Global Well-posedness and soliton resolution for the Derivative Nonlinear Schr\"{o}dinger equation
Global Well-posedness and soliton resolution for the Derivative Nonlinear Schr\"{o}dinger equation
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发表时间：
2017-06
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通讯作者：
R. Jenkins;Jiaqi Liu;P. Perry;C. Sulem
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作者：
R. Jenkins;Jiaqi Liu;P. Perry;C. Sulem
在支持亮孤子(但不包括谱奇异性)的加权Soblev空间中，我们研究了一般初始条件下的导数非线性薛定谔方程。我们证明了解的整体适定性，并以局域孤子和色散分量的有限和的形式给出了解的长时间行为的完整描述。在前导阶解和时空锥中，解具有多孤子的形式，其参数受孤子-孤子和孤子-辐射相互作用的影响比初值略有改变。我们的分析给出了修正色散项的显式表达式。我们使用Deift和周的非线性最陡下降法，Dieng-McLaughlin的部分分析和Borghese-Jenkins-McLaughlin最近关于聚焦非线性薛定谔方程的孤子分辨的工作是补充的。
We study the Derivative Nonlinear Schr\"odinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and give a full description of the long- time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. At leading order and in space-time cones, the solution has the form of a multi-soliton whose parameters are slightly modified from their initial values by soliton-soliton and soliton-radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou revisited by the $\bar{\partial}$-analysis of Dieng-McLaughlin and complemented by the recent work of Borghese-Jenkins-McLaughlin on soliton resolution for the focusing nonlinear Schr\"odinger equation.