非线性演化方程的孤子-椭圆周期波解及其准孤立子行为研究

Recently, the interaction between soliton and small amplitude periodic waves has aroused a notably growing interest of physicists and mathematicians. This phenomenon has also been named as nanopternon structure by Boyd，it can be found in many branches of physics, such as condense matter physics, nonlinear optics and hydrodynamics. In the past decades, the nanopteron structure in both continuous and discrete systems has been studied extensively by numerical approach. However, the analytical study has progressed slowly over a long time and needs further development. Our preliminary study shows that the quasisoliton behavior of the soliton-cnoidal wave solution is a nanopteron. We aim to take advantages of various methods in our soliton theory, such as the symmetry method and the generalized function expansion method to obtain soliton-cnoidal wave solution and its quasisoliton behavior of some classical nonlinear evolution equation. The nanopteron exciting pattern in plasma physics and the influence of plasma parameters on wave shape and phase shift will also be taken into consideration in our further study. It is hoped that our results might be able to explain some experimental phenomena, and to provide some theoretical basis for numerical simulations and experimental designs.

目前，孤子-小振幅周期波相互作用现象引起了物理学家和数学家的广泛兴趣。这种现象也被Boyd命名为nanopteron结构，它广泛存在于物理学的诸多分支学科中，如等离子体物理、凝聚态物理、非线性光学、流体力学等等。在过去几十年中，人们对连续系统或离散系统中的nanopteron结构进行了大量的数值研究。然而，关于nanopteron结构的解析解研究却进展缓慢， 有待深入。我们的初步研究表明孤子-椭圆周期波的准孤立子行为即是nanopteron结构。本项目将运用我们擅长的数学物理方法，特别是孤子理论中的一些方法，如对称法、推广的函数展开法等，给出若干经典非线性演化方程的孤子-椭圆周期波解及其准孤立子行为。我们还将考虑等离子中的nanopteron激发模式，并讨论等离子参数对波形和相移的影响。争取解释一些实际观实验现象，为数值模拟或实验设计提供一些理论依据。

本项目主要开展了以下几方面的研究。一是应用推广的tanh函数展开法，给出了Korteweg-de Vries方程具有准孤立子行为的两组孤子-椭圆周期波解，并指出该解在合适的参数条件下具有准孤立子行为，此时, 孤子-椭圆周期波解的孤子核越接近于满足零边界条件的经典孤立子, 而围绕在其周围的椭圆周期波也越接近于在零附近振荡的正余弦波。二是推导了若干等离子体系统中描述离子声波动力学行为的Korteweg-de Vries方程, 发现等离子体参数对离子声准孤立子的波形具有显著影响。三是超对称可积系统的高维推广研究，应用形式级数对称方法讨论了超负KP系统及(2+1)维MKdV系统的对称可积性，证明了两系统在无穷多广义对称意义下是可积的，并在此基础上提出了一个关于(2+1)-维超对称可积系统猜想，该研究成果对超对称系统的高维扩展具有一定的开拓意义。