ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR

doi:10.1016/S0252-9602(11)60405-0

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (6): 2343-2377.doi: 10.1016/S0252-9602(11)60405-0

ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR

Jeffery C. DiFranco, Peter D. Miller, Benson K. Muite

1.Department of Mathematics, Seattle University, 901 12th Avenue, Seattle, WA 98122, USA|2.Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109, USA

收稿日期:2011-10-27 出版日期:2011-11-20 发布日期:2011-11-20
基金资助:
J.C. DiFranco and P.D. Miller were partially supported by the National Science Foundation under grant DMS-0807653.

ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR

Jeffery C. DiFranco, Peter D. Miller, Benson K. Muite

1.Department of Mathematics, Seattle University, 901 12th Avenue, Seattle, WA 98122, USA|2.Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109, USA

Received:2011-10-27 Online:2011-11-20 Published:2011-11-20
Supported by:
J.C. DiFranco and P.D. Miller were partially supported by the National Science Foundation under grant DMS-0807653.

摘要/Abstract

摘要：

The purpose of this paper is to present a comparison between the modified nonlinear Schr¨odinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr¨odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.

关键词: semiclassical limits, dispersionless limits, modulational instability, focusing, defocusing, modified nonlinear Schr¨odinger equations

Abstract:

中图分类号:

35M10

Jeffery C.DiFranco, Peter D.Miller, Benson K.Muite. ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR[J]. 数学物理学报(英文版), 2011, 31(6): 2343-2377.

Jeffery C.DiFranco, Peter D.Miller, Benson K.Muite. ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR[J]. Acta mathematica scientia,Series B, 2011, 31(6): 2343-2377.

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ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR

ON THE MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN THE SEMICLASSICAL LIMIT: SUPERSONIC, SUBSONIC, AND TRANSSONIC BEHAVIOR

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