Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases.(English)Zbl 1187.35238

The authors study the initial value problem in statistical physics: $$i\partial_t u(t,x)= [-\Delta+x+ V(x)] u+ g(u)$$, $$t\in\mathbb{R}$$, $$x\in\mathbb{R}^3$$, $$u(0,x)= u_0(x)$$. Here $$g:\mathbb{R}\to\mathbb{R}$$ is a real-valued odd $$C^2$$ function satisfying $$g(0)= g'(0)= 0$$, $$|g''(s)|\leq C(|s|^{\alpha_1}+ |x|^{\alpha_2})$$, $$0<\alpha_1\leq\alpha_2< 3$$, which is extended to a complex function via $$g(e^{i\theta} s)= e^{i\theta}g(s)$$.
The center manifold is formed by the collection of solutions:
$u_E(t, x)= e^{-iEt}\psi_E(x),\quad (-\Delta+ V)\psi_E+ g(\psi_E)= E\psi_E.$
Let $$\psi_E= a\psi_0+ h$$, $$a=\langle\psi_0, \psi_E\rangle$$, $$h= P_c\psi_E$$, $$P_c$$: projection onto the continuous part of the spectrum of $$H=-\Delta+ V$$. $$H_a= \{-i\partial\psi_E/\partial a_2, i\partial\psi_E/\partial a_1\}^\perp$$, $$a= a_1+ ia_2$$.
Theorem. Let $$p_1= 3+\alpha_1$$, $$p_2= 3+\alpha_2$$. One finds an $$\varepsilon_0> 0$$ such that, for $$u_0(x)$$ satisfying
$\max\{\| u_0\|_{L^{p_2'}},\,\| u_0\|_{H^1}\}\leq\varepsilon_0,$
there exists a $$C^1$$ function $$a(t):\mathbb{R}\to C$$ such that, for all $$t\in\mathbb{R}$$, $u(t,x)= \psi_E(x,t)+ \eta(t,x)\equiv a(t)\psi_0(x)+ h(a(t))+ \eta(t,x)$ holds. $$\psi_E(x,t)$$, $$t\in\mathbb{R}$$, are on the center manifold, and $$\eta(t,x)\in H_{a(t)}$$.
There exist $$\Psi_{E\pm}(x)$$ and $$\theta(t)\in C^1(\mathbb{R})$$ such that $$\lim_{t\to\pm\infty}\theta(t)= 0$$ and
$\lim_{t\to\pm\infty} \|\psi_E(x, t)- e^{-it(E\pm-\theta(t))}\psi_{E\pm}(x)\|_{H^2\cap L_\sigma^2}= 0$
hold. Radiative part $$\eta(t,x)$$ satisfies the decay estimates.
Finally, the authors try to extend the estimates of $$\| e^{-iHt}P_c u_0\|_{L^p}$$.

MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
Full Text:

References:

