高阶b族方程研究

This proposal focuses on wave propagation modeled by the higher order b-family equation, the two-component b-family system and the short-pulse equation. The higher order b-family equation and the two-component b-family system are the new models for the propagation of shallow-water waves, the short-pulse equation is a model of the ultra-short light pulses in silica optical fibres. So they have great meaning on theory and in practice. Firstly, it is considered that the geometric investigation on the higher order b-family equation and it is shown that the equation can be realized as an Euler equation on the Lie group. Then, applying Kato's theory and the method of the Littlewood-Paley decomposition for the initial-value problem of the periodic higher order b-family equation, the existence and uniquess of the strong global solutions are established. Moreover, the existence, uniquness and continuous for the weak solutions of the two-component b-family system are considered. Forthermore, it is shown that the features for the b-family equation persist in the two-component b-family system. Finally， the existence and uniquess of the weak solutions for the short-pulse equation are proved by the method of a suitable priori estimates together with an application of the compensated compactness.

本项目主要研究高阶b族方程，二维b族方程组及其short-pulse方程中波的传播问题。高阶b族方程和二维b族方程组为新型的浅水波传播模型,而short-pulse方程是石英光纤中的超短光脉冲模型。因此该研究具有重要的应用背景和理论价值。本项目首先，从几何角度研究高阶b族方程，证明高阶b族方程对应李群上的欧拉方程。然后，用Kato理论和Littlewood-paley分解方法研究高阶b族方程周期初值问题整体强解的存在唯一性。进一步考虑二维b族方程组弱解的存在唯一和连续性，同时将b族方程的性质推广到二维b族方程组中。最后,运用先验估计结合补偿紧性的方法证明short-pulse方程弱解的存在唯一性。

本项目主要研究四个新型的浅水波模型的解的传播性质。第一个模型是带有线性色散项的周期modified Camassa-Holm方程。我们首先研究了当线性色散项为零时，运用解和解的梯度之间的比率关系得到解在有限时间爆破的结果。进一步运用解的连续性和通过适当的变量变换，可以得到当线性色散项的参数为负数时解的爆破准则，并证明当初动量的密度下有界于某个与线性色散项和初值相关的常数时会发生解的爆破。最后研究了当线性色散项大于零时，初值的斜率充分大会引起解的爆破。第二个模型是带有线性色散项的generalized modified Camassa-Holm方程。我们通过寻找新的守恒律和 的有界性得到了解的爆破机制。第三个模型是modified b-family方程。我们通过Riccati型微分不等式的带到了解的爆破，并在weighted空间中证明了解的延迟性质。第四个模型是generalized two-component Camassa-Holm方程组。运用解的单调性和矫顽性研究得出该方程组的光滑孤立波在能量空间中是稳定的。