Exactly Solvable Models and Applications

1)本文将孤立子理论推广到不稳定物理系统中非线性波的分析。介绍了不稳定的非线性薛定谔（UNLS）方程。解释了电子束等离子体中的局域模。此外，还提出了一种新的非线性演化。2）研究了拉多姆介质中孤立子在外界噪声作用下的传播。3）将二维统计力学模型与一维四元系统的巴克斯特公式推广到有限温度情况。结合有限尺寸修正理论，我们计算了量子自旋系统的自由能和卷积长度。4）我们发展了一个从二维统计力学中精确可解模型得到链多项式的一般理论。应用理论。对于不同的模型，我们得到了新的链接多项式和已知的链接多项式。这一理论不仅在近代数学物理学中，而且在量子引力理论中，都有着重要的作用。

1) We extend the theory of soliton to the analysis of non-linear waves in unstable physical system. The unstable Nonlinear Schrodinger (UNLS) equation is introduced, whereby propagation of. a localized mode in electron-beam plasma is explained. In addition, a new nonlinear evolution. equation which encompass chaos and soliton phenomena is proposed.2) We investigate the solillo-n propagations under external noise and in radom media. Using a one-dimensional lattice with radom mass distribution, we clarify the localized wave propagations and the modulations of plane waves.3) We extend the Baxter's formula which relates 2-D statistical mechanical model and 1-D quatum system into finite temperature case. Combining this extension with the theory of finite size corrections, we evaluate the free energies and the covelation lengths of quantum spin systems.4) We develop a general theory to obtain link polgnomials from exactly solvable models in 2-D statistical mechanics. Applying the theory. to various model, we get new link polynomials and the known link polynomials. This theory may play an important role not only in recent mathematical' physics but also in quatum gravitation theory.