Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrödinger equation

SIAM Journal on Scientific Computing

Volumn 25, Issue 1, 2003, Pages 186-212

  Akrivis, G D ^a   Dougalis, V A ^b,c   Karakashian, O A ^d   McKinney, W R ^e  

^a UNIVERSITY OF IOANNINA   (Greece)

^b UNIVERSITY OF ATHENS   (Greece)

^c INSTITUTE OF COMPUTER SCIENCE   (Greece)

^d UNIVERSITY OF TENNESSEE   (United States)

^e NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Adaptive mesh refinement; Finite element methods; Nonlinear Schr dinger equation; Point blow up

Indexed keywords

APPROXIMATION THEORY; FINITE ELEMENT METHOD; FUNCTIONS; GALERKIN METHODS; INITIAL VALUE PROBLEMS; PIECEWISE LINEAR TECHNIQUES; PROBLEM SOLVING; THEOREM PROVING;

ADAPTIVE MESH REFINEMENT; CUBIC NONLINEARITY; MATHEMATICAL PHYSICS; NONLINEAR SCHRODINGER EQUATION;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 1842459598     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827597332041     Document Type: Article

References (54)

1. G. D. Akrivis, V. A. Dougalis, and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), pp. 31-53.
2. G. Akrivis, V. A. Dougalis, and O. A. Karakashian, Solving the systems of equations arising in the discretization of some nonlinear p.d.e.'s by implicit Runge-Kutta methods, Modél. Math. Anal. Numér., 31 (1997), pp. 251-288.
3. G. D. Akrivis, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, Galerkin-finite element methods for the nonlinear Schrödinger equation, in Advances on Computer Mathematics and Its Applications, E. Lipitakis, ed., World Scientific, Singapore, 1993, pp. 85-106.
4. G. D. Akrivis, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, Numerical approximation of singular solutions of the damped nonlinear Schrödinger equation, in ENUMATH 97, H. G. Bock et al., eds., World Scientific, River Edge, NJ, 1998, pp. 117-124.
5. G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31 (1997), pp. 45-59.
6. G. A. Baker, W. N. Jureidini, and O. A. Karakashian, Piecewise solenoidal vector fields and the Stokes problem, SIAM J. Numer. Anal., 27 (1990), pp. 1466-1485.
7. M. Berger and R. V. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41 (1988), pp. 841-863.
8. A. L. Bertozzi, M. P. Brenner, T. F. Dupont, and L. P. Kadanoff, Singularities and similarities in interface flows, in Trends and Perspectives in Applied Mathematics, L. Sirovich, ed., Springer-Verlag, New York, 1994, pp. 155-208.
9. J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, Fully-discrete methods with grid refinement for the generalized Korteweg-de Vries equation, in Viscous Profiles and Numerical Methods for Shock Waves, M. Shearer, ed., SIAM, Philadelphia, 1991, pp. 1-11.
10. J. B. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Roy Soc. London Ser. A, 351 (1995), pp. 107-164.
11. C. J. Budd, S. Chen, and R. D. Russell, New self-similar solutions of the nonlinear Schrödinger equation with moving mesh computations, J. Comput. Phys., 152 (1999), pp. 756-789.
12. T. Cazenave, Blow Up and Scattering in the Nonlinear Schrödinger Equation, Textos de Métodos Matemáticos 30, Instituto de Matématica, UFRJ, Rio de Janeiro, 1994.
13. T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Mathématiques et Applications 1, SMAI-Ellipses, Paris, 1990.
14. R. Y. Chiao, E. Garmire, and C. Townes, Self-trapping of optical beams, Phys. Rev. Lett., 73 (1964), pp. 479-482.
15. T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, and S.-M. Zhou, Finite time singularity formation in Hele-Shaw systems, Phys. Rev. E, 47 (1993), pp. 4182-4196.
16. G. Fibich, Self-focusing in the Nonlinear Schrödinger Equation for Ultrashort Laser-Tissue Interactions, Ph.D. Thesis, New York University, New York, 1994.
17. G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), pp. 1680-1705.
18. G. Fibich and G. Papanicolaou, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (1999), pp. 183-240.
19. G. M. Fraiman, Asymptotic stability of manifold of self-similar solutions in self-focusing, Soviet Physics JETP, 61 (1985), pp. 228-233.
20. J. Ginibre, Introduction aux équations de Schrödinger non linéaires, Paris Onze édition L161, Université de Paris-Sud, Orsay, France, 1998.
21. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations I: The Cauchy problem, general case, J. Funct. Anal., 32 (1979), pp. 1-32.
22. R. T. Glassey, On the blowing up of the solutions in the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), pp. 1974-1977.
23. O. Karakashian, G. D. Akrivis, and V. A. Dougalis, On optimal order error estimates for the nonlinear Schrödinger equations, SIAM J. Numer. Anal., 30 (1993), pp. 377-400.
