Study on the instability of Benjamin-Feir type concerned with nonlinear strongly dispersive systems

构造了典型的强色散非线性系统sin - gordon方程的傅里叶型一阶近似解，并用Floquet理论阐明了其本杰明-费尔型不稳定性。此外，为了阐明一般强色散非线性系统的不稳定现象，首先对弱色散非线性系统（如KdV型非线性方程）提出了各种构造第一积分的方法。进一步阐明了高阶稳态KdV方程与示踪公式之间的关系，并统一证明了速降巴格曼势和周期有限纬向势可解高阶稳态KdV方程。同时，为了找出给定微观系统的色散性质，研究了一种称为婴儿洗澡水格式的数值方法。一方面，用数值方法阐明了复杂网络系统临界现象的机理。

The first order approximate solution of Fourier type is constructed for the Sine-Gordon equation, which is the typical example of strongly dispersive nonlinear system, and its instability of Benjamine-Feir type is clarified by Floquet theory. Furthermore, to clarify the instability phenomena of general strongly dispersive nonlinear system, to begin with, various methods of constructing first integrals have been developed for weakly dispersive nonlinear system such as nonlinear equations of KdV type. Moreover, the relations between the higher order stationary KdV equation and the trace formulas have been clarified, and it is uniformly proved that the rapidly decreasing Bargmann potentials and the periodic finite zonal potentials solve the higher order stationary KdV equations. Simultaneously, to find the dispersive property for the given microscopic system, a numerical method called Baby-Bathwater scheme is studied. On the one hand, mechanism of critical phenomena has been clarified for complex network system by numerical methods.