非线性波动模型的相干结构及其相互作用
摘要
孤立波，作为一种特殊的相干结构，反映了一类稳定的自然波动现象。本学位论文以源于实际水波和其它物理问题的2+1维(二个空间变量，一个时间变量)非线性波动模型(方程)为研究对象，深入研究了它们的求解问题，揭示出2+1维非线性波动模型不仅蕴涵比1+1维非线性波动模型更加丰富的相干结构，而且具有多种不同方式的相互作用性质。主要工作如下：
第一部分：研究2+1维非线性波动模型的求解问题。在三个方面展开了工作：(1) 利用Hirota双线性方法进一步研究了2+1维Schrdinger方程，2+1维Maccari方程和2+1维破裂非线性波动方程，获得了具有不同特征并带有任意函数的广义dromion结构。(2) 反散射方法可以看成是非线性波动问题中的Fourier变换法，但分离变量法对非线性波动问题的推广直到最近才有所突破。变量分离法有效地解决了2+1维非线性波动模型的求解问题。我们分别利用Painlevé截断展开方法和齐次平衡方法实现了多类2+1维非线性波动模型，如2+1维Nizhnik-Novikov-Veselov方程，2+1维色散长波方程，2+1维长波短波共振作用方程，2+1维长色散波方程，2+1维广义Nizhnik-Novikov-Veselov方程等的变量分离求解，给出了一些新的结果，(3) 直接代数法在孤立波的求解中具有重要的地位，由于计算机符号软件的广泛应用，对直接代数法的完善、发展和应用起了很好的作用。我们对近年创立的Jacobi椭圆函数方法和形变映射方法作了介绍和讨论，并具体研究了多类2+1维非线性波动模型，如2+1维Nizhnik-Novikov-Veselov方程，2+1维长短波共振方程，2+1维破裂孤子方程，2+1维非线性Burgers方程，2+1维Wu-Zhang方程，2+1维Schr(o|¨)dinger方程，2+1维Boussinesq方程，给出了双周期波结构和迭加椭圆函数周期解。
第二部分：研究不同类型的2+1维相干结构的构造和它们相互作用问题。在诸多专家学者相关研究工作的启示和基础上，从2+1维非线性波动方程的变量分离通用解表示出发，通过适当地选择这些任意函数，构造起2+1维非线性波动模型具有十分丰富的相干结构，包括多线孤立波解，多lump解，多solitoff解，多dormion解，多compacton解，多peakon解，多foldon解，格子dormion解，振荡型dormion解，环孤子解，运动和静止呼吸子解，瞬子解，双周期波斑图结构，混沌斑图结构，分形斑图结构等等。通过利用Maple，Mathematica等符号软件处理复杂的求解运算，极大地提高了研究工作的深入和效率。借助图形分析和数学解析法，对各种相干结构进行了分析研究，总结归纳了已经揭示的各类相干结构的特征和数学机理，而且许多是1+1维情形所没有的现象。Dromion为
所有方向都呈指数衰减的相千局域结构，可以由直线孤子，也可以由曲线孤子
形成，不仅局域在直线或曲线的交点，也可以存在与曲线的近邻点上.Solito仔是
非局域的，除了一个特定方向外，其他所有方向都为指数衰减，二直线孤子由
于共振的影响变为一半直线孤子一sollto任解.Dro而on格子为多dromion点阵.振
荡形dromion解在空间某一方向上产生振荡.多团解，环孤子为非点状的局域激
发，环孤子解在闭合曲线的内部不为零，闭合曲线外部指数衰减呼吸子无论幅
度、形状、峰间的距离，峰的数目都进行了呼吸.瞬子的幅度随时间的变化而变
化.周期性孤子解在空间分别呈现周期性特征.尖峰孤立子解在波峰处有一个尖
点，其一阶导数不连续.紧孤立子解就是那些在某个有限区域上高度不为零，而
在这个有限区域之外其高度为零的一类特殊行波.褶皱孤立波，可以在两个方向
同时褶皱，也可以在一个方向褶皱.双周期波斑图结构在两个空间方向都表现为
周期性，而且发现周期、振幅都可以变化.混沌斑图结构和分形斑图结构展示出
孤立波形态中的混沌和分形特征.通过图形分析方法，本文还考察了多种2斗一1维
相干结构之间的相互作用过程，发现2十1维相干结构的相互作用性质要比1+1维
孤子丰富的多，其相互作用可以是弹性碰撞、非弹性碰撞和完全非弹性碰撞，
并具有产生、湮没、分裂和聚合等等现象.
