（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

（3＋1）维破碎孤子方程的变量分离解和局域激发模式

多线性分离变量法已成功地应用于诸多（2＋1）维非线性可积系统.将该方法拓展运用于（3＋1）维破碎孤子方程中,获得了含任意函数的变量分离解.通过适当地设定任意函数的形式,得到了（3＋1）维破碎孤子方程丰富的局域激发模式.     

周期系数的高维Riccati方程的周期解

本文研究了周期系数的高维Riccati方程X’＝X·A（t）·X＋B（t）·X＋C（t）,其中X∈R(n×1)A（t）∈R(1×n),B（t）∈R(n×n),C（t）E∈R(n×1);A（t）,B（t）,C（t）均是以2π为周期的实连续矩阵或向量函数,建立了该方程存在广义周期解的一个充要条件和存在周期解的两个充分条件,推广了周期系数的Riccati方程存在周期解的一些结论．     

（2＋1）维非线性Schroedinger型方程的同宿轨道

研究了几类（2＋1）维非线性Schroedinger型方程同宿轨道的问题．利用Hirota双线性算子方法，通过给出的相关变换，得到了包括（2＋1）维的长短波相互作用方程，广义Zakharov方程，Mel’nikov方程和g-Schroedinger方程的同宿轨道解的显式解析表达式，从而讨论了这些方程的同宿轨道．     

1998年全国高中数学联合竞赛试题及参考答案

一、选择题（本题满分36分，每小题6分）1．若a＞1，b＞l，且ig（a＋b）一lga＋lgb，则ig（a—1）十ig（b—l）的值（）．（A）等于lgZ（B）等于1（C）等于0（D）不是与a，b无关的常数解’．“ig（a＋b）一lga＋lgb，a＋b—ah即（a—1）（b—1）＝1．因此ig（a—1）十ig（b—1）一0．答：（C）．2．若非空集合A一｛Xlb十互＜X＜u一引，B一《X口＜X＜22），则能使A二AnB成立的所有a的集合是（）．（A）（aDI＜a＜9｝（B）（】6＜a＜9）（C川ala＜9｝（D）o解由题意得AMB，答：（B）．3．各项均为实数的等比数列K）前n项之和记为S…     

Konopelchenko-Dubrovsky方程组的对称,精确解和守恒律

通过利用修正的CK直接方法,建立了Konopelchenko-Dubrovsky（KD）方程组的新旧解之间的关系．利用李群分析方法,得到了（2＋1）维KD方程的对称、相似约化和新的精确解,包括指数函数解、双曲函数解、和三角函数解．同时找到了此方程的无穷多守恒律．     

(2+1)-维广义Benney-Luke方程的精确行波解

用平面动力系统方法研究（2＋1）＋维，“义Benney-Luke方程的精确行波解，获得了该方程的扭波解，不可数无穷多光滑周期波解和某些无界行波解的精确的参数表达式，以及上述解存在的参数条件．     

海仑数组的探求

如果三角形的三边长为整数且面积亦为整数，则称之为海仑三角形．海仑三角形的三边长所构成的数组（a，b，c）称之为海会数组．本文对海会数组进行新的探索．假定D＞0，D不是平方数，c是非0整数．设x=u，y=V是不定方程x~2-Dy~2=c的一个解，那么就称u＋v是它的一个解．其中当u≥0，v≥0时，最小的一个叫做基本解．再设x＋y是Pell方程X~2-Dy~2=1的任意一个解，则容易验证（u十v）（X十y）（=ux＋uyD＋（ux＋ut）也是x~2-Dy~2=c的解．设三角形三边长分别为a，从一a十…，C，其中p为奇数（可正可负）．则其面积为由于这个关于C’的M次方程…     

一类时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数，结合上、下解方法及相应的单调迭代方法研究了一类时滞反应扩散方程，证明了在反应项非单调时，如果一雏边值问题存在一对周期（或概周期）上、下解，则方程一定存在唯一的周期（或概周期）解．并给出了二维边值问题周期（或概周期）解存在唯一性的充分条件．推广了已有的一些结果。     

一类非线性薛定谔方程的多解的存在性

本文讨论了如下一类非线性薛定谔方程：-△u＋V（x）u=f（u）,x∈R^N,在H^1（R^N）中无穷多解的存在性,其中N≥3,V（x）是RN上的实值连续函数并且满足对（A）x∈R^N,V（z）≥V0＞0.     

