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

#### Abstract

**Abstract.**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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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.

Buslaev, V. S. and Perelman, G. S., Scattering for the nonlinear Schr¨ odinger equation: states that are close to a soliton, Algebra i Analiz, 4(6):63–102, 1992.

Buslaev, V. S. and Perelman, G. S., On the stability of solitary waves for nonlinear Schr¨ odinger equations. In Nonlinear evolution equations, volume 164 of Amer. Math. Soc. Transl. Ser. 2, pages 75–98. Amer. Math. Soc., Providence, RI, 1995.

Buslaev, Vladimir S. and Sulem, Catherine, On asymptotic stability of solitary waves for nonlinear Schr¨ odinger equations, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 20(3):419–475, 2003.

Cazenave Thierry. Semilinear Schr¨ odinger equations, volume 10 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2003.

Cuccagna Scipio, Stabilization of solutions to nonlinear Schr¨ odinger equations, Comm. Pure Appl. Math., 54(9):1110–1145, 2001.

Goldberg, M. and Schlag, W., Dispersive estimates for Schr¨ odinger operators in dimensions one and three, Comm. Math. Phys., 251(1):157–178, 2004.

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.

Grillakis, Manoussos and Shatah, Jalal and Strauss, Walter, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94(2):308–348, 1990.

Gustafson, Stephen and Nakanishi, Kenji and Tsai, Tai-Peng, Asymptotic stability and completeness in the energy space for nonlinear Schr¨ odinger equations with small solitary waves, Int. Math. Res. Not., (66):3559–3584, 2004.

Kirr,E and Mızrak, ¨ O., On the stability of ground states in 4D 5D nonlinear Schr¨ odinger equation including subcritical cases, submitted to IMRN. Kirr, E. and Mızrak, ¨ O., Asymptotic stability of ground states in 3D nonlinear Schr¨ odinger equation including subcritical cases, J. Funct. Anal., 257(12):3691–3747, 200

Kirr, E. and Zarnescu, A., On the asymptotic stability of bound states in 2D cubic Schr¨ odinger equation, Comm. Math. Phys., 272(2):443–468, 2007.

Kirr,E and Zarnescu, A., Asymptotic stability of ground states in 2d nonlinear Schr¨ odinger equation including subcritical cases., J. Diff. Eq., 247(3):710–735, 2009. Murata, Minoru, Asymptotic expansions in time for solutions of Schr¨ odinger-type equations, J. Funct. Anal., 49(1):10–56, 1982. 41

Nirenberg, Louis. Topics in nonlinear functional analysis, volume 6 of the Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 2001.

Pillet, Claude-Alain and Wayne, C. Eugene, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differential Equations, 141(2):310– 326, 1997.

Rose, Harvey A. and Weinstein, Michael I., On the bound states of the nonlinear Schr¨ odinger equation with a linear potential, Phys. D, 30(1-2):207–218, 1997.

Schlag, W., Dispersive estimates for Schr¨ odinger operators in dimension two, Comm. Math. Phys., 257(1), 87–117, 2005.

Shatah, Jalal and Strauss, Walter, Instability of nonlinear bound states, Comm. Math. Phys., 100(2):173–190, 1985.

Soffer, A. and Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133(1):119–146, 1990.

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.

Strauss, Walter A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55(2):149–162, 1997.

Weder, Ricardo, Center manifold for nonintegrable nonlinear Schr¨ odinger equations on the line, Comm. Math. Phys., 215(2):343–356, 2000.

Weinstein, Michael I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39(1):51–67, 1986. 42