Multiphase weakly nonlinear geometric optics for schrödinger equations

SIAM Journal on Mathematical Analysis

Volumn 42, Issue 1, 2010, Pages 489-518

  Carles, Rémi ^a,b   Dumas, Eric ^c   Sparber, Christof ^d  

^a UNIV MONTPELLIER   (France)

^b CNRS   (France)

^c INSTITUT FOURIER   (France)

^d UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Dinger equation; High frequency limit; Instability; Nonlinear schr ; Resonances

Indexed keywords

CONTROL NONLINEARITIES; PLASMA STABILITY; RESONANCE; SOBOLEV SPACES;

CUBIC NONLINEARITIES; DINGER EQUATION; GEOMETRIC INTERPRETATION; HIGH FREQUENCY HF; NONLINEAR INTERACTIONS; NONLINEAR WAVE MIXING; RESONANCE STRUCTURE; WEAKLY NON-LINEAR;

NONLINEAR EQUATIONS;

EID: 78649753049     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/090750871     Document Type: Article

References (29)

1. W. BAO, S. JIN, AND P. A. MARKOWICH, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487-524.
2. W. BAO, S. JIN, AND P. A. MARKOWICH, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Cornput., 25 (2003), pp. 27-64.
3. K. BARRAILH AND D. LANNES, A general framework for diffractive optics and its applications to lasers with large spectrums and short pulses, SIAM J. Math. Anal., 34 (2003), pp. 636-674.
4. d, Math. Res. Lett., 9 (2002), pp. 323-335.
5. R. CARLES, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific, Hackensack, NJ, 2008.
6. R. CARLES, P. A. MARKOWICH, AND C. SPARBER, Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Stat. Phys., 117 (2004), pp. 343-375.
7. M. CHRIST, J. COLLIANDER, AND T. TAO, Instability of the Periodic Nonlinear Schrödinger Equation, preprint; available online from http://arxiv.org/abs/math/0311227v1.
8. M. COLIN AND D. LANNES, Short pulses approximations in dispersive media, SIAM J. Math. Anal., 41 (2009), pp. 708-732.
9. J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO, Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation, Invent. Math.
10. S. DENG ET AL., Nonlinear atom optics: Multi-wave mixing of matter waves, Nature, 398 (1999), pp. 218-224.
11. L. FRIEDLAND AND A. G. SHAGALOV, Excitation of multiphase waves of nonlinear Schrödinger equation by capture into resonances, Phys. Rev. E, 71 (2005), article 036206.
12. J. GIANNOULIS, A. MlELKE, AND C. SPARBER, Interaction of modulated pulses in the nonlinear Schrödinger equation with periodic potential, J. Differential Equations, 245 (2008), pp. 939-963.
13. J. GIANNOULIS, A. MIELKE, AND C. SPARBER, High-Frequency Averaging in Semi-classical Hartree-Type Equations, preprint; available online from http://arxiv.org/abs/0909.5640v2.
14. O. GUÈS, Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal., 6 (1993), pp. 241-269.
15. D. L. HART, A. F. JUDY, R. ROY, AND J. W. BELETIC, Dynamical evolution of multiple fourwave mixing processes in an optical fibre, Phys. Rev. E, 57 (1998), pp. 4757-4774.
16. E. INFELD AND M. TRIPPENBACH, Useful models of four-wave mixing in Bose-Einstein condensates, J. Phys. B: At. Mol. Opt. Phys., 36 (2003), pp. 4327-4337.
17. A. R. ITS, Inversion of hyperelliptic integrals, and integration of nonlinear differential equations, Vestnik Leningrad. Univ., 7 (1976), pp. 39-46, 162.
18. J.-L. JOLY, G. MÉTIVIER, AND J. RAUCH, Coherent nonlinear waves and the Wiener algebra, Ann. Inst. Fourier (Grenoble), 44 (1994), pp. 167-196.
19. D. LANNES, Nonlinear geometrical optics for oscillatory wave trains with a continuous oscillatory spectrum, Adv. Differential Equations, 6 (2001), pp. 731-768.
20. P. D. LAX, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), pp. 627-646.
21. Y. C. MA AND M. J. ABLOWITZ, The periodic cubic Schrödinger equation, Stud. Appl. Math., 65 (1981), pp. 113-158.
22. L. MOLINET, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett., 16 (2009), pp. 111-120.
23. J. RAUCH, Lectures on Nonlinear Geometric Optics, lecture notes; available from www.math. lsa.umich.edu/-rauch/courses.html.
24. J. RAUCH AND M. KEEL, Lectures on geometric optics, in Hyperbolic Equations and Frequency Interactions (Park City, UT, 1995), AMS, Providence, RI, 1999, pp. 383-466.
25. S. RAUF AND J. A. TATARONIS, Resonant four-wave mixing of finite-amplitude Alfvén waves, J. Plasma Phys., 55 (1996), pp. 173-180.
26. S. SCHOCHET, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), pp. 476-512.
27. C. SULEM AND P.-L. SULEM, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Springer-Verlag, New York, 1999.
28. M. TRIPPENBACH, Y. B. BAND, AND P. S. JULIENNE, Theory of four-wave mixing of matter waves from a Bose-Einstein condensate, Phys. Rev. A, 62 (2000), article 023608.
29. J. ZWECK AND C. R. MENYUK, Analysis of four-wave mixing between pulses in high-data rate quasi-linear subchannel-multiplexed systems, Optics Lett., 27 (2002), pp. 51-113.