Singular ring solutions of critical and supercritical nonlinear Schrödinger equations

Physica D: Nonlinear Phenomena

Volumn 231, Issue 1, 2007, Pages 55-86

  Fibich, Gadi ^a   Gavish, Nir ^a   Wang, Xiao Ping ^b  

^a TEL AVIV UNIVERSITY   (Israel)

^b HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

Author keywords

Blowup rate; Collapse; Nonlinear Schr dinger equation; Ring profile; Self similar solution; Singularity; Supercritical collapse

Indexed keywords

ASYMPTOTIC STABILITY; NONLINEAR EQUATIONS;

BLOWUP RATES; RING PROFILES; SELF SIMILAR SOLUTION; SUPERCRITICAL COLLAPSE;

SCHRODINGER EQUATION;

EID: 34249993538     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2007.04.007     Document Type: Article

References (47)

1. Fibich G., and Papanicolaou G. Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension. SIAM J. Appl. Math. 60 (1999) 183-240
2. Strauss W. Nonlinear Wave Equations (1989), American Mathematical Society, Providence, RI
3. Sulem C., and Sulem P. The Nonlinear Schrödinger Equation (1999), Springer, New-York
4. N. J. Differential Equations 85 (1990) 367-400
5. Chiao R., Garmire E., and Townes C. Self-trapping of optical beams. Phys. Rev. Lett. 13 (1964) 479-482
6. Weinstein M. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983) 567-576
7. Talanov V. Focusing of light in cubic media. JETP Lett. 11 (1970) 199-201
8. Landman M., Papanicolaou G., Sulem C., Sulem P., and Wang X. Stability of isotropic singularities for the nonlinear Schrödinger equation. Physica D 47 (1991) 393-415
9. McLaughlin D., Papanicolaou G., Sulem C., and Sulem P. Focusing singularity of the cubic Schrödinger equation. Phys. Rev. A 34 (1986) 1200-1210
10. Fraiman G. Asymptotic stability of manifold of self-similar solutions in self-focusing. Sov. Phys. JETP 61 (1985) 228-233
11. Landman M., Papanicolaou G., Sulem C., and Sulem P. Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A 38 (1988) 3837-3843
12. LeMesurier B., Papanicolaou G., Sulem C., and Sulem P. Local structure of the self-focusing singularity of the nonlinear Schrödinger equation. Physica D 32 (1988) 210-226
13. LeMesurier B., Papanicolaou G., Sulem C., and Sulem P. Focusing and multi-focusing solutions of the nonlinear Schrödinger equation. Physica D 31 (1988) 78-102
14. Weinstein M. The nonlinear Schrödinger equation - singularity formation, stability and dispersion. Contemp. Math. 99 (1989) 213-232
15. Nawa H. Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity. J. Math. Soc. Japan 46 (1994) 557-586
16. Nawa H. Asymptotic and limiting profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power. Comm. Pure Appl. Math. 52 (1999) 193-270
17. Merle F., and Raphael P. Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13 (2003) 591-642
18. 2 critical nonlinear Schrödinger equation. Invent. Math. 156 (2004) 565-672
19. Merle F., and Raphael P. Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. Comm. Math. Phys. 253 (2005) 675-704
20. Merle F., and Raphael P. On a sharp lower bound on the blow-up rate for the critical nonlinear Schrödinger equation. J. Amer. Math. Soc. 19 (2006) 37-90
21. Moll K., Gaeta A., and Fibich G. Self-similar optical wave collapse: Observation of the Townes profile. Phys. Rev. Lett. 90 (2003) 203902
22. LeMesurier B., Papanicolaou G., Sulem C., and Sulem P. The focusing singularity of the nonlinear Schrödinger equation. In: Grandall M., Rabinovitz P., and Turner R. (Eds). Directions in Partial Differential Equations (1987), Academic Press, New-York 159-201
23. N. Kosmatov, I. Petrov, V. Shvets, V. Zakharov, Large scale amplitude simulation of wave collapse in nonlinear Schrödinger equations, Academy of Sciences of the USSR, Space Research Institute. Preprint
24. Kosmatov N., Shvets V., and Zakharov V. Computer simulation of wave collapses in the nonlinear Schrödinger equation. Physica D 52 (1991) 16-35
25. Budd C.J. Asymptotics of multibump blow-up self-similar solutions of the nonlinear Schrödinger equation. SIAM J. Appl. Math. 62 (2001) 801-830
26. Fibich G., Gavish N., and Wang X. New singular solutions of the nonlinear Schrödinger equation. Physica D 211 (2005) 193-220
27. 2 supercritical non linear Schrödinger equation. Duke Math. J. 134 2 (2006) 199-258
28. Merle F. Construction of solutions with exactly k blow-up points for the Schrödinger equation with the critical power nonlinearity. Comm. Math. Phys. 129 (1990) 223-240
29. Degtiarev L., Zakharov V.E., and Rudakov L.I. Two examples of langmuir wave collapse. JETP 41 1 (1975) 57-61
30. s. Nonlinear Anal. 14 (1990) 807-836
31. Merle F. Limit of the solution of a nonlinear Schrödinger equation at blow-up time. J. Funct. Anal. 84 (1989) 201-214
32. 1. Manuscripta Math. 61 (1988) 477-498
33. Merle F. Lower bounds for the blow-up rate of solutions of the Zakharov equation in dimension two. Comm. Pure Appl. Math. 49 (1996) 765-794
34. Perelman G. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2 (2001) 605-673
35. Atai J., Chen Y., and Soto-Crespo J. Stability of three-dimensional self-trapped beams with a dark spot surrounded by bright rings of varying intensity. Phys. Rev. A 49 (1994) 3170-3173
36. Bergé L., Gouédard C., Schjodt-Eriksen J., and Ward H. Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states. Physica D 176 (2003) 181-211
37. Feit M., and Fleck J. Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams. J. Opt. Soc. Amer. B 5 (1988) 633-640
38. Fibich G., and Ilan B. Vectorial and random effects in self-focusing and in multiple filamentation. Physica D 157 (2001) 112-146
39. Konno K., and Suzuki H. Self-focusing of a laser bean in nonlinear media. Phys. Scripta 20 (1979) 382-386
40. Soto-Crespo J., Heatley D., Wright E., and Akhmediev N. Stability of the higher-bound states in a saturable self-focusing medium. Phys. Rev. A 44 (1991) 636-644
41. Grow T.D., Ishaaya A.A., Vuong L.T., Gaeta A.L., Gavish N., and Fibich G. Collapse dynamics of super-Gaussian beams. Opt. Express 14 (2006) 5468-5475
42. Ren W., and Wang X. An iterative grid redistribution method for singular problems in multiple dimensions. J. Comput. Phys. 159 (2000) 246-273
43. Fibich G., Ren W., and Wang X. Numerical simulations of self focusing of ultrafast laser pulses. Phys. Rev. E 67 (2003) 056603
44. Wang D., and Wang X. A three-dimensional adaptive method based on the iterative grid redistribution. J. Comput. Phys. 199 (2004) 423-436
45. G. Fibich, Self-focusing in the nonlinear Schrödinger equation for ultrashort laser-tissue interactions, Ph.D. Thesis, Courant Institute, NYU, 1994
46. Dubietis A., Tamos ̌auskas G., Fibich G., and Ilan B. Multiple filamentation induced by input-beam ellipticity. Optim. Lett. 29 (2004) 1126-1128
47. Weinstein M. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985) 472-490