TY - JOUR
T1 - A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit
AU - Assainova, O.
AU - Klein, C.
AU - McLaughlin, K. D.T.R.
AU - Miller, P. D.
N1 - Funding Information:
The authors benefited from participation in a Focused Research Group on “Inverse Problems, Nonlinear Waves, and Random Matrices” at the Banff International Research Station in 2012, the “Exceptional Circle” workshop at the University of Helsinki in 2013, a conference on “Scattering and Inverse Scattering in Multi-Dimensions” at the University of Kentucky in 2014 (funded by the National Science Foundation under grant DMS-1408891), and a Research in Pairs meeting
Funding Information:
Acknowledgments. OA and CK acknowledge support by the program PARI and the FEDER 2016 and 2017, the I-QUINS project, and the ANR-FWF project ANuI as well as by the RISE network IPaDEGAN. The research of KDTRM was supported by the National Science Foundation under grants DMS-1401268 and DMS-1733967. The research of PDM was supported by the National Science Foundation under grants DMS-1206131 and DMS-1513054 and by the Simons Foundation under grant 267106. The authors are grateful to Kari Astala, Sarah Hamilton, Michael Music, Peter Perry, Samuli Siltanen, Johannes Sjöstrand, and Nikola Stoilov for useful discussions.
PY - 2019/7
Y1 - 2019/7
N2 - The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.
AB - The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.
UR - http://www.scopus.com/inward/record.url?scp=85063225844&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85063225844&partnerID=8YFLogxK
U2 - 10.1002/cpa.21822
DO - 10.1002/cpa.21822
M3 - Article
AN - SCOPUS:85063225844
VL - 72
SP - 1474
EP - 1547
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
SN - 0010-3640
IS - 7
ER -