Rational Normal Forms and Stability of Small Solutions to Nonlinear Schrödinger Equations
Rational Normal Forms and Stability of Small Solutions to Nonlinear Schrödinger Equations
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DOI：
10.1007/s40818-020-00089-5
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发表时间：
2018-12
期刊：
影响因子：
2.8
通讯作者：
J. Bernier;E. Faou;B. Grébert
中科院分区：
文献类型：
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作者：
J. Bernier;E. Faou;B. Grébert
考虑了圆上具有非平凡立方部分且无外参数的一般类非线性薛定谔方程。在高Sobolev正则性的开集上构造了一种新的规范形，即有理规范形。有了这个新的工具，我们证明了，给定一个大的constantM和一个足够小的参数，通用的初始数据的大小，流共轭到一个可积流到一个任意小的剩余的顺序。这意味着对于这样的初始数据au（0），我们控制解u（t）的Sobolev范数。此外，这个性质是局部稳定的：如果v（0）足够接近tou（0）（阶），那么解v（t）也是阶时间控制的。
We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constantMand a sufficiently small parameter, for generic initial data of size, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order. This implies that for such initial datau(0) we control the Sobolev norm of the solutionu(t) for time of order. Furthermore this property is locally stable: ifv(0) is sufficiently close tou(0) (of order) then the solutionv(t) is also controled for time of order.