## A centre-stable manifold for the focussing cubic NLS in $${\mathbb{R}}^{1+3\star}$$.(English)Zbl 1148.35082

Summary: Consider the focussing cubic nonlinear Schrödinger equation in $${\mathbb{R}}^3$$: $i\psi_t+\Delta\psi = -|\psi|^2 \psi. \tag{0.1}$ It admits special solutions of the form $$e^{it \alpha} \varphi$$, where $$\varphi \in {\mathcal{S}}({\mathbb{R}}^3)$$ is a positive $$(\varphi > 0)$$ solution of $-\Delta \varphi + \alpha\varphi = \varphi^3. \tag{0.2}$ The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional manifold that consists of functions of the form $$e^{i(v \cdot + \Gamma)} \varphi(\cdot - y, \alpha)$$. We prove that any solution starting sufficiently close to a standing wave in the $$\Sigma = W^{1, 2}({\mathbb{R}}^3) \cap |x|^{-1}L^2({\mathbb{R}}^3)$$ norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that $${\mathcal{N}}$$ is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of P. W. Bates and C. K. R. T. Jones [in Dynamics reported, Vol. 2, 1–38, Wiley, Chichester (1989; Zbl 0674.58024)]. The proof is based on the modulation method introduced by Soffer and Weinstein for the $$L^2$$-subcritical case and adapted by Schlag to the $$L^{2}-supercritical$$ case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in $${\mathbb{R}}^3$$ for the nonselfadjoint Schrödinger operator obtained by linearizing $$(0.1)$$ around a standing wave solution. All results in this paper depend on the standard spectral assumption that the Hamiltonian $\mathcal H = \left (\begin{matrix}\Delta + 2\varphi(\cdot, \alpha)^2 - \alpha & \varphi(\cdot, \alpha)^2 \\ -\varphi(\cdot, \alpha)^2 & -\Delta - 2 \varphi(\cdot, \alpha)^2 + \alpha \end{matrix}\right) \tag{0.3}$ has no embedded eigenvalues in the interior of its essential spectrum $$(-\infty, -\alpha) \cup (\alpha, \infty)$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems

Zbl 0674.58024
Full Text:

