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Asymptotic stability of ground states of quadratic nonlinear Schrödinger equation with potential in 4D

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Abstract (2. Language): 
In this paper a class of quadratic nonlinear Schrödinger equation in four space dimensions with an attractive potential is considered. We investigate asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. We show that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the non-linear bound states are asymptotically stable.
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REFERENCES

References: 

[1] Berestycki, H. and Lions, P.-L., Nonlinear scalar field equations. I. Existence of a
ground state, Arch. Rational Mech. Anal., 82(4):313-345, 1983. [2] Buslaev, V. S. and Perelman, G. S., Scattering for the nonlinear Schrödinger equation:
states that are close to a soliton, Algebra i Analiz, 4(6):63-102, 1992. [3] Buslaev, V. S. and Perelman, G. S., On the stability of solitary waves for nonlinear
Schrödinger
equations
. In Nonlinear evolution equations, volume 164 of Amer. Math.
Soc. Transl. Ser. 2, pages 75-98. Amer. Math. Soc., Providence, RI, 1995. [4] Buslaev, Vladimir S. and Sulem, Catherine, On asymptotic stability of solitary waves
for nonlinear Schrodinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire,
20(3):419-475, 2003.
[5] Cazenave Thierry. Semilinear Schroödinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2003.
[6] Cuccagna Scipio, Stabilization of solutions to nonlinear Schrödinger equations, Comm.
Pure Appl. Math., 54(9):1110-1145, 2001.
[7] Goldberg, M. and Schlag, W., Dispersive estimates for Schrödinger operators in di¬mensions one and three, Comm. Math. Phys., 251(1):157-178, 2004.
[8] Grillakis, Manoussos and Shatah, Jalal and Strauss, Walter, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74(1):160-197, 1987.
[9] Grillakis, Manoussos and Shatah, Jalal and Strauss, Walter, Stability theory ofsolitary waves in the presence of symmetry. II, J. Funct. Anal., 94(2):308-348, 1990. [10] Gustafson, Stephen and Nakanishi, Kenji and Tsai, Tai-Peng, Asymptotic stability and completeness in the energy space for nonlinear Schröodinger equations with small
solitary waves, Int. Math. Res. Not., (66):3559-3584, 2004.
[11] Kirr,E and Mızrak, O., On the stability of ground states in 4D 5D nonlinear Schröodinger equation including subcritical cases, submitted to IMRN.
[12] Kirr, E. and Mızrak, Oö., Asymptotic stability of ground states in 3D nonlinear Schröodinger equation including subcritical cases, J. Funct. Anal., 257(12):3691-3747,
2009.
[13] Kirr, E. and Zarnescu, A., On the asymptotic stability of bound states in 2D cubic
Schröodinger
equation
, Comm. Math. Phys., 272(2):443-468, 2007.
[14] Kirr,E and Zarnescu, A., Asymptotic stability of ground states in 2d nonlinear
Schrodinger equation including subcritical cases., J. Diff. Eq., 247(3):710-735, 2009. [15] Murata, Minoru, Asymptotic expansions in time for solutions of Schroödinger-type
equations, J. Funct. Anal., 49(1):10-56, 1982.
4 1
O. MıZRAK
[16] Nirenberg, Louis. Topics in nonlinear functional analysis, volume 6 of the Courant Lec¬ture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2001.
[17] Pillet, Claude-Alain and Wayne, C. Eugene, Invariant manifolds for a class of disper¬sive, Hamiltonian, partial differential equations, J. Differential Equations, 141(2):310-
326, 1997.
[18] Rose, Harvey A. and Weinstein, Michael I., On the bound states of the nonlinear
Schröodinger
equatio
n with a linear potential, Phys. D, 30(1-2):207-218, 1997. [19] Schlag, W., Dispersive estimates for Schröodinger operators in dimension two, Comm.
Math. Phys., 257(1), 87-117, 2005.
[20] Shatah, Jalal and Strauss, Walter, Instability of nonlinear bound states, Comm. Math.
Phys., 100(2):173-190, 1985. [21] Soffer, A. and Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable
equations, Comm. Math. Phys., 133(1):119-146, 1990.
[22] Soffer, A. and Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data, J. Differential Equations,
98(2):376-390, 1992.
[23] Strauss, Walter A., Existence of solitary waves in higher dimensions, Comm. Math.
Phys., 55(2):149-162, 1997.
[24] Weder, Ricardo, Center manifold for nonintegrable nonlinear Schröodinger equations
on the line, Comm. Math. Phys., 215(2):343-356, 2000.
[25] Weinstein, Michael I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39(1):51-67, 1986.

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