Structure of Solutions and Geometric Symmetry for Nonlinear Evolution Equations

在2003- 2004学年，我们研究了具有周期边界条件的修正KdV方程Cauchy问题的时间局部适定性。1993年，Bourgain证明了这个柯西问题在H^s$，s <$1/2中是时间局部适定的。此外，在1996年，Bourgain还证明了解映射在H^s$，s<1/2中不属于C^3 $，尽管它在H^s$，s <1/2中是解析的。高冈和Tsutsumi做了一个密切的调查问题的区别是什么情况下$H^s$，s <1/2和$H^s$，s<1/2。结果表明，Cauchy问题在H^s$，s>1/3时仍然是局部适定的，但由于非线性振动的出现，其解映射不是一致连续的. 2005学年，Tsutsumi与Akihiro Shimomura研究了二维二次非线性Schrodinger方程解的渐近性态.在二维空间中 ...更多信息 ns，二次非线性介于短程情况和长程情况之间，从非线性散射理论的观点来看，二次非线性有着特殊的意义。结果表明，对于未知的平方模的非线性，解不接近自由解，而已知的其余二次非线性属于短程相互作用。2006学年，Tsutsumi研究了非线性Schrodinger方程Cauchy问题解的无条件唯一性。当解被构造时，人们通常施加一个条件，即该解属于一个辅助空间，该辅助空间与Kahrtz估计相关联。无条件唯一性是指解是唯一的，即使它不在这种辅助空间中。证明了非线性薛定谔方程的解在临界Sobolev空间中的无条件唯一性，并与标度不变性相联系。这比Furioli和Terraneo的结果有了很大的改进。少

In what follows, the main research results of this project are described.In the academic years of 2003-4, we studied the time local well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. In 1993, Bourgain proved that this Cauchy problem is time locally well-posed in $H^s$, s ≧1/2. Furthermore, in 1996, Bourgain also proved that the solution map fails to belong to $C^3$ in $H^s$, s<1/2, though it is analytic in $H^s$, s≧ 1/2. Takaoka and Tsutsumi made a close investigation into the problem of what is the difference between the cases $H^s$, s ≧1/2 and $H^s$, s<1/2. It was showed that the Cauchy problem is still locally well-posed in $H^s$, s>1/3, but the solution map is not uniformly continuous because of the occurrence of nonlinear oscillation.In the academic year of 2005, Tsutsumi studied the asymptotic behavior of solution for the quadratic nonlinear Schrodinger equation in two space dimensions with Akihiro Shimomura. In two space dimensio … More ns, the quadratic nonlinearity is a boader between the short renge case and the long range case and the quadratic nonlinearity has a special interest from a viewpoint of nonlinear scattering theory. It was showed that for a nonlinearity of squared modulous of the unkown, the solution does not approach a free solution, while it is already known that the rest of other quadratic nonlinearities belong to the shourt range interaction. casse.In the academic year of 2006, Tsutsumi studied the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation. When the solution is constructed, one usually impose a condition that the solution belongs to an auxiliary space associated with the Stricahrtz estimate. The unconditional uniqueness means that the solution is unique even though it is not in this kind of auxiliary space. It was proved that the unconditional uniqueness holds for the solution in the critical Sobolev space associated with the scaling invariance of nonlinear SchrOdinger equations. This is a substantial improvement over the results by Furioli and Terraneo. Less