Wave collapse in a class of nonlocal nonlinear Schrödinger equations

Physica D: Nonlinear Phenomena

Volumn 207, Issue 3-4, 2005, Pages 230-253

  Ablowitz, Mark ^a   Bakirtaş, Ilkay ^b   Ilan, Boaz ^a  

^a UNIVERSITY OF COLORADO   (United States)

^b ISTANBUL TECHNICAL UNIVERSITY   (Turkey)

Author keywords

Blowup; Modulated nonlinear waves; Singularity formation

Indexed keywords

FLUID DYNAMICS; GROUND STATE; INTEGRATION; NONLINEAR OPTICS; PLASMAS; THEOREM PROVING; WATER WAVES;

BLOWUP; GLOBAL-EXISTENCE THEORY; MODULATED NONLINEAR WAVES; SINGULARITY FORMATION;

NONLINEAR EQUATIONS;

EID: 21644459325     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2005.06.001     Document Type: Article

References (36)

1. M.J. Ablowitz, G. Biondini, and S. Blair Multi-dimensional pulse propagation in resonant materials χ (2 ) materials Phys. Lett. A 236 1997 520
2. M.J. Ablowitz, G. Biondini, and S. Blair Nonlinear Schrödinger equations with mean terms in nonresonant multidimensional quadratic materials Phys. Rev. E 63 2001 605
3. M.J. Ablowitz, and P.A. Clarkson Solitons, Nonlinear Evolution Equations and Inverse Scattering 1991 Cambridge University Press Cambridge
4. M.J. Ablowitz, and R. Haberman Nonlinear evolution equations - two three dimensions Phys. Rev. Lett. 35 1975 1185
5. M.J. Ablowitz, and Z. Musslimani Dark and gray strongly-dispersion managed solitons Phys. Rev. E 67 2003 025601(R)
6. M.J. Ablowitz, and H. Segur On the evolution of packets of water waves J. Fluid Mech. 92 1979 691
7. M.J. Ablowitz, and H. Segur Solitons and the Inverse Scattering Transform 1981 SIAM Philadelphia
8. D.J. Benney, and G.J. Roskes Wave instabilities Stud. Appl. Math. 48 1969 377
9. R.Y. Chiao, E. Garmire, and C.H. Townes Self-trapping of optical beams Phys. Rev. Lett. 13 1964 479
10. R. Cipolatti On the existence of standing waves for the Davey-Stewartson system Commun. Part. Diff. Eq. 1992 967
11. B.I. Cohen, B.F. Lasinski, A.B. Langdon, and J.C. Cummings Dynamics of ponderomotive self-focusing in plasmas Phys. Fluids B-Plasma Phys. 3 1992 766
12. L.C. Cravosan, J.P. Torres, D. Mihalache, and L. Torner Arresting wave collapse by self-rectification Phys. Rev. Lett. 91 2003 063904
13. A. Davey, and K. Stewartson On three-dimensional packets of surface waves Proc. R. Soc. Lond. A 338 1974 101
14. V.D. Djordevic, and L.G. Reddekopp On two-dimensional packets of capillary gravity waves J. Fluid Mech. 79 1977 703
15. G. Fibich Small beam nonparaxiality arrests self-focusing of optical beams Phys. Rev. Lett. 76 1996 4356
16. G. Fibich, and A.L. Gaeta Critical power for self-focusing in bulk media and in hollow waveguides Opt. Lett. 25 2000 335
17. G. Fibich, and B. Ilan Self-focusing of elliptic beams: an example for the failure of the aberrationless approximation J. Opt. Soc. Am. B 17 2000 1749
18. G. Fibich, and B. Ilan Vectorial and random effects in self-focusing and in multiple filamentation Physica D 157 2001 113
19. G. Fibich, and G.C. Papanicolaou Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension SIAM J. Appl. Math. 60 1999 183
20. G. Fibich, W. Ren, and X.P. Wang Numerical simulations of self-focusing of ultrafast laser pulses Phys. Rev. E 67 2003 056603
21. A.S. Fokas, and M.J. Ablowitz On a method of solution for a class of multidimensional nonlinear evolution equations Phys. Rev. Lett. 51 1983 7
22. J.M. Ghidaglia, and J.C. Saut On the initial value problem for the Davey-Stewartson systems Nonlinearity 3 1990 475
23. P.L. Kelley Self focusing of optical beams Phys. Rev. Lett. 15 1965 1005
24. K.D. Moll, A.L. Gaeta, and G. Fibich Self-similar optical wave collapse: observation of the Townes profile Phys. Rev. Lett. 90 2003 203902
25. F. Merle, and Y. Tsutsumi L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity J. Diff. Eq. 84 1990 205
26. F. Merle, and P. Raphael On the universality of blow-up profile for L 2 critical nonlinear Schrödinger equation Inventiones Mathematicae 156 2004 565
27. G.C. Papanicolaou, D. McLaughlin, and M. Weinstein Focusing singularity for the nonlinear Schrödinger equation Lec. No. Numer. Appl. Anal. 5 1982 253
28. G.C. Papanicolaou, C. Sulem, P.L. Sulem, and X.P. Wang The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves Physica D 72 1994 61
29. V.I. Petviashvili Equation of an extraordinary soliton Sov. J. Plasma Phys. 2 1976 257
30. C. Sulem, and P.L. Sulem The Nonlinear Schrödinger Equation 1999 Springer-Verlag New York
31. V.I. Talanov Self focusing of wave beams in nonlinear media Sov. Phys. JETP Lett. 2 1965 138
32. J.P. Torres, L. Torner, I. Biaggio, and M. Segev Tunable self-action of light in optical rectification Opt. Commun. 213 2002 351
33. F. Vidal, and T.W. Johnston Electromagnetic beam breakup: Multiple filaments, single beam equilibria, and radiation Phys. Rev. Lett. 77 1996 1282
34. S. Vlasov, V. Petrishchev, and V. Talanov Averaged description of wave beams in linear and nonlinear media Radiophys. Quant. Elec. 14 1971 1062 (1070 in English)
35. M.I. Weinstein Nonlinear Schrödinger equations and sharp interpolation estimates Commun. Math. Phys. 87 1983 567
36. G.B. Whitham Linear and Nonlinear Waves 1974 Wiley New York