一个2+1维变形Boussinesq方程的N孤子解

研究了一个2＋1维变形Boussinesq非线性发展方程：utt-uxx-uyy-3（u^2）xx-uxxxx=0,运用Hirota双线性方法得到它的N孤子解.     

Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions

One of the more interesting solutions of the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in 2+1 dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in 2+1 dimensions. It is interesting that neither of the (2+1)-dimensional integrable systems considered admit Virasoro-type subalgebras.     

RLW-Burgers方程的一类解析解

本文给出了 RLW-Burgers方程及 Kd V-Burgers方程的一类解析解 ,且可得到 RLW-Burgers方程的振荡激波解 .这些解可以表示为 Burgers方程和 Kd V方程解的线性组合 ,文末还对文 [8]作了讨论 .     

Traveling-Wave Solutions of the Calogero-Degasperis-Fokas Equation in 2+1 Dimensions

Soliton solutions are among the more interesting solutions of the (2+1)-dimensional integrable Calogero-Degasperis-Fokas (CDF) equation. We previously derived a complete group classiffication for the CDF equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. The solutions consequently exhibit a rich variety of qualitative behaviors. Choosing the arbitrary functions appropriately, we exhibit solitary waves and bound states.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 44–55, July, 2005.     

(2+1)维拟线性扩散方程的不变集和精确解

研究(2+1)维拟线性扩散方程的精确解问题.运用推广的不变集方法,给出(2+1)维拟线性扩散方程的一些特殊解.此方法是(1+1)维拟线性扩散方程的推广.     

New exact solutions to the CDF equations

New exact soliton solutions to the Cologero–Degasperies–Fokas (CDF) equations in (1+1)-dimension and (2+1)-dimension by using the improved tanh method are investigated. First, the (1+1)-dimensional CDF equation is analyzed. By the improved tanh method, the corresponding nonlinear partial differential equation is reduced to the nonlinear ordinary differential equations and then the different types of exact solutions to the original equation are obtained based on the solutions of the Riccati equation. For the case of (2+1)-dimensional CDF equation the same computation procedure is carried out. It is presented that one could obtain new exact explicit solutions, which are traveling wave solutions, to (2+1)-dimensional CDF equation. Additionally, some graphical representations of the solitary and periodic solutions are presented.     

A new note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation

Exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili by Zhang [Huiqun Zhang, A note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation, Appl. Math. Comput. 216 (2010) 2771-2777] are considered. To look for “new types of exact solutions travelling wave solutions” of equation Zhang has used the G′/G-expansion method. We demonstrate that there is the general solution for the reduction by Zhang from the (2+1)-dimensional Kadomtsev-Petviashvili equation and all solutions by Zhang are found as partial cases from the general solution.     

New multisoliton solutions of the(2+1)-dimensional dispersive long wave equations

IntroductionSoliton is a complicated mathematical structure based on the nonlinear evolution equation.(1+ 1)-dimensional soliton and solitary wave solutions have been studied we1l and widely appliedto many physics fields like the condense matter physics, fluid mechanics, plasma physics, optics,etc. However, to find some exact physically significant soliton solutions in (2+l)-dimensions ismuch more difficult than in (1+1)-dimensions. Recently, by using some different approashes,one special type…     

Abundant soliton structures of (2+1)-dimensional NLS equation

In this paper, we study the possible localized coherent solutions of a (2+1)-dimensional nonlinear Schrödinger (NLS) equation. Using a Bäcklund transformation and the variable separation approach, we find that there exist much more abundant localized structures for the (2+1)-dimensional NLS equation because of the entrance of an arbitrary function of the seed solution. Some special types of the dromion solutions, breathers, instantons and dromion solutions with oscillated tails are discussed by selecting the arbitrary functions appropriately. The dromion solutions can be driven by some sets of straight-line and curved line ghost solitons. The breathers may breath both in amplitudes and in shapes.     

(2+1)维广义Burgers 方程的Lie点对称, 相似约化和精确解

讨论了(2+1)维广义Burgers方程.通过Lie群方法求出了该方程的李点对称,并利用李点对称将方程进行相似约化,求出了(2+1)维广义Burgers方程的几种精确解.该方法可以用于研究更高阶的偏微分方程.