Solitary wave, which is a special coherent structures, describes a kind of stable nature wave phenomena. In this dissertation, the exact solutions of the 2+1-dimensional ( two spatial- dimensions and one time dimension) nonlinear wave models (equations) which originate from practical water wave and other physical problems are investigated by means of symbolic computation and the more abundant localized and non-localized coherent structures in 2+1-dimensional nonlinear wave models are revealed as well as the rich interaction properties for these structures are discussed.
Part I devoted to investigated exact solutions of 2+1-dimensional nonlinear wave equations. (1) The Hirota bilinear method is improved and investigated several 2+1 dimensional nonlinear wave models, such as 2+1-dimensional Schrodinger equation, 2+1-dimensional Maccari equation and 2+1-dimensional breaking soliton equation, the generalized dromion structures with arbitrary functions are obtained and some characteristics are found. (2) The variable separation method is established to deal with 2+1-dimensional nonlinear wave models. The Painlevetruncated expansion approach and homogenous balance approach are employed respectively to explore the variables separation solution of 2+1-dimensional nonlinear wave models , such as 2+1-dimensional Nizhnik-Novikov-Veselov equation, 2+1-dimensional dispersive long wave equation, 2+1-dimensional resonant interaction between long and short wave equation, 2+1-dimensional long dispersive wave equation, 2+1-dimensional generalized Nizhnik-Novikov-Veselov equation, and a quite
"universal " variable separation solution formula with several arbitrary function which is valid for a large classes of 2+1- dimensional nonlinear wave models is obtained. (3) The direct algebraic methods are generalized to solving 2+1-dimensional nonlinear wave models. The Jacobi elliptic function method and formal mapping method are introduced and discussed respectively and several class of 2+1-dimensional nonlinear wave models , such as 2+1-dimensional breaking soliton equation, 2+1-dimensional resonant interaction between long and short wave equation, 2+1-dimensional Nizhnik-Novikov-Veselov equation^ 2+1-dimensional Burgers equation, 2+1-dimensional Wu-Zhang equation, 2+1-dimensional Schrodinger equation, 2+1-dimensional Boussinesq equa-
tion are studied by making use of Maple and mathematica. Their doubly periodic wave structures and line superposition periodic solutions of Jacobi elliptic functions which will change in their amplitudes, shapes and period are obtained.
