Soliton dynamics for fractional Schrödinger equations

Applicable Analysis

Volumn 93, Issue 8, 2014, Pages 1702-1729

  Secchi, Simone ^a   Squassina, Marco ^b  

^a UNIVERSITY OF MILANO BICOCCA   (Italy)

^b UNIVERSITY OF VERONA   (Italy)

Author keywords

fractional Schr dinger equation; ground states; soliton dynamics

Indexed keywords

EID: 84902652640     PISSN: 00036811     EISSN: 1563504X     Source Type: Journal    
DOI: 10.1080/00036811.2013.844793     Document Type: Article

References (29)

1. Hilfer R. Application of fractional calculus in physics. Singapore: World Scientific; 2000.
2. Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000;339:1-77.
3. Metzler R, Klafter J. The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. 2004;37:161-208.
4. Laskin N. Fractional Schrödinger equations. Phys. Rev. E. 2002;66:056108.
5. Rozmej P, Bandrowski B. On fractional Schrödinger equation. Comput. Methods Sci. Technol. 2010;16:191-194.
6. Bronski JC, Jerrard RL. Soliton dynamics in a potential. Math. Res. Lett. 2000;7:329-342.
7. Keraani S. Semiclassical limit for nonlinear Schrödinger equation with potential II. Asymptot. Anal. 2006;47:171-186.
8. Benci V, Ghimenti MG, Micheletti AM. The nonlinear Schrödinger equation: soliton dynamics. J. Differ. Equ. 2010;249:3312-3341.
9. Benci V, Ghimenti MG, Micheletti AM. On the dynamics of solitons in the nonlinear Schrödinger equation. Arch. Ration. Mech. Anal. 2012;205:467-492.
10. Fröhlich J, Gustafson S, Jonsson BLG, Sigal IM. Dynamics of solitary waves external potentials. Commun. Math. Phys. 2004;250:613-642.
11. Weinstein MI. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 1985;16:472-491.
12. Weinstein MI. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 1986;39:51-67.
13. Tao T. Why are solitons stable? Bull. Am. Math. Soc. 2009;46:1-33.
14. Guo B, Huang D. Existence and stability of standing waves for nonlinear fractional Schrödinger equations. J. Math. Phys. 2012;53:083702.
15. Guo B, Huo Z. Global well-posedness for the fractional nonlinear Schrodinger equation. Commun. Partial Differ. Equ. 2011;36:247-255.
16. Frank RL, Lenzmann E. Uniqueness of non-linear ground states for fractional Laplacians in ℝ. Acta Mathematica 210, 2013;2:261-318.
17. Frank RL, Lenzmann E, Silvestre L. Uniqueness of radial solutions for the fractional Laplacian. preprint. Available from: http://arxiv.org/abs/1302.2652.
18. Cabre X, Sire Y. Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. in press.
19. Felmer P, Quaas A, Tan J. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. Royal Soc. Edinburgh A. 2012;142:1237-1262.
20. D'Avenia P, Squassina M. Soliton dynamics for the Schrödinger-Newton system. Math. Models Methods Appl. Sci. in press.
21. Lenzmann E. Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE. 2009;2:1-27.
22. Killip R, Visan M. Nonlinear Schrödinger equations at critical regularity, lectures delivered at Clay Institute. 2008. Available from: http://www.claymath.org/programs/summer_school/2008/visan.pdf.
23. Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. math. 2012;136:512-573.
24. N when s is close to 1. Communications in Mathematical Physics. in press.
25. Weidmann J. Linear operators in Hilbert spaces. Vol. 68, Graduate Texts in Mathematics. New York, NY, Berlin: Springer; 1980.
26. Ros-Oton X, Serra J. The Pohozaev identity for the fractional laplacian. preprint. Available from: http://arxiv.org/abs/1207.5986.
27. Teschl G. Mathematical methods in Quantum Mechanics, with applications to Schrödinger operators. Vol. 99, Graduate Studies in Mathematics. Providence, RI: American Mathematical Society; 2009.
28. Edmunds DE, Evans WD. Spectral theory and differential operators. New York, NY: Clarendon Press; 1987.
29. Cazenave T, Lions PL. Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 1982;85:549-561.