### References:

 [1] Agmon, S., Spectral properties of Schrdinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2, 2, 151-218 (1975) · Zbl 0315.47007 [2] Bates, P. W.; Jones, C. K.R. T., Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2, 1-38 (1989) [3] Berestycki, H.; Cazenave, T., Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math., 293, 9, 489-492 (1981) · Zbl 0492.35010 [4] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rat. Mech. Anal., 82, 4, 313-345 (1983) · Zbl 0533.35029 [5] Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Berlin-Heidelberg-New York: Springer-Verlag, 1976 · Zbl 0344.46071 [6] Buslaev, V.S., Perelman, G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton (Russian). Algebra i Analiz 4(6), 63-102 (1992); translation in St. Petersburg Math. J. 4(6), 1111-1142 (1993) [7] Buslaev, V.S., Perelman, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, 164, Providence, RI: Amer. Math. Soc. 1995, pp. 75-98 · Zbl 0841.35108 [8] Buslaev, V.S., Perelman, G.S.: Nonlinear scattering: states that are close to a soliton (Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov, (POMI) 200 (1992), Kraev. Zadachi Mat. Fiz. Smezh. Voprosy Teor. Funktsii, 24, 38-50, 70, 187; translation in J. Math. Sci. 77(3), 3161-3169 (1995) [9] Cazenave, T.: Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, Providence, RI: Amer. Math. Soc., 2003 · Zbl 1055.35003 [10] Cazenave, T.; Lions, P. L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85, 549-561 (1982) · Zbl 0513.35007 [11] Coffman, C. V., Uniqueness of positive solutions of Δu−u + u^3 = 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal., 46, 81-95 (1972) · Zbl 0249.35029 [12] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54, 9, 1110-1145 (2001) · Zbl 1031.35129 [13] Cuccagna, S.; Pelinovsky, D.; Vougalter, V., Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math., 58, 1, 1-29 (2005) · Zbl 1064.35181 [14] Demanet, L.; Schlag, W., Numerical verification of a gap condition for a linearized NLS equation, Nonlinearity, 19, 829-852 (2006) · Zbl 1106.35044 [15] Erdogan, B., Schlag, W.: Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II. To appear in Journal d’Analyse Mathematique, available at http://arxiv.org/list/math/0504585, 2005 · Zbl 1146.35324 [16] Gesztesy, F.; Jones, C. K.R. T.; Latushkin, Y.; Stanislavova, M., A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations, Indiana Univ. Math. J., 49, 1, 221-243 (2000) · Zbl 0969.35123 [17] Glassey, R. T., On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys., 18, 1794-1797 (1977) · Zbl 0372.35009 [18] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74, 1, 160-197 (1987) · Zbl 0656.35122 [19] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94, 1, 308-348 (1990) · Zbl 0711.58013 [20] Hundertmark, D.; Lee, Y.-R., Exponential decay of eigenfunctions and generalized eigenfunctions of a non self-adjoint matrix Schrödinger operator related to NLS, Bull. London Math. Soc., 39, 5, 709-720 (2007) · Zbl 1155.35065 [21] Keel, M.; Tao, T., Endpoint Strichartz estimates, Amer. Math. J., 120, 955-980 (1998) · Zbl 0922.35028 [22] Kenig, C.; Merle, F., Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Inv. Math., 166, 3, 645-675 (2006) · Zbl 1115.35125 [23] Kelley, P. L., Self-focusing of optical beams, Phys. Rev. Lett., 15, 1005-1008 (1965) [24] Krieger, J.; Schlag, W., Stable manifolds for all monic supercritical NLS in one dimension, J. AMS, 19, 4, 815-920 (2006) · Zbl 1281.35077 [25] Krieger, J., Schlag, W.: Non-generic blow-up solutions for the critical focusing NLS in 1-d. http://arxiv.org/list/math/0508576, 2005 · Zbl 1163.35035 [26] Krieger, J.; Schlag, W., On the focusing critical semi-linear wave equation, Amer. J. Math., 129, 3, 843-913 (2007) · Zbl 1219.35144 [27] Kwong, M. K., Uniqueness of positive solutions of Δu−u + u^p = 0 in $${\mathbb{R}}^n$$, Arch. Rat. Mech. Anal., 65, 243-266 (1989) · Zbl 0676.35032 [28] McLeod, K.; Serrin, J., Nonlinear Schrödinger equation. Uniqueness of positive solutions of Δu + f (u) = 0 in $${\mathbb{R}}^n$$, Arch. Rat. Mech. Anal., 99, 115-145 (1987) · Zbl 0667.35023 [29] Merle, F., Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69, 2, 427-454 (1993) · Zbl 0808.35141 [30] Merle, F.; Raphael, P., On a sharp lower bound on the blow-up rate for the L^2 critical nonlinear Schrödinger equation, J. Amer. Math. Soc., 19, 1, 37-90 (2006) · Zbl 1075.35077 [31] Perelman, G., On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2, 4, 605-673 (2001) · Zbl 1007.35087 [32] Pillet, C. A.; Wayne, C. E., Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Diff. Eq., 141, 2, 310-326 (1997) · Zbl 0890.35016 [33] Rodnianski, I.; Schlag, W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155, 3, 451-513 (2004) · Zbl 1063.35035 [34] Rodnianski, I.; Schlag, W.; Soffer, A., Dispersive analysis of charge transfer models, Comm. Pure and Appl. Math., 58, 2, 149-216 (2004) · Zbl 1130.81053 [35] Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS. http://arxiv.org/list/math/0309114, 2003 [36] Schlag, W.: Stable Manifolds for an orbitally unstable NLS. http://arxiv.org/list/math/0405435 , 2004 [37] Schlag, W., Spectral theory and nonlinear partial differential equations: a survey, Discrete Contin. Dyn. Syst., 15, 3, 703-723 (2006) · Zbl 1121.35121 [38] Soffer, A.; Weinstein, M. I., Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133, 119-146 (1990) · Zbl 0721.35082 [39] Soffer, A.; Weinstein, M. I., Multichannel nonlinear scattering, II, The case of anisotropic potentials and data. J. Diff. Eq., 98, 376-390 (1992) · Zbl 0795.35073 [40] Stein, E., Harmonic Analysis (1994), Princeton, NJ: Princeton University Press, Princeton, NJ [41] Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse. Applied Mathematical Sciences, 139, New York: Springer-Verlag, 1999 · Zbl 0928.35157 [42] Taylor, M.E.: Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials. Mathematical Surveys and Monographs, 81, Providence, RI: Amer. Math. Soc., 2000 · Zbl 0963.35211 [43] Weinstein, M. I., Lyapunov stability of ground states of nonlinear dispersive equations, Comm. Pure Appl. Math., 39, 1, 51-67 (1986) · Zbl 0594.35005 [44] Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 3, 472-491 (1985) · Zbl 0583.35028 [45] Yajima, K., Dispersive estimate for Schrödinger equations with threshold resonance and eigenvalue, Commun. Math. Phys., 259, 2, 475-509 (2005) · Zbl 1079.81021
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.