Partll is devoted to reveals the abundant coherent structures and interaction properties contained in 2+1-dimensional nonlinear wave equations. Prom the 'universal'variable separation solution of 2+1-dimensional nonlinear wave models and by introducing suitably these arbitrary functions, we constructed the considerable abundant coherent structures, including multi-line solitary wave solutions, multi-lump solutions, multi-solitoff solutions, multi-dromion solutions, multi-compacton solution, multi-peakon solution, multi-foldon solution, lattice dromiln solution, oscillating dromion solutions, ring-soliton solutions, motive and static breather solutions, instanton solutions, doubly periodic wave solutions, chaos pattern structures, fractal pattern structures and so on. The development of computer algebra and the application of Maple and Mathematica improve our study and enhance efficiency greatly. Based on the plots and mathematical analysis, we explored all this exotic coherent structures. Dromions are localized solutions decaying exponentially in all directions, which can be driven not only by straight line solitons but also by curved line solitons and can be located not only at the cross points of the lines but also at the closed points of the curves. The solitoff solution decays exponentially in all directions except for a preferred one, two straight line soliton become only one half straight line soliton because of the resonance effect. The dromion lattice is
第一部分：研究2+1维非线性波动模型的求解问题。在三个方面展开了工作：(1) 利用Hirota双线性方法进一步研究了2+1维Schrdinger方程，2+1维Maccari方程和2+1维破裂非线性波动方程，获得了具有不同特征并带有任意函数的广义dromion结构。(2) 反散射方法可以看成是非线性波动问题中的Fourier变换法，但分离变量法对非线性波动问题的推广直到最近才有所突破。变量分离法有效地解决了2+1维非线性波动模型的求解问题。我们分别利用Painlevé截断展开方法和齐次平衡方法实现了多类2+1维非线性波动模型，如2+1维Nizhnik-Novikov-Veselov方程，2+1维色散长波方程，2+1维长波短波共振作用方程，2+1维长色散波方程，2+1维广义Nizhnik-Novikov-Veselov方程等的变量分离求解，给出了一些新的结果，(3) 直接代数法在孤立波的求解中具有重要的地位，由于计算机符号软件的广泛应用，对直接代数法的完善、发展和应用起了很好的作用。我们对近年创立的Jacobi椭圆函数方法和形变映射方法作了介绍和讨论，并具体研究了多类2+1维非线性波动模型，如2+1维Nizhnik-Novikov-Veselov方程，2+1维长短波共振方程，2+1维破裂孤子方程，2+1维非线性Burgers方程，2+1维Wu-Zhang方程，2+1维Schr(o|¨)dinger方程，2+1维Boussinesq方程，给出了双周期波结构和迭加椭圆函数周期解。
第二部分：研究不同类型的2+1维相干结构的构造和它们相互作用问题。在诸多专家学者相关研究工作的启示和基础上，从2+1维非线性波动方程的变量分离通用解表示出发，通过适当地选择这些任意函数，构造起2+1维非线性波动模型具有十分丰富的相干结构，包括多线孤立波解，多lump解，多solitoff解，多dormion解，多compacton解，多peakon解，多foldon解，格子dormion解，振荡型dormion解，环孤子解，运动和静止呼吸子解，瞬子解，双周期波斑图结构，混沌斑图结构，分形斑图结构等等。通过利用Maple，Mathematica等符号软件处理复杂的求解运算，极大地提高了研究工作的深入和效率。借助图形分析和数学解析法，对各种相干结构进行了分析研究，总结归纳了已经揭示的各类相干结构的特征和数学机理，而且许多是1+1维情形所没有的现象。Dromion为
所有方向都呈指数衰减的相千局域结构，可以由直线孤子，也可以由曲线孤子
形成，不仅局域在直线或曲线的交点，也可以存在与曲线的近邻点上.Solito仔是
非局域的，除了一个特定方向外，其他所有方向都为指数衰减，二直线孤子由
于共振的影响变为一半直线孤子一sollto任解.Dro而on格子为多dromion点阵.振
荡形dromion解在空间某一方向上产生振荡.多团解，环孤子为非点状的局域激
发，环孤子解在闭合曲线的内部不为零，闭合曲线外部指数衰减呼吸子无论幅
度、形状、峰间的距离，峰的数目都进行了呼吸.瞬子的幅度随时间的变化而变
化.周期性孤子解在空间分别呈现周期性特征.尖峰孤立子解在波峰处有一个尖
点，其一阶导数不连续.紧孤立子解就是那些在某个有限区域上高度不为零，而
在这个有限区域之外其高度为零的一类特殊行波.褶皱孤立波，可以在两个方向
同时褶皱，也可以在一个方向褶皱.双周期波斑图结构在两个空间方向都表现为
周期性，而且发现周期、振幅都可以变化.混沌斑图结构和分形斑图结构展示出
孤立波形态中的混沌和分形特征.通过图形分析方法，本文还考察了多种2斗一1维
相干结构之间的相互作用过程，发现2十1维相干结构的相互作用性质要比1+1维
孤子丰富的多，其相互作用可以是弹性碰撞、非弹性碰撞和完全非弹性碰撞，
并具有产生、湮没、分裂和聚合等等现象.
