离散孤立子的动力学性质和稳定性研究

In this proposal, we study explicit solutions to several semidiscrete and fulldiscrete integrable systems. Continuous limits of discrete integrability and discrete soliton solutions, and rigorous theoretical analysis of discrete soliton interactions are obtained. We investigate the asymptotical stability of discrete solitons for an important integrable discrete Hamiltonian system. These topics are very important and significative in the theory of integrable system and analysis of PDE. In addition, the work related to the topics in the proposal are not much..The topic list includes:.(1) Aiming to get more insight on the relation between the semidiscrete integrable systems and continuous ones, we construct a semidiscrete Boussinesq system based on a spectral problem. We not only give Darboux transformation, determinantal representation of explicit solutions and infinitely many conservation laws, but also show that their continuous limits yield the corresponding results of the continuous Boussinesq system. These results will be significative for further studies of the relationship between discrete and continuous integrable systems..(2) interaction analysis of solutions to a discrete KdV-type system and a nonintegrable nonlinear Schrodinger equation are studied, including some 1-soliton solutions with different properties and interaction of 2-soliton solutions. We find that, by rigorous theoretical analysis, discrete integrable systems indicate some new features and possess some different properties from continuous ones, for example, the shorter waves can travel faster than or advance with the taller ones and the elastic properties of collisions between two solitary waves are sensitively dependent on the choices of some parameters in solutions. This shows that in some cases the solitary waves of the discrete integrable systems can interact elastically, i.e., the shapes, the amplitudes, and the velocities of waves preserve after undergoing collision. However, soliton interactions in the fulldiscrete integrable systems possess different properties from the continuous solitons..(3) The asymptotical stabilities of Volterra system as being an important integrable discretization of KdV equation will be investigated in the third part in the proposal. We applied the Mizumachi’s method which is used to study Toda soliton successfully to investigate the stability of discrete solitons of Voterra. We expect that our results could be the new fact for the basical importance of solitons in ingrable Hamiltonian systems.

本项目研究若干离散(包括半离散和全离散) 可积系统的精确解，离散可积系统可积性和精确解的连续极限，离散孤立子的相互作用以及一个半离散可积哈密顿系统的离散孤立子的渐近稳定性。具体包括以下几个方面的内容：（1）研究有重要物理背景的半离散Boussinesq方程的精确解以及解的行列式表示；（2）研究半离散Boussinesq方程的可积性，构造其Darboux 变换并求出精确解，建立半离散可积方程的可积性（包括Lax对，Darboux变换，无穷守恒律，Hamiltonian结构和精确解等）与相应的连续方程的可积性之间的联系；（3）研究可积全离散系统和一个不可积半离散系统（例如不可积离散非线性Schrodinger方程）的离散孤子解的相互作用，给出严格的理论分析，揭示离散孤立子不同于连续孤立子的新特征。（4）通过将可积的Volterra系统的哈密顿形式的线性化，在附加正交条件下研究其初值问题解的渐近分解性质。

孤立子与可积系统理论是非线性科学中的重要研究内容，在流体力学、等离子体物理、非线性光学、光纤通讯等物理领域中都有着重要的应用。人们还发现很多非线性系统有着丰富的可积性质，例如它是一对线性谱问题的相容性条件，具有无穷守恒律、哈密尔顿结构和无穷多对称，具有多种形式的精确解等；同时，人们提出了很多有效的研究方法（包括逆散射方法，Darboux变换方法，双线性方法，非线性化方法，超对称方法等）来研究非线性可积系统的问题。最近几年，离散可积系统和非局域可积系统称为非线性科学研究的热点之一。本项目的主要研究内容是关于连续型和离散型可积系统的精确求解和动力学性质，非局域等谱和非等谱系统的可积性及精确求解问题，以及可积动力系统精确解的渐近稳定性问题。我们通过对一个全离散KdV型方程的精确解的研究，详细研究了这一方程的多种结构的精确解，发现离散的孤立波解有着更加丰富的动力学性质，例如1-孤立波解具有某种意义下的周期性，2-孤立波解具有波幅与速度正相关的特征，这些新现象在连续可积系统中没有出现过，对全离散可积系统特别是解的动力学性质的研究增添了新的有意义的内容。对于一个四分量的向量形式的非线性薛定谔系统，我们给出了线性谱问题和相应的N次达布变换，借助达布变换得到了这一向量形式的非线性系统的若干具有不同特征的精确解，包括亮孤子解，暗孤子解和混合孤子解，取得了一些有意义的研究成果。我们对于非局域的可积系统也做了相关的研究工作。对一个非局域矩阵型的非线性薛定谔方程，我们给出了线性谱问题和相应的达布矩阵，并给出了一系列的新的矩阵形式精确解，包括周期解，暗-亮孤子混合解，呼吸子解和有理解。此外我们还研究了一个非局域非等谱的可积系统，给出了它的Lax对，达布矩阵和位势之间的变换关系，并求出了精确解，同时我们还研究了这一系统的无穷守恒律和对称。这些研究结果为我们更清晰地理解非局域可积等谱和非等谱系统的性质和应用提供了有意义的帮助。