广义修正Boussinesq方程的孤波解与周期解

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3.0 牛悦 2024-11-19 4 4 696.46KB 60 页 15积分
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摘 要
本文利用假设待定法，讨论了广义修正 Boussinesq 方程
0)( 3
3
2
21 xxxxtttt ubububuu
)1(
具Jacobi 椭圆函数分式形式的周期解，分别就
0
3b
和
0
2b
给出了这种周期解的
具体形式.求出了广义修正 Boussinesq 方程型如
)(sec
)(sec
)( 2
10
2
10


hBB
hAA
us
,
)( vtx
)2(
的钟状孤波解和型如
)(cos
)(cos
)( 2
10
2
10


BB
AA
up
,
)( vtx
)3(
的余弦函数周期波解，并分别讨论了它们的有界性，揭示了行波波速的改变对上
述钟状孤波解和余弦函数周期波解波形变化的相关性.
本文分为五章.第一章，着重介绍了非线性发展方程求解的概况，广义修正
Boussinesq 方程的研究状况和本文所需的预备知识.
在第二章中，我们具体研究广义修正 Boussinesq 方程
)1(
的具 Jacobi 椭圆函数
分式形式的精确周期解.首先运用积分方法把求方程（1）的行波解问题化为非线性
常微分方程：
)4(
的求解问题.通过假设方程
)4(
具有型如
)(
)(
)( 2
10
2
10


cnBB
cnAA
u
)5(
形式的 Jacobi 椭圆函数分式形式的周期解，我们把求型如
)5(
解的问题化为了一个
求12 次代数方程根的问题.文中对
0
3b
情形，求出了 6个型如
)5(
的精确周期解；
而对
0
2b
情形，求出了 12 个型如
)5(
的精确周期解.
第三章，我们求出了广义修正 Boussinesq 方程型如
)2(
的6个显式钟状孤波解.
其中 2个与以往文献中求出的结果一致，而其余 4个是本文给出的新解.文中经详
细讨论，给出了这 6个钟状孤波解有界的条件.
我们在第四章中，求出了广义修正 Boussinesq 方程型如
)3(
的6个余弦函数有
理分式形式的周期波解，并分别详细研究了它们的有界性，以定理形式给出了这些
解的有界性条件.
在前面几章的基础上，我们在第五章中，对比所求的孤波解和余弦函数周期
波解的有界性条件，揭示了行波波速的改变对这两种解形状变化的相关性.
关键词：广义修正 Boussinesq 方程 Jacobi 椭圆函数 孤波解
精确周期解
ABSTRACT
In this paper, exact periodic solutions with Jacobi elliptic functions fractional form
for the generalized modified Boussinesq equation
0)( 3
3
2
21 xxxxtttt ubububuu
)1(
are obtained by use of undetermined assumption method，in terms of the method，
some
new exact solitary-wave solutions are found for the equation. Respectively,
0
3b
and
0
2b
, we give the detail form. We obtained the bell profile solitary wave solutions for
the generalized modified Boussinesq equation as following
)(sec
)(sec
)( 2
10
2
10


hBB
hAA
us
,
)( vtx
,
)2(
and the periodic wave solutions with cosine function form
)(cos
)(cos
)( 2
10
2
10


BB
AA
up
,
)( vtx
,
)3(
besides, we discussed its bounded property, as well as the problem of solitary-wave
solution. Furthermore, we discovered the correlative characteristic between the exact the
bell-type solitary wave and the periodic wave solutions with cosine function form as the
traveling wave velocity varying.
The paper is separated into 5 chapters: chapter 1 is devoted to reviewing the epitome
about looking for the solutions to many nonlinear evolution equations and introducing
the research achievements for generalized modified Boussinesq equation and required
preliminary knowledge.
In chapter 2, we principally investigated exact periodic solutions with Jacobi elliptic
functions fractional form for the generalized modified Boussinesq equation in detail.
First, by integrating Eq
)1(
, the problem of finding the solutions of Eq
)1(
leads to
solving the following nonlinear ODE:
.0)()()()( 32
husuluu
)4(
we assume that the periodic solution of Eq
)4(
has the following form
)(
)(
)( 2
10
2
10