 [1] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. existence of a ground state, Arch. ration. mech. anal., 82, 4, 313-345, (1983) · Zbl 0533.35029 [2] Buslaev, V.S.; Perel’man, G.S., Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, Algebra i analiz, 4, 6, 63-102, (1992) · Zbl 0853.35112 [3] Buslaev, V.S.; Perel’man, G.S., On the stability of solitary waves for nonlinear Schrödinger equations, (), 75-98 · Zbl 0841.35108 [4] Buslaev, Vladimir S.; Sulem, Catherine, On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 20, 3, 419-475, (2003) · Zbl 1028.35139 [5] Cazenave, Thierry, Semilinear Schrödinger equations, Courant lect. notes math., vol. 10, (2003), New York University Courant Institute of Mathematical Sciences New York · Zbl 1055.35003 [6] Cuccagna, Scipio, Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., 54, 9, 1110-1145, (2001) · Zbl 1031.35129 [7] Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S., Theory of bose – einstein condensation in trapped gases, Rev. modern phys., 71, 3, 463-512, (1999) [8] Deift, P.; Simon, B.; Hunziker, W.; Vock, E., Pointwise bounds on eigenfunctions wave packets in n-body quantum systems IV, Comm. math. phys., 64, 1-34, (1978) · Zbl 0419.35079 [9] Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. math., 167, 3, 515-614, (2007) · Zbl 1123.35066 [10] Goldberg, M.; Schlag, W., Dispersive estimates for Schrödinger operators in dimensions one three, Comm. math. phys., 251, 1, 157-178, (2004) · Zbl 1086.81077 [11] Grillakis, Manoussos; Shatah, Jalal; Strauss, Walter, Stability theory of solitary waves in the presence of symmetry. I, J. funct. anal., 74, 1, 160-197, (1987) · Zbl 0656.35122 [12] Grillakis, Manoussos; Shatah, Jalal; Strauss, Walter, Stability theory of solitary waves in the presence of symmetry. II, J. funct. anal., 94, 2, 308-348, (1990) · Zbl 0711.58013 [13] Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng, Asymptotic stability completeness in the energy space for nonlinear Schrödinger equations with small solitary waves, Int. math. res. not., 66, 3559-3584, (2004) · Zbl 1072.35167 [14] Journé, J.-L.; Soffer, A.; Sogge, C.D., $$L^p \rightarrow L^{p^\prime}$$ estimates for time-dependent Schrödinger operators, Bull. amer. math. soc. (N.S.), 23, 2, 519-524, (1990) · Zbl 0751.35011 [15] Keel, Markus; Tao, Terence, Endpoint Strichartz estimates, Amer. J. math., 120, 5, 955-980, (1998) · Zbl 0922.35028 [16] Kirr, E.; Zarnescu, A., On the asymptotic stability of bound states in 2D cubic Schrödinger equation, Comm. math. phys., 272, 2, 443-468, (2007) · Zbl 1194.35416 [17] Kirr, E.; Zarnescu, A., Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases, J. differential equations, 247, 710-735, (2009) · Zbl 1171.35112 [18] Lieb, E.; Seiringer, R.; Yngvason, J., The ground state energy and density of interacting bosons in a trap, (), 101-110 · Zbl 1043.82515 [19] Murata, Minoru, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. funct. anal., 49, 1, 10-56, (1982) · Zbl 0499.35019 [20] Nirenberg, Louis, Topics in nonlinear functional analysis, Courant lect. notes math., vol. 6, (2001), New York University Courant Institute of Mathematical Sciences New York, Chapter 6 by E. Zehnder, Notes by R.A. Artino, revised reprint of the 1974 original · Zbl 0992.47023 [21] Pillet, Claude-Alain; Wayne, C. Eugene, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. differential equations, 141, 2, 310-326, (1997) · Zbl 0890.35016 [22] Rose, Harvey A.; Weinstein, Michael I., On the bound states of the nonlinear Schrödinger equation with a linear potential, Phys. D, 30, 1-2, 207-218, (1988) · Zbl 0694.35202 [23] Shatah, Jalal; Strauss, Walter, Instability of nonlinear bound states, Comm. math. phys., 100, 2, 173-190, (1985) · Zbl 0603.35007 [24] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering for nonintegrable equations, Comm. math. phys., 133, 1, 119-146, (1990) · Zbl 0721.35082 [25] Soffer, A.; 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) · Zbl 0795.35073 [26] Soffer, A.; Weinstein, M.I., Selection of the ground state for nonlinear Schrödinger equations, Rev. math. phys., 16, 8, 977-1071, (2004) · Zbl 1111.81313 [27] Strauss, Walter A., Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 2, 149-162, (1977) · Zbl 0356.35028 [28] Weder, Ricardo, Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Comm. math. phys., 215, 2, 343-356, (2000) · Zbl 1003.37045 [29] Weinstein, Michael I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. pure appl. math., 39, 1, 51-67, (1986) · Zbl 0594.35005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.