Dispersive Asymptotics for Linear and Integrable Equations by the \(\overline {\partial }\) Steepest Descent Method

  • Momar Dieng
  • Kenneth D. T.-R. McLaughlin
  • Peter D. MillerEmail author
Part of the Fields Institute Communications book series (FIC, volume 83)


We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-t limit, based on a generalization of steepest descent techniques for Riemann-Hilbert problems to the setting of \({\overline {\partial }}\)-problems. Expanding upon prior work (Dieng and McLaughlin, Long-time asymptotics for the NLS equation via \({\overline {\partial }}\) methods, arXiv:0805.2807, 2008) of the first two authors, we develop the method in detail for the linear and defocusing nonlinear Schrödinger equations, and show how in the case of the latter it gives sharper asymptotics than previously known under essentially minimal regularity assumptions on initial data.



The first two authors were supported in part by NSF grants DMS-0451495, DMS-0800979, and the second author was supported by NSF Grant DMS-1733967. The third author was supported in part by NSF grant DMS-1812625.


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Authors and Affiliations

  • Momar Dieng
    • 1
  • Kenneth D. T.-R. McLaughlin
    • 1
    • 2
  • Peter D. Miller
    • 3
    Email author
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsColorado State UniversityFort CollinsUSA
  3. 3.Department of MathematicsUniversity of Michigan–Ann ArborAnn ArborUSA

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