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

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

Abstract

Abstract:

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.

CLC Number:

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].Acta mathematica scientia,Series B, 2011, 31(6): 2343-2377.

Trendmd

References

[1] Deift P, Venakides S, Zhou X. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat Math Res Notices, 1997, 1997: 286–299

[2] DiFranco J C, Miller P D. The semiclassical modified nonlinear Schr¨odinger equation I: Modulation theory and spectral analysis. Physica D, 2008, 237: 947–997

[3] DiFranco J C, Miller P D. The semiclassical modified nonlinear Schr¨odinger equation II: Asymptotic analysis of the Cauchy problem. In preparation, 2011

[4] Gurevich A V, Pitaevskii L P. Nonstationary structure of a collisionless shock wave. Sov Phys JETP, 1974, 38: 291–297

[5] Kamvissis S, McLaughlin K T -R, Miller P D. Semiclassical Soliton Ensembles for the Focusing Nonlinear Schr¨odinger Equation. Annals of Mathematics Studies 154. Princeton: Princeton University Press, 2003

[6] Klein C. Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schr¨odinger equations. Electron Trans Numer Anal, 2008, 29: 116–135

[7] Kuvshinov B N, Lakhin V P. The Riemann invariants and characteristic velocities of Whitham equations for the derivative nonlinear Schr¨odinger equation. Phys Scr, 1994, 49: 257–260

[8] Lax P D. On dispersive difference schemes. Solitons and coherent structures (Santa Barbara, Calif, 1985). Physica D, 1986, 18: 250–254

[9] Lax P D, Levermore C D. The small dispersion limit of the Korteweg-de Vries equation. Comm Pure Appl Math, 1983, 36: 253–290 (Part I), 571–593 (Part II), 809–929 (Part III)

[10] Madelung E. Quantum theory in hydrodynamic form. Zeitschrift f¨ur Physik, 1926, 40: 322–326

[11] Miller P D. Some remarks on a WKB method for the nonselfadjoint Zakharov-Shabat eigenvalue problem with analytic potentials and fast phase. Physica D, 2001, 152-153: 145-162

[12] Pathria D, Morris J. Pseudo-spectral solution of nonlinear Schr¨odinger equations. J Comp Phys, 1990, 87: 108–125

[13] Reed M, Simon B. Methods of Modern Mathematical Physics, Volume III: Scattering Theory. San Diego, CA: Academic Press, 1979

[14] Reed M, Simon B. Methods of Modern Mathematical Physics, Volume IV: Analysis of Operators. San Diego, CA: Academic Press, 1979

[15] Tappert F. Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split step Fourier method//Newell A C, ed. Nonlinear Wave Motion. Lectures in Applied Mathematics, Volume 15. Amer Math Soc, 1974: 215–216

[16] Tovbis A, Venakides S, Zhou X. On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schr¨odinger equation. Comm Pure Appl Math, 2004, 57: 877–985

[17] Yang J. Nonlinear Waves in Integrable and Nonintegrable Systems. Mathematical Modeling and Computation, Volume 16. Philadelphia: SIAM, 2010

[18] Zakharov V E, Shabat A B. Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media. Sov Phys JETP, 1972, 34: 62–69