一个具有新型孤立子的非线性水波方程的若干问题研究

This project is concerned with several aspects for a nonlinear water wave equation with new shape solitons. We mainly investigate the initial value problem and the initial boundary problem value for the equation. The local well-posedness of the equation is established, and the blow-up phenomena as well as the global existence for strong solutions to the equation are studied. The existence and the uniqueness of global weak solutions for the equation are proved, and the orbitally stability of peakon solutions and solitons is discussed. Furthermore, the numerical simulations of interactions of the solitons to the equation are considered. Researches on the above issues for the equation will help us deeply understand the phenomena of solitons in water wave from the view of mathematics, so the project is very important for both mathematical theories and real applications.

本项目对一个具有新型孤立子的非线性水波方程的若干问题进行研究。我们主要研究方程的初值问题和初边值问题的局部适定性，强解的爆破与整体存在，整体弱解的存在性和唯一性，以及孤立子解的轨道稳定性。对本项目上述问题的研究，将有助于我们从数学角度对水波领域的孤立子现象加以深刻描述和刻划，因此本项目的研究在数学理论和实际应用方面都有重要的意义。

本项目完成情况：2012年12月，研究了具有耗散项的非线性浅水波方程的初值问题的局部适定性和和强解的爆破，并得到了几个的强解爆破的结果。该成果在SCI收录的杂志Acta. Math.Appl.Sin., 上接收：Qiaoyi Hu and Z. Yin, Local well-posedness and blow-up phenomena for a weakly dissipative rod equation, Acta. Math.Appl.Sin., (2012) 。另外我们正在对一个具有新型孤立子的非线性水波方程的若干问题进行研究。我们主要研究方程的强解的整体存在，以及整体弱解的存在性和唯一性，目前研究进展顺利。