Solitary wave, which is a special coherent structures, describes a kind of stable nature wave phenomena. In this dissertation, the exact solutions of the 2+1-dimensional ( two spatial- dimensions and one time dimension) nonlinear wave models (equations) which originate from practical water wave and other physical problems are investigated by means of symbolic computation and the more abundant localized and non-localized coherent structures in 2+1-dimensional nonlinear wave models are revealed as well as the rich interaction properties for these structures are discussed.
Part I devoted to investigated exact solutions of 2+1-dimensional nonlinear wave equations. (1) The Hirota bilinear method is improved and investigated several 2+1 dimensional nonlinear wave models, such as 2+1-dimensional Schrodinger equation, 2+1-dimensional Maccari equation and 2+1-dimensional breaking soliton equation, the generalized dromion structures with arbitrary functions are obtained and some characteristics are found. (2) The variable separation method is established to deal with 2+1-dimensional nonlinear wave models. The Painlevetruncated expansion approach and homogenous balance approach are employed respectively to explore the variables separation solution of 2+1-dimensional nonlinear wave models , such as 2+1-dimensional Nizhnik-Novikov-Veselov equation, 2+1-dimensional dispersive long wave equation, 2+1-dimensional resonant interaction between long and short wave equation, 2+1-dimensional long dispersive wave equation, 2+1-dimensional generalized Nizhnik-Novikov-Veselov equation, and a quite
"universal " variable separation solution formula with several arbitrary function which is valid for a large classes of 2+1- dimensional nonlinear wave models is obtained. (3) The direct algebraic methods are generalized to solving 2+1-dimensional nonlinear wave models. The Jacobi elliptic function method and formal mapping method are introduced and discussed respectively and several class of 2+1-dimensional nonlinear wave models , such as 2+1-dimensional breaking soliton equation, 2+1-dimensional resonant interaction between long and short wave equation, 2+1-dimensional Nizhnik-Novikov-Veselov equation^ 2+1-dimensional Burgers equation, 2+1-dimensional Wu-Zhang equation, 2+1-dimensional Schrodinger equation, 2+1-dimensional Boussinesq equa-
tion are studied by making use of Maple and mathematica. Their doubly periodic wave structures and line superposition periodic solutions of Jacobi elliptic functions which will change in their amplitudes, shapes and period are obtained.
Partll is devoted to reveals the abundant coherent structures and interaction properties contained in 2+1-dimensional nonlinear wave equations. Prom the 'universal'variable separation solution of 2+1-dimensional nonlinear wave models and by introducing suitably these arbitrary functions, we constructed the considerable abundant coherent structures, including multi-line solitary wave solutions, multi-lump solutions, multi-solitoff solutions, multi-dromion solutions, multi-compacton solution, multi-peakon solution, multi-foldon solution, lattice dromiln solution, oscillating dromion solutions, ring-soliton solutions, motive and static breather solutions, instanton solutions, doubly periodic wave solutions, chaos pattern structures, fractal pattern structures and so on. The development of computer algebra and the application of Maple and Mathematica improve our study and enhance efficiency greatly. Based on the plots and mathematical analysis, we explored all this exotic coherent structures. Dromions are localized solutions decaying exponentially in all directions, which can be driven not only by straight line solitons but also by curved line solitons and can be located not only at the cross points of the lines but also at the closed points of the curves. The solitoff solution decays exponentially in all directions except for a preferred one, two straight line soliton become only one half straight line soliton because of the resonance effect. The dromion lattice is
引文
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