cnBB
cnAA
u
,
)5(
thus, the problem of finding the solutions with the form
)5(
leads to seek the root of a
12-order algebraic equation. As
0
3b
, six exact periodic solutions as the pattern
)5(
are obtained; As
0
2b
，twelve exact periodic solutions as the pattern
)5(
are obtained
in the text.
In chapter 3, we obtained six explicit bell profile solitary-wave solutions with the
form
)2(
for Eq
)1(
, 2 of which are uniform with the results of the ancient literature,
whereas the other 4 solutions are the new solitary wave solutions. Moreover, we give
the condition of the 6 solitary-wave solutions bounded property detailedly.
In chapter 4, we obtained six exact periodic solutions with Jacobi elliptic functions
fractional form with the form
)3(
for the Eq
)1(
, and we discussed their boundary
problem in detail and the bounded conditions are stated in theorem
On the basis of several former chapters, comparing the bounded conditions of the
solitary-wave solutions and the periodic wave solutions with cosine function form, we
discovered the correlative characteristic between the exact the bell profile solitary wave
and the periodic wave solutions with cosine function form as the traveling wave
velocity varying.
Key word: generalized modified Boussinesq equation，Jacobi elliptic
function，solitary-wave solution，exact periodic solution
目 录
摘 要
ABSTRACT
第一章 绪 论
§1.1 非线性发展方程求解的概况 ....................................... 1
§1.2 广义修正 BOUSSINESQ 方程的研究状况 ................................ 2
§1.3 预备知识 ....................................................... 3
§1.3.1 JACOBI 椭圆函数 ............................................... 3
第二章 广义修正 Boussinesq 方程的精确周期解
§2.1 B
3
=0 的情形 .................................................... 9
§2.2 B
2
=0 的情形 ................................................... 12
第三章 精确周期解的极限情形与钟状孤波解有界性讨论
§3.1 精确周期解的极限情形 .......................................... 15
§3.2 钟状孤波解有界性讨论 .......................................... 20
§3.2.1 对于
)(),( 21
ss uu
............................................. 20
§3.2.2 对于
)(),( 43
ss uu
.............................................20
§3.2.2.1
0
2
4,3
的条件 ............................................. 20
§3.2.2.2
1
Q
有意义的条件 ...........................................21
§3.2.2.3
)(),( 43
ss uu
分母不为零的条件 ............................... 23
§3.2.3 对于
)(),( 65
ss uu
.............................................25
§3.2.3.1
0
2
6,5
的条件 ............................................. 25
§3.2.3.2
2
Q
有意义的条件 .......................................... 26
§3.2.3.3
)(),( 65
ss uu
分母不为零的条件 ............................... 28
第四章 余弦函数周期波解及其有界性证明
§4.1 余弦函数有理分式形式的周期波解 ................................ 31
§4.2 关于余弦函数周期波解的有界性讨论 .............................. 32
§4.2.1 对
)(
1
p
u
有界性的讨论 ......................................... 32
§4.2.2 对
)(
2
p
u
有界性的讨论 .........................................34
§4.2.3 对
)(
3
p
u
的有界性讨论 ........................................ 35
§4.2.4 对
)(
4
p
u
的有界性讨论 ........................................ 40
§4.2.5 对
)(
5
p
u
有界性的讨论 .........................................41
§4.2.6 对
)(
6
p
u
的有界性讨论 ........................................ 45
第五章 有界孤波解与有界余弦函数周期解之间的关系
§5.1 孤波解与余弦函数周期解之间的关联性 ............................ 49
参考文献 ............................................................ 52
在读期间公开发表的论文和承担科研项目及取得成果 ...................... 55
致 谢 ............................................................. 55
摘要：

摘要本文利用假设待定法，讨论了广义修正Boussinesq方程0)(33221xxxxttttubububuu)1(具Jacobi椭圆函数分式形式的周期解，分别就03b和02b给出了这种周期解的具体形式.求出了广义修正Boussinesq方程型如)(sec)(sec)(210210hBBhAAus,)(vtx)2(的钟状孤波解和型如)(cos)(cos)(210210BBAAup,)(vtx)3(的余弦函数周期波解，并分别讨论了它们的有界性，揭示了行波波速的改变对上述钟状孤波解和余弦函数周期波解波形变化的相关性.本文分为五章.第一章，...

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作者：牛悦 分类：高等教育资料 价格：15积分 属性：60 页 大小：696.46KB 格式：PDF 时间：2024-11-19

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