24. O. A. Karakashian and W. N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal., 35 (1998), pp. 93-120.
25. O. A. Karakashian and M. Plexousakis, Adaptive methods for nonradially symmetric solutions of the nonlinear Schrödinger equation, to appear.
26. T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théoret., 46 (1987), pp. 113-129.
27. N. E. Kosmatov, V. F. Shvets, and V. E. Zakharov, Computer simulation of wave collapses in the nonlinear Schrödinger equation, Phys. D, 52 (1991), pp. 16-35.
28. M. J. Landman, G. C. Papanicolaou, C. Sulem and P.-L. Sulem, Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimensions, Phys. Rev. A, 38 (1988), pp. 3837-3843.
29. M. J. Landman, G. C. Papanicolaou, C. Sulem, P.-L. Sulem, and X. P. Wang, Stability of isotropic singularities for the nonlinear Schrödinger equation, Phys. D, 47 (1991), pp. 393-415.
30. B. Le Mesurier, G. Papanicolaou, C. Sulem, and P.-L. Sulem, The focusing singularity of the nonlinear Schrödinger equation, in Directions in Partial Differential Equations, M. G. Crandall, P.H. Rabinowitz, and R.E. Turner, eds., Academic Press, New York, 1987, pp. 159-201.
31. B. J. Le Mesurier, G. Papanicolaou, C. Sulem, and P.-L. Sulem, Focusing and multifocusing solutions of the nonlinear Schrödinger equation, Phys. D, 31 (1988), pp. 78-102.
32. B. J. Le Mesurier, G. Papanicolaou, C. Sulem, and P.-L. Sulem, Local structure of the self-focusing singularity of the nonlinear Schrödinger equation, Phys. D, 32 (1988), pp. 210-226.
33. V. M. Malkin, On the analytical theory for stationary self-focusing of radiation, Phys. D, 64, (1993), pp. 251-266.
34. D. W. McLaughlin, G. C. Papanicolaou, C. Sulem, and P.-L. Sulem, Focusing singularity of the cubic Schrödinger equation, Phys. Rev. A, 34 (1986), pp. 1200-1210.
35. F. Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, in Proceedings of the International Congress of Mathematicians (Berlin 1998), Documenta Mathematica 3, University of Bielefeld, Bielefeld, Germany, 1998, pp. 57-66.
36. J. Rasmussen and K. Rypdal, Blow-up in nonlinear Schrödinger equations, parts I and II, Phys. Scripta, 33 (1986), pp. 481-504.
37. P. A. Robinson, Nonlinear wave collapse and strong turbulence, Rev. Modern Phys., 69 (1997), pp. 507-573.
38. M. J. Shelley, A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method, J. Fluid Mech., 244 (1992), pp. 493-526.
39. A. I. Smirnov and G. M. Fraiman, The interaction representation in the self-focusing theory, Phys. D, 52 (1991), pp. 2-15.
40. W. A. Strauss, Nonlinear Wave Equations, CBMS Reg. Conf. Ser. Math. 73, AMS, Providence, RI, 1989.
41. C. Sulem and P.-L. Sulem, Focusing nonlinear Schrödinger equation and wave-packet collapse, Nonlinear Anal., 30 (1997), pp. 833-844.
42. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, New York, 1999.
43. P.-L. Sulem, C. Sulem, and A. Patera, Numerical simulation of singular solutions to the two-dimensional cubic Schrödinger equation, Comm. Pure Appl. Math., 37 (1984), pp. 755-778.
44. V. I. Talanov, Self-focusing of wave beams in nonlinear media, JETP Lett., 2 (1965), pp. 138-141.
45. Y. Tourigny and J. M. Sanz-Serna, The numerical study of blow-up with application to a nonlinear Schrödinger equation, J. Comput. Phys., 102 (1992), pp. 407-416.
46. Y. Tourigny, J. M. Sanz-Serna, and J. Ll. Morris, Approximation of radial functions by piecewise polynomials on arbitrary grids, Numer. Methods Partial Differential Equations, 7 (1991), pp. 1-8.
47. M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal., 15 (1984), pp. 357-366.
48. S. N. Vlasov, L. V. Piskunova, and V. I. Talanov, Structure of the field near a singularity arising from self-focusing in a cubically nonlinear medium, Soviet Physics JETP, 48, (1978), pp. 808-812.
49. M. I. Weinstein, Nonlinear Schrödinger equation and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), pp. 567-576.
50. D. Wood, The self-focusing singularity in the nonlinear Schrödinger equation, Stud. Appl. Math., 71 (1984), pp. 103-115.
51. V. E. Zakharov, Collapse of Langmuir waves, Soviet Physics JETP, 35 (1972), pp. 908-922.
52. V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, Hamiltonian approach to the description of non-linear plasma phenomena, Phys. Rep., 129 (1985), pp. 286-366.
53. V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP, 34 (1972), pp. 62-69.
54. V. E. Zakharov and V. S. Synakh, The nature of the self-focusing singularity, Soviet Physics JETP, 41 (1976), pp. 465-468.