Global well-posedness for cubic NLS with nonlinear damping

Communications in Partial Differential Equations

Volumn 35, Issue 12, 2010, Pages 2310-2328

  Antonelli, Paolo ^a   Sparber, Christof ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Dissipation; Energy space; Nonlinear damping; Nonlinear schr dinger equation; Three body recombination

Indexed keywords

EID: 78049525386     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300903540943     Document Type: Article

References (38)

1. Adhikari, S.K. (2002). Mean-field description of collapsing and exploding Bose-Einstein condensates. Phys. Rev. A 66:13611-13619.
2. Antonelli, P., Marcati, P. (2009). On the infinite energy weak solutions to a system in quantum fluid dynamics. Comm. Math. Phys. 287:657-686.
3. Aranson, I.S., Kramer, L. (2002). The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74:99-143.
4. Bao, W., Jaksch, D. (2003). An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity. SIAM J. Numer. Anal. 41:1406-1426.
5. Bao, W., Jaksch, D., Markowich, P. (2004). Three dimensional simulation of jet formation in collapsing condensates. J. Phys. B: At. Mol. Opt. Phys. 37:329-343.
6. Biswas, E.A. (2001). Optical soliton perturbation with nonlinear damping and saturable amplifiers. Math. Comput. Simul. 56:521-537.
7. Carles, R. (2002). Remarks on nonlinear Schrödinger equations with harmonic potential. Ann. Henri Poincare 3:757-772.
8. Carles, R. (2003). Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation. Annales I.H.P. Analyse non Linéaire 20:501-542.
9. Carles, R. (2003). Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J. Math. Anal. 35:823-843.
10. Carles, R. (2005). Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential. Comm. Contemp. Math. 7:483-508.
11. Carles, R. (2005). Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete Contin. Dyn. Syst. 13:385-398.
12. Cazenave, T. (2003). Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, Vol. 10. New York: New York University, Courant Institute of Mathematical Sciences.
13. [13] Chen, T., Pavlovic, N. The quintic NLS as the mean field limit of a Boson gas with three-body interactions. Preprint arXiv:0812.2740v1.
14. 3. Ann. Math. 167:767-865.
15. Constantin, P., Saut, J.C. (1988). Local smoothing properties of dispersive equations. J. Amer. Math. Soc. 1:413-439.
16. Fibich, G. (2001). Self focusing in the damped nonlinear Schrödinger equation. SIAM J. Appl. Math. 61:1680-1705.
17. Fujiwara, D. (1979). A construction of the fundamental solution for the Schrödinger equation. J. Analyse Math. 35:4-96.
18. Gasser, I., Markowich, P.A. (1997). Quantum hydrodynamics, Wigner transforms and the classical limit. Asympt. Anal. 14:97-116.
19. 2. J. Diff. Eqs. 165:96-122.
20. Jüngel, U., Pinnau, R. (2000). Inviscid limits of the complex Ginzburg-Landau equation. Comm. Math. Phys. 214:201-226.
21. Kagan, Y., Muryshev, A.E., Shlyapnikov, G.V. (1998). Collapse and Bose-Einstein condensation in a trapped Bose gas with negative scattering length. Phys. Rev. Lett. 81:933-937.
22. Keel, M., Tao, T. (1998). Endpoint Strichartz estimates. Amer. J. Math. 120:955-980.
23. Killip, R., Visan, M. (2009), Energy-critical NLS with quadratic potentials. Comm. Part. Diff. Eqs. 34:1531-1565.
24. Kitaa, N., Shimomura, A. (2007). Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity. J. Diff. Eqs. 242:192-210.
25. N, N ≤ 3. NoDEA Nonlin. Diff. Equ. Appl. 2:357-369.
26. Oh, Y.-G. (1989). Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials. J. Diff. Eqs. 81:255-274.
27. Ohta, M., Todorova, G. (2009). Remarks on global existence and blow-up for damped nonlinear Schrödinger equations. Discrete Contin. Dyn. Syst. 23:1313-1325.
28. Passot, T., Sulem, C., Sulem, P.L. (2005). Linear versus nonlinear dissipation for critical NLS equation. Physica D 203:167-184.
29. Pereira, N.R., Steno, L. (1977). Nonlinear Schrödinger equation including growth and damping. Phys. Fluids 20:1733-1734.
30. Rakotoson, J.-M., Temam, R. (2001). An optimal compactness theorem and application to elliptic-parabolic systems. Appl. Math. Lett. 14:303-306.
31. Saito, H., Ueda, M. (2001). Intermittent implosion and pattern formation of trapped Bose-Einstein condensates with attractive interaction. Phys. Rev. Lett. 86:1406-14011.
32. Sakaguchi, H., Malomed, B.A. (2009). Two-dimensional dissipative gap solitons. Phys. Rev. E, Stat. Nonlin. Soft. Matter Phys. 80:026606-026614.
33. Shimomura, A. (2006). Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities. Comm. Part. Diff. Eqs. 31:1407-1423.
34. Sulem, C., Sulem, P.L. (1999). The Nonlinear Schrödinger Equation. Applied Math. Sciences, Vol. 139. New York: Springer.
35. Tao, T. (2006). Nonlinear Dispersive Equations: Local and Global Analysis. CBMS Regional Conference Series in Mathematics. Providence, RI: AMS.
36. Tao, T., Visan, M., Zhang, X. (2007). The nonlinear schrödinger equation with combined power-type nonlinearities. Comm. Part. Diff. Eqs. 32:1281-1343.
37. Tsutsumi, M. (1984). Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations. SIAM J. Math. Anal. 15:357-366.
38. Zhang, J. (2000). Stability of attractive Bose-Einstein condensates. J. Statist. Phys. 101:731-746.