlinear Schrodinger(NLS)equation,which is used to describe the propagation of the solitons in many real physical systems like fiber and plasma,as a simple example,a direct perturbation method is established.Up to the adiabatic(zero order)approximation,any waves of the NLS equation decay in the same rate.Especially,different from the known claims in literature,the decay rate of the dark soliton in fiber is the same as that of the bright soliton.Starting from any one of the infinitely many adiabatic symmetries(or conservation laws)of the nonperturbative NLS equation,one can get the same adiabatic solutions.An adiabatic symmetry by multiplying a decay factor is just the first order modification.Higher order modifications can be obtained by solving linear equations.

Dark Parameterizations,Equivalent Partner Fields and Integrable Systems

After introducing dark parameters into the traditional physical models, some types of new phenomena may be found. An important difficult problem is how to directly observe this kind of physical phenomena. An alternative treatment is to introduce equivalent multiple partner fields. If use this ideal to integrable systems, one may obtain infinitely many new coupled integrable systems constituted by the original usuM field and partner fields. The idea is illustrated via the celebrate KdV equation. From the procedure, some byproducts can be obtained： A new method to find exact solutions of some types of coupled nonlinear physical problems, say, the perturbation KdV systems, is provided; Some new localized modes such as the staggered modes can be found and some new interaction phenomena like the ghost interaction are discovered.

Lie Symmetry Groups of(2+1)-Dimensional BKP Equation and Its New Solutions

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Furthermore,the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.

A New Coupled KdV Equation： Painlevé Test

A new type of coupled Korteweg de-Vries equation is found to be Painlevé-integrable. The new model is a special case which can be used to describe two-layer fluids with different dispersion relations.

Multi-soliton Solutions of the Konopelchenko-Dubrovsky Equation

By using the standard truncated Painlevé analysis, a Backlund transformation is used to obtain some new types of multi-soliton solutions of the (2+ 1)-dimensional integrable Konopelchenko-Dubrovsky equation from the trivial vacuum solution.

Some Special Types of Multisoliton Solutions of the Nizhnik-Novikov-Vesselov

Using the standard truncated Painleve analysis,we have obtained some new special types of soliton solutions of a(2-pl)-dimensional integrable model,the Nizhnik-Novikov-Vesselov equation.Starting from the standard truncation approach in the Painleve analysis,one can obtain a Backlund transformation to find a new solution from a known one.Usually,one can obtain only a single solitary wave solution from the Backlund transformation related to the truncated Painleve analysis starting from the trivial vacuum solution.In this paper,we find some special types of the multisoliton solutions from the truncated Painleve analysis and the trivial vacuum solution.

New Types of Exact Solutions for （N＋1）-Dimensional φ^4 Model

New types of exact solutions of the （N ＋ 1）-dimensional φ^4-model are studied in detail. Some types of interaction solutions such as the periodic-periodic interaction waves and the periodic-solitary wave interaction solutions are found.