Multibump, blow-up, self-similar solutions of the complex Ginzburg-Landau equation

SIAM Journal on Applied Dynamical Systems

Volumn 4, Issue 3, 2005, Pages 649-678

  Buddt, C J ^a   Rottschafer V ^b   Williams, J F ^b  

^a University of Bath   (United Kingdom)

^b LEIDEN UNIVERSITY   (Netherlands)

Author keywords

Asymptotic; Blow up; Complex Ginzburg Landau; Multibump solutions; Self similar

Indexed keywords

ALGEBRA; ASYMPTOTIC STABILITY;

BLOW-UP; COMPLEX GINZBURG-LANDAU; MULTIBUMP SOLUTIONS; SELF-SIMILAR;

NONLINEAR EQUATIONS;

EID: 33644914522     PISSN: 15360040     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/040610866     Document Type: Article

References (24)

1. I. S. ARANSON AND L. KRAMER, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), pp. 99-143.
2. G. I. BARENBLATT, Scaling, Sell-Similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, UK, 1996.
3. H. R. BRAND, P. S. LOMDAHL, AND A. C. NEWELL, Benjamir-Feir turbulence in convective binary fluid mixtures, Phys. D, 23 (1986), pp. 345-362.
4. C. J. BUDD, Asymptotics of multibump blow-up self-similar solutions of the nonlinear Schrödinger equation, SIAM J. Appl. Math., 62 (2001), pp. 801-830.
5. 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.
6. C. J. BUDD, W. HUANG, AND R. D. RUSSELL, Moving mesh methods for problems with blow-up, SIAM J. Sci. Comput., 17 (1996), pp. 305-327.
7. C. J. BUDD AND J. F. WILLIAMS, Optimal Grids and Uniform Error Estimates for PDEs with Blow-Up, in preparation, 2004.
8. A. DAVEY, L. M. HOCKING, AND K. STEWARTSON, On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow, J. Fluid Mech., 63 (1974), pp. 529-536.
9. R. C. DIPRIMA AND H. L. SWINNEY, Instabilities and transition in flow between concentric cylinders, in Hydrodynamic Instabilities and the Transition to Turbulence, H. Swinney and J. Gollub, eds., Topics in Applied Physics 45, Springer-Verlag, New York, 1981, pp. 139-180.
10. E. J. DOEDEL, A. R. CHAMPNEYS, T. F. FAIRGRIEVE, Y. A. KUZNETSOV, B. SANDSTEDE, AND X.-J. WANG, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations, Tech. report, Department of Computer Science, Concordia University, Montreal, Canada, 1997, available at ftp://ftp.cs. concordia.ca/pub/doedel/auto.
11. G. FIBICH AND D. LEVY, Self-focusing in the complex Ginzburg-Landau limit of the critical nonlinear Schrödinger equation, Phys. Lett. A, 249 (1998), pp. 286-294.
12. 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.
13. E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd ed., Springer Ser. Comput. Math. 8, Springer-Verlag, 1993, available at http://www.unigo.ch/rnath/folks/ hairer/software.html.
14. W. HUANG, Y. REN, AND R. RUSSELL, Moving mesh methods based on moving mesh partial differential equations, J. Comput. Phys., 113 (1994), pp. 279-290.
15. N. J. KOPELL AND M. LANDMAN, Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: A geometrical analysis, SIAM J. Appl. Math., 55 (1995), pp. 1297-1323.
16. n, Arch. Ration. Mech. Anal., 99 (1987), pp. 115-145.
17. A. MIELKE, The Ginzburg-Landau equation in its role as a modulation equation, in Handbook of Dynamical Systems 2, North-Holland, Amsterdam, 2002, pp. 759-834.
18. A. C. NEWELL AND J. A. WHITEHEAD, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 28 (1969), pp. 279-303.
19. P. PLECHÁČ AND V. ŠVERÁK, On self-similar singular solutions of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 54 (2001), pp. 1215-1242.
20. M. J. D. POWELL, A hybrid method for nonlinear equations, in Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Gordon and Breach, New York, 1988, pp. 87-114.
21. V. ROTTSCHÄFER AND T. KAPER, Blow-up in the nonlinear Schrödinger equation near critical dimension, J. Math. Anal. Appl., 268 (2002), pp. 517-549.
22. V. ROTTSCHÄFER AND T. KAPER, Geometric theory for multi-bump, self-similar, blow-up solutions of the cubic nonlinear Schrödinger equation, Nonlinearity, 16 (2003), pp. 929-961.
23. K. STEWARTSON AND J. STUART, A nonlinear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech., 48 (1971), pp. 529-545.
24. C. SULEM AND P. L. SULEM, The Nonlinear Schrödinger Equation, Appl. Math. Sci. 139, Springer-Verlag, Berlin, 1999.