Symmetric ground state solutions of m-coupled nonlinear Schrödinger equations

Nonlinear Analysis, Theory, Methods and Applications

Volumn 71, Issue 10, 2009, Pages 4696-4704

  Hajaiej, Hichem ^a  

^a KING SAUD UNIVERSITY   (Saudi Arabia)

Author keywords

Cauchy problem; Ground state; Schr dinger equations

Indexed keywords

CAUCHY PROBLEM; DINGER EQUATION; NON-LINEARITY; STATE SOLUTION;

CONTROL NONLINEARITIES; GROUND STATE;

NONLINEAR EQUATIONS;

EID: 67651083516     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2009.03.035     Document Type: Article

References (37)

1. Sukhorukov A.A., and Kivshar Y.S. Stability of spatial optical solitons. In: Torruellas T., and Trillo S. (Eds). Spatial Optical Solitons (2001), Springer, New York
2. Bronski J.C., Carr L.D., Deconink B., and Kutz J.N. Bose-Einstein condensates in standing waves. Phys. Rev. Lett. 86 (2001) 1402-1405
3. Bennery D.J., and Newell A.C. The propagation of nonlinear wave envelopes. J. Math. Phys. 46 (1967) 133-139
4. Mankov S.V. On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phgs. JETP38 (1974) 248-253
5. Menyuk C.R. Nonlinear pulse propagation in birefringent optical fibers. IEEE J. Quantum Electron. 23 (1987) 174-176
6. Akhmediev, and Ankiewicz N. Partially coherent solitons on a finite background. Phys. Rev. Lett. 82 (1999) 2661
7. Akhmediev, and Ankiewicz N. Solitons, Nonlinear Pulses and Beams (1997), Chapman and Hall, London
8. Christodoulides D.N., Singh S.R., Carvalho M.I., and Segev M. Observation of bound-states of interacting vector solitons. Opt. Lett. 25 (2000) 561-582
9. Haelterman M., and Sheppard A.P. Bifurcation phenomena and multiple soliton-bound states in isotropic Kerr media. Phy. Rev. E 49 (1994) 3376-3381
10. Kivshar Y.S., and Agrawal G.P. Optical solitons: from fibers to photonic crystals (2003), Academic Press, San Diego
11. Cipolatti R., and Zumpichiatti W. On the existence and regularity of ground states for nonlinear system of coupled Schrödinger equations. Comput. Appl. Math. 18 (1999) 19-36
12. Cipolatti R., and Zumpichiatti W. Orbital stable standing waves for a system of coupled nonlinear Schrödinger equations. Nonlinear Anal. 42 (2000) 445-461
13. Ostrovskaya E.A., Kivshar Y.S., Skrybin D.V., and Firth W.J. Stability of multihump optical solitons. J. Opt. B1 (1999) 77-83
14. Synder A.W., Hewlett S.J., and Mitchell D.J. Dynamic spatial solitons. Phys. Rev. Lett. 72 (1994)
15. Stubbe J. Linear stability theory of solitary waves arising in Hamiltonian systems with symmetry. Portugal Math. 46 (1989) 17-32
16. Tratnik M.V., and Sipe J.E. Bound solitary waves in optical fibers. Phys. Rev. A38 (1988) 2011-2017
17. A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations (227) 258-281
18. Ambrosetti A., and Colorado E. Standing waves of some coupled nonlinear Schrödinger equations. J. London Math. Soc. 75 (2007) 67-82
19. Bartsch T., and Wang Z.Q. Note on ground state of nonlinear Schrödinger systems. J. Partial Differential Equations 19 (2006) 200-207
20. n, n ≤ 3. Comm. Math. Phy. 255 (2005) 629-653
21. Maia L.A., Montefusco E., and Pellacci B. Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differential Equations 229 (2006) 743-767
22. L.A. Maia, E. Montefusco, B. Pellacci, Orbital stability of ground state solutions of coupled nonlinear Schrödinger equations. Preprint
23. n. Comm. Math. Phys. 271 (2007) 199-221
24. Troy W.C. Symmetry properties in systems of semilinear elliptic equations. J. Diff. Eq. 42 3 (1981) 400-413
25. Burchard A., and Hajaiej H. Rearrangement inequalities for functionals with monotone integrands. J. Funct. Anal. 233 2 (2006) 561-582
26. Brezis H., and Lieb E.H. Minimum action solutions of some vector field equations. Commun. Math. Phys. 96 (1984) 97-113
27. Berestycki H., Gallouet T., and Kavian O. Equations de champs scalaires Euclidiens non linéaires dans le plan. Compt. Rend. Acad. Sci. 297 (1983) 307-310
28. 2 (in preparation)
29. Berestycki H., and Lions P.-L. Existence of stationary states of non-linear scalar field equations. In: Bardos C., and Bessis D. (Eds). Bifurcation Phenomena in Mathematical Physics and Related Topics. Proc. NATO ASI, Cargese, 1979 (1980), Reidel
30. Berestycki H., and Lions P.-L. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 4 (1983)
31. Berestycki H., and Lions P.-L. Existence d'états multiples dans les équations de champs scalaires non linéaires dans le cas de masse nulle. Compt. Rend. Acad. Sci. 297 1 (1983) 267-270
32. Strauss W.A. Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55 (1977) 149-162
33. Strauss W.A., and Vazquez L. Existence of localized solutions for certain model field theories. J. Math. Phys. 22 (1981) 1005-1009
34. Hardy G.H., Littlewood J.E., and Polya G. Inequalities (1934), Cambridge Univ. Press, London
35. Stuart C.A. Bifurcation for Dirichlet problems without eigenvalues. Proc. London Math. Soc. 45 (1982) 169-192
36. H. Hajaiej, Inégalité de symétrisation et application, Thèse 2465, EPFL
37. Hajaiej H., and Stuart C.A. Existence and non-existence of Schwarz symmetric ground states for eigenvalue problems. Ann. Mat. Pura. App. 3 (2004) 297-314