Numerical methods for computing nonlinear eigenpairs: Part II. Non-iso-homogeneous cases

SIAM Journal on Scientific Computing

Volumn 30, Issue 2, 2007, Pages 937-956

  Yao, Xudong ^a,b   Zhou, Jianxin ^c,d  

^a UNIVERSITY OF CONNECTICUT   (United States)

^b UNIVERSITY OF ARKANSAS   (United States)

^c TEXAS A AND M UNIVERSITY   (United States)

^d NSF DMS 0311905

Author keywords

Active Lagrange multiplier; Critical points; Minimax method; Nonlinear eigenproblems

Indexed keywords

BANACH SPACES; BOSE-EINSTEIN CONDENSATION; EIGENVALUES AND EIGENFUNCTIONS; NON NEWTONIAN FLOW; NUMERICAL METHODS; PROBLEM SOLVING; STATISTICAL MECHANICS;

ACTIVE LAGRANGE MULTIPLIER; CRITICAL POINT; EIGEN-PROBLEMS; EIGENPAIRS; LOCAL MINIMAX METHODS; MINIMAX METHODS; NONLINEAR EIGENPROBLEM; NONLINEAR WAVE MOTION; RAYLEIGH; STEADY STATE SOLUTION;

LAGRANGE MULTIPLIERS;

EID: 55549114717     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060656425     Document Type: Article

References (29)

1. V. A. ALEXSHKEVICH, Y. V. KAETASHOV, A. ZELENINIA, V. A. VYSLOUKH, J. P. TORRES, AND L. TORNER, Eigenvalue control and switching by fission of multisoliton bound states in planar waveguides, Optics Letters, 29 (2004), pp. 483-485.
2. 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.
3. V. BENCI, P. D'AVENIA, AND L. PISANI, Solitons in several space dimensions: Derrick's problem and infinitely many solitons, Arch. Ration. Mech. Anal., 154 (2000), pp. 297-324.
4. V. BENCI, D. FORTUNATO, AND L. PISANI, Soliton-like solutions of Lorentz invariant equation in dimension 3, Rev. Math. Phys., 3 (1998), pp. 315-344.
5. M. CUESTA, Eigenvalue problems for the p-Laplacian with indefinite weights, Electron. J. Differential Equations, 33 (2001), pp. 1-9.
6. J. I. DIAZ, Nonlinear Partial Differential Equations and Free Boundaries, Vol. I, Elliptic Equations, Res. Notes Math. 106, Pitman Publishing, Boston, 1985.
7. A. FERRANDO, M. ZACARES, P. F. DE CORADOBA, D. BINOSI, AND J. A. MONSORIU, Spatial soliton formation in photonic crystal fibers, Optics Express, 11 (2003), pp. 452-459.
8. J. J. GARCIA-RIPOLL, V. V. KONOTOP, B. M. MALOMED, AND V. M. PEREZ-GARCIA, A quasilocal Gross-Pitaevskii equation for attractive Bose-Einstein condensate, Math. Comput. Simulation, 62 (2003), pp. 21-30.
9. V. M. PEREZ-GARCIA, H. MIOHINEL, J. I. CIRAC, M. LEWENSTEIN, AND P. ZOLLER, Low energy excitations of a Bose-Einstein condensate: A time-dependent variational analysis, Phys. Rev. Lett., 77 (1996), p. 5320.
10. J. J. GARCIA-RIPOLL AND V. M. PEREZ- GARCIA, Optimizing Schrodinger functionals using Sobolev gradients: Applications to quantum mechanics and nonlinear optics, SIAM J. Sci. Comput., 23 (2001), pp. 1316-1334.
11. E. P. GROSS, Structure of a quantized vortex in, boson systems, Nuovo Cimento, 20 (1961), pp. 454-477.
12. M. HOLLAND AND J. COOPER, Expansion of a Bose-Einstein condensate in a harmonic potential, Phys. Rev. A, 53 (1996), p. R1954.
13. S. V. IZBASH, O Filtracii V Kropnozernstom Materiale, Izv. Nauchnoissled, Inst. Gidrotechniki (NIIG), Leningrad, USSR, 1931.
14. Y. Q. LI AND Z. L. LIU, Solutions of an elliptic eigenvalue problem involving subcritical or critical exponents, Comm. Partial Differential Equations, 26 (2001), pp. 2227-2248.
15. Y. LI AND J. ZHOU, A minimax method for finding multiple critical points and its applications to semilinear PDEs, SIAM J. Sci. Comput., 23 (2001), pp. 840-865.
16. Y. LI AND J. ZHOU, Convergence results of a local minimax method for finding multiple critical points, SIAM J. Sci. Comput., 24 (2002), pp. 865-885.
17. E. MONTEFUSCO AND V. RADULESCU, Nonlinear eigenvalue problems for quasilinear operators on bounded domains, Nonlinear Diff. Equat. Appl., 8 (2001), pp. 481-497.
18. L. P.PITAEVSKII, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), pp. 451-454.
19. P. A. RUPRECHT, M. J. HOLLAND, AND K. BURNETT, Time-dependent solution of the nonlinear Schrdinger equation for Bose-condensed trapped neutral atoms, Phys. Rev. A, 51 (1995), p. 4704.
20. K. TSO, On a real Monge-Ampére functional, Invent. Math., 101 (1990), pp. 425-448.
21. Z. WANG AND J. ZHOU, An efficient and stable method for computing multiple saddle points with symmetries, SIAM J. Numer. Anal., 43 (2005), pp. 891-907.
22. Z. WANG AND J. ZHOU, A local minimax-Newton method for finding multiple saddle points with symmetries, SIAM J. Numer. Anal., 42 (2004), pp. 1745-1759.
23. J. YANG AND T. I. LAKOBA, Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations, Stud. Appl. Math., 118 (2007), pp. 153-197.
24. X. YAO AND J. ZHOU, A minimax method for finding multiple critical points in Banach spaces and its application to quasi-linear elliptic PDE, SIAM J. Sci. Comput., 26 (2005), pp. 1796-1809.
25. X. YAO AND J. ZHOU, Unified convergence results on a minimax algorithm for finding multiple critical points in Banach spaces, SIAM J. Numer. Anal., 45 (2007), pp. 1330-1347.
26. X. YAO AND J. ZHOU, Numerical methods for computing nonlinear eigenpairs: Part I. Isohomogeneous cases, SIAM J. Sci. Comput., 29 (2007), pp. 1355-1374.
27. E. ZEIDLER, Nonlinear Functional Analysis and, Its Applications III, Springer-Verlag, New York, 1985.
28. J. ZHOU, A local min-orthogonal method for finding multiple saddle points, J. Math. Anal. Appl., 291 (2004), pp. 66-81.
29. J. ZHOU, Instability analysis of saddle points by a local minimax method, Math. Comp., 74 (2005), pp. 1391-1411.