Abstract
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.
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Notes
- 1.
Since
implies that (1 + |x|)q
0(x) is square-integrable, it follows by Cauchy-Schwarz that 
, which in turn implies that the reflection coefficient r(z) is well-defined for each 
. - 2.
In many works on long-time asymptotics for the Cauchy problem (1)–(2) written before the Digital Library of Mathematical Functions was freely available (e.g., [8, 9]), the solution of Riemann-Hilbert Problem 3 was developed in terms of the related function \(D_\nu (y):=U(-\tfrac {1}{2}-\nu ,y)\). Since most formulæ in [18, §12] are phrased in terms of U(⋅, ⋅), we favor the latter.
- 3.
All L p norms of matrix-valued functions in this section depend on the choice of matrix norm, which we always take to be induced by a norm on
.
implies that (1 + |x|)q
0(x) is square-integrable, it follows by Cauchy-Schwarz that 
, which in turn implies that the reflection coefficient r(z) is well-defined for each 
.
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Acknowledgements
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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Dieng, M., McLaughlin, K.D.TR., Miller, P.D. (2019). Dispersive Asymptotics for Linear and Integrable Equations by the \(\overline {\partial }\) Steepest Descent Method. In: Miller, P., Perry, P., Saut, JC., Sulem, C. (eds) Nonlinear Dispersive Partial Differential Equations and Inverse Scattering. Fields Institute Communications, vol 83. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9806-7_5
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