Stability of Traveling Waves of Nonlinear Schrödinger Equation with Nonzero Condition at Infinity

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• 作者：Zhiwu Lin ; Zhengping Wang ; Chongchun Zeng
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：222
• 期：1
• 页码：143-212
• 全文大小：1,152 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Electromagnetism, Optics and Lasers
  Mathematical and Computational Physics
  Complexity
  Fluids
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0673
• 卷排序：222

文摘

We study the stability of traveling waves of the nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models for this are the Gross–Pitaevskii (GP) equation and the cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For dimension two, the non-degeneracy condition is also proved for these slow traveling waves. For general traveling waves without vortices (that is nonvanishing) and with general nonlinearity in any dimension, we find a sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable, and we find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of all of the above cases, we construct unstable and stable invariant manifolds.Communicated by F. Lin