ON THE MODIFIED NONLINEAR SCHRDINGER EQUATION IN THE SEMICLASSICAL LIMIT:SUPERSONIC,SUBSONIC,AND TRANSSONIC BEHAVIOR

作     者：Jeffery C. DiFranco Peter D. Miller Benson K. Muite Jeffery C. DiFranco Department of Mathematics, Seattle University, 901 12th Avenue, Seattle, WA 98122, USA Peter D. Miller Benson K. Muite Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109, USA

作者机构：Department of MathematicsSeattle University Department of MathematicsUniversity of Michigan 

出 版 物：《Acta Mathematica Scientia》 (数学物理学报（B辑英文版）)

年 卷 期：2011年第31卷第6期

页      面：2343-2377页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0805[工学-材料科学与工程（可授工学、理学学位）] 0704[理学-天文学] 0701[理学-数学] 

基　　金：supported by the National Science Foundation under grant DMS-0807653 

主　　题：semiclassical limits dispersionless limits modulational instability focusing,defocusing, and modified nonlinear SchrSdinger equations 

摘      要：The purpose of this paper is to present a comparison between the modified nonlinear SchrSdinger （MNLS） equation and the focusing and defocusing variants of the （unmodified） nonlinear SchrSdinger （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.