Liapunov-schmidt reduction and continuation for nonlinear Schrödinger equations

SIAM Journal on Scientific Computing

Volumn 29, Issue 2, 2007, Pages 729-755

  Chang, S L ^a   Chien, C S ^b   Jeng, B W ^b,c  

^a SOUTHERN TAIWAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Taiwan)

^b NATIONAL CHUNG HSING UNIVERSITY   (Taiwan)

^c NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

Author keywords

Bifurcation; Bose Einstein condensates; Mass conservation; Mode interactions; Time independent system

Indexed keywords

BIFURCATION (MATHEMATICS); BOSE-EINSTEIN CONDENSATION; LYAPUNOV FUNCTIONS; NUMERICAL METHODS; SCHRODINGER EQUATION; THREE DIMENSIONAL;

MASS CONSERVATION; MODE INTERACTIONS; TIME-INDEPENDENT SYSTEM;

NONLINEAR EQUATIONS;

EID: 34548436335     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/050642861     Document Type: Article

References (40)

1. M. H. ANDERSON, J. R. ENSHER, M. R. MATTHEWS, C. E. WIEMAN, AND E. A. CORNELL, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), pp. 198-201.
2. J. R. ANGLIN AND W. KETTERLE, Bose-Einstein condensation of atomic gases, Nature, 416 (2002), pp. 211-218.
3. C. C. BRADLEY, C. A. SACKETT, AND R. G. HULET, Bose-Einstein condensation of Lithium: Observation of limited condensate number, Phys. Rev. Lett., 78 (1997), pp. 985-989.
4. G. P. AGRAWAL, Nonlinear Fiber Optics, 2nd ed., Academic Press, New York, 1995.
5. E. P. GROSS, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), pp. 454-477.
6. L. P. PITAEVSKII, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), pp. 451-454.
7. F. T. HIOE, Solitary waves for N coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), pp. 1152-1155.
8. T. KANNA AND M. LAKSHMANAN, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 86 (2001), pp. 5043-5046.
9. A. GAMMAL., T. FREDERICO, AND L. TOMIO, Improved numerical approach for the time-independent Gross-Pitaevskii nonlinear Schrödinger equation, Phys. Rev. E, 60 (1999), pp. 2421-2424.
10. M. MITCHELL, Z. CHEN. M. SHIH, AND M. SEGEV, Self-trapping of partially spatially incoherent light, Phys. Rev. Lett., 77 (1996), pp. 490-493.
11. W. BAO AND W. TANG, Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187 (2003), pp. 230-254.
12. W. BAO AND Q. DU, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), pp. 1674-1697.
13. W. BAO. D. JAKSCH, AND P. A. MARKOWICH, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), pp. 318-342.
14. Y.-C. KUO, W.-W. LIN, AND S.-F. SHIEH, Bifurcation analysis of a two-component Bose-Einstein condensate, Phys. D, 21 (2005), pp. 311-346.
15. n, n ≤ 3, Comm. Math. Phys., 255 (2005), pp. 629-653.
16. D. S. HALL, M. R. MATTHEWS. J. R. ENSHER, C. E. WIEMAN, AND E. A. CORNELL, Dynamics oj component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), pp. 1539-1542.
17. H. PU AND N. P. BIGELOW, Properties of two-species Bose condensates, Phys. Rev. Lett., 80 (1998), pp. 1130-1133.
18. W. BAO, Ground states and dynamics of multicomponents Bose-Einstein condensates, Multiscale Model. Simul., 2 (2004), pp. 210-236.
19. P. K. GHOSH, Exact results on the dynamics of a multicomponent Bose-Einstein condensate, Rhys. Rev. A, 65 (2002), 053601(6).
20. S.-L. CHANG AND C.-S. CHIEN, Numerical continuation for nonlinear Schrödinger equations, Inter. J. Bifurcation and Chaos, to appear.
21. E. L. ALLGOWER AND K. GEORG, Introduction to Numerical Continuation Methods, Classics in Appl. Math. 45, SIAM, Philadelphia, 2003.
22. H. B. KELLER, Lectures on Numerical Methods in Bifurcation Problems, Springer-Verlag, Berlin, 1987.
23. M. GOLUBITSKY AND D. G. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. I, Appl. Math. Sci. 51, Springer-Verlag, 1985.
24. K. BÖHMER, On a numerical Liapunov-Schmidt method for operator equations, Computing, 51 (1993), pp. 237-269.
25. K. BÖHMER AND Z. MEI, On a numerical Liapunov-Schmidt method, in Computational Solution of Nonlinear Systems of Equations, E. L. Allgower and K. Georg, eds., Lectures in Appl. Math. 26, AMS, Providence, RI, 1990, pp. 79-98.
26. P. ASHWIN, K. BÖHMER, AND Z. MEI, A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square, Math. Comp., 64 (1995), pp. 649-670.
27. K. BÖHMER AND Z. MEI, Mode interactions of an elliptic system on the square, Int. Ser. Numer. Math. 104, Birkhäuser Verlag, Basel, 1992, pp. 49-58.
28. Z. MEI, Path following around corunk-2 biforcation points of a semilinear elliptic problem with symmetry, Computing, 47 (1991), pp. 69-85.
29. L. BAUER, H. B. KELLER, AND E. L. REISS, Multiple eigenvalues lead to secondary bifurcation, SIAM Rev., 17 (1975), pp. 101-122.
30. C.-S. CHIEN, S.-Y. GONG, AND Z. MEI, Mode jumping in the von Kármán equations, SIAM J. Sci. Comput., 22 (2000), pp. 1354-1385.
31. E. ISAACSON AND H. B. KELLER, Analysis of Numerical Methods, John Wiley, New York, 1965.
32. S.-L. CHANG, C.-S. CHIEN, AND B.-W. JENG, An efficient algorithm for the Schrödinger-Poisson eigenvalue problem, J. Comput. Appl. Math., to appear.
33. E. L. ALLGOWER, C.-S. CHIEN, AND W.-C. LEE, Bifurcations of corank greater than two, Comput. Math. Appl., 25 (1993), pp. 51-64.
34. M. GOLUBITSKY, I. STEWART, AND D. G. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. 2, Appl. Math. Sci., 69. Springer-Verlag, 1988.
35. C.-S. CHIEN, Z. MEI. AND C.-L. SHEN, Numerical continuation at double bifurcation points of a reaction-diffusion problem, Int. J. Bifurcation and Chaos, 8 (1998), pp. 117-139.
36. C.-S. CHIEN AND S.-L. CHANG, Application of the Lanczos algorithm for solving the linear systems that occur in continuation problems, Numer. Linear Algebra Appl., 10 (2003), pp. 335-355.
37. W. J. F. GOVAERTS, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000.
38. S. H. LUI AND G. H. GOLUB, Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators, Numer. Algorithms, 10 (1995), pp. 363-378.
39. B. N. PARLETT, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
40. C.-S. CHIEN AND B.-W. JENG, A two-grid discretization scheme for semilinear elliptic eigen-value problems, SIAM J. Sci. Comput., 27 (2006), pp. 1287-1304.