# American Institute of Mathematical Sciences

November  2016, 36(11): 5837-5879. doi: 10.3934/dcds.2016057

## Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes

 1 Dipartimento di Scienze Matematiche, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino 2 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 53, 20125 Milano 3 PriceWaterhouseCoopers Italia, Via Monte Rosa 91, 21049, Milano, Italy

Received  September 2015 Revised  May 2016 Published  August 2016

In this paper the study of asymptotic stability of standing waves for a model of Schrödinger equation with spatially concentrated nonlinearity in dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point $x=0$ obtained considering a contact (or $\delta$) interaction with strength $\alpha$, which consists of a singular perturbation of the Laplacian described by a selfadjoint operator $H_{\alpha}$, and letting the strength $\alpha$ depend on the wavefunction in a prescribed way: $i\dot u= H_\alpha u$, $\alpha=\alpha(u)$. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $\Phi_\omega$, and the linearization around them admits two imaginary eigenvalues (neutral modes, absent in the range $(0,\frac{1}{\sqrt 2})$ previously treated by the same authors) which in principle could correspond to non decaying states, so preventing asymptotic relaxation towards an equilibrium orbit. We prove that, in the range $(\frac{1}{\sqrt 2},\sigma^*)$ for a certain $\sigma^* \in (\frac{1}{\sqrt{2}}, \frac{\sqrt{3} +1}{2 \sqrt{2}}]$, the dynamics near the orbit of a standing wave asymptotically relaxes in the following sense: consider an initial datum $u(0)$, suitably near the standing wave $\Phi_{\omega_0},$ then the solution $u(t)$ can be asymptotically decomposed as $$u(t) = e^{i\omega_{\infty} t +i b_1 \log (1 +\epsilon k_{\infty} t) + i \gamma_\infty} \Phi_{\omega_{\infty}} +U_t*\psi_{\infty} +r_{\infty}, \quad \textrm{as} \;\; t \rightarrow +\infty,$$ where $\omega_{\infty}$, $k_{\infty}, \gamma_\infty > 0$, $b_1 \in \mathbb{R}$, and $\psi_{\infty}$ and $r_{\infty} \in L^2(\mathbb{R}^3)$ , $U(t)$ is the free Schrödinger group and $$\| r_{\infty} \|_{L^2} = O(t^{-1/4}) \quad \textrm{as} \;\; t \rightarrow +\infty\ .$$ We stress the fact that in the present case and contrarily to the main results in the field, the admitted nonlinearity is $L^2$-subcritical.
Citation: Riccardo Adami, Diego Noja, Cecilia Ortoleva. Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5837-5879. doi: 10.3934/dcds.2016057
##### References:
 [1] R. Adami, G. Dell'Antonio, R. Figari and A. Teta, The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity,, Ann. I. H. Poincaré, 20 (2003), 477.  doi: 10.1016/S0294-1449(02)00022-7.  Google Scholar [2] R. Adami, G. Dell'Antonio, R. Figari and A. Teta, Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity,, Ann. I. H. Poincaré, 21 (2004), 121.  doi: 10.1016/j.anihpc.2003.01.002.  Google Scholar [3] R. Adami, D. Noja, Stability and Symmetry-Breaking Bifurcation for the Ground States of a NLS with a $\delta'$ Interaction,, Commun. Math. Phys., 318 (2013), 247.  doi: 10.1007/s00220-012-1597-6.  Google Scholar [4] R. Adami, D. Noja and C. Ortoleva, Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4772490.  Google Scholar [5] R. Adami, D. Noja, N. Visciglia, Constrained energy minimization and ground states for NLS with point defects,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1155.  doi: 10.3934/dcdsb.2013.18.1155.  Google Scholar [6] S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, Solvable Models in Quantum Mechanics,, American Mathematical Society, (2005).   Google Scholar [7] D. Bambusi, Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry,, Comm. Math. Phys. 320 (2013), 320 (2013), 499.  doi: 10.1007/s00220-013-1684-3.  Google Scholar [8] V. S. Buslaev, A. I. Komech, A.E. Kopylova and D. Stuart, On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator,, Comm. PDE, 33 (2008), 669.  doi: 10.1080/03605300801970937.  Google Scholar [9] V. S. Buslaev and G. Perelman, Scattering for the nonlinear Schrödinger equation: states close to a soliton,, St.Petersbourg Math J., 4 (1993), 1111.   Google Scholar [10] V. S. Buslaev and G. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations,, Amer.Math.Soc.Transl., 164 (1995), 75.  doi: 10.1090/trans2/164/04.  Google Scholar [11] V. S. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equation,, Ann. I. H. Poincaré, 20 (2003), 419.  doi: 10.1016/S0294-1449(02)00018-5.  Google Scholar [12] C.Cacciapuoti, D.Finco, D.Noja and A.Teta, The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit,, Lett. Math. Phys., 104 (2014), 1557.  doi: 10.1007/s11005-014-0725-y.  Google Scholar [13] C. Cacciapuoti, D.Finco, D.Noja and A.Teta, The point-like limit for a NLS Equation with concentrated nonlinearity in dimension three,, , (2015).   Google Scholar [14] S. Cuccagna, Stabilization of solution to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 1110.  doi: 10.1002/cpa.1018.  Google Scholar [15] S. Cuccagna and T. Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations,, Comm. Math. Phys., 284 (2008), 51.  doi: 10.1007/s00220-008-0605-3.  Google Scholar [16] S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states,, Comm. Math. Phys., 305 (2011), 279.  doi: 10.1007/s00220-011-1265-2.  Google Scholar [17] Z. Gang and I. M. Sigal, Relaxation of solitons in nonlinear Schrödinger equations with potentials,, Adv. Math., 216 (2007), 443.  doi: 10.1016/j.aim.2007.04.018.  Google Scholar [18] Z. Gang and M. Weinstein, Dynamics of nonlinear Schrödinger-Gross-Pitaevskii equations; mass transfer in systems with solitons and degenerates neutral modes,, Analysis & PDE, 1 (2008), 267.  doi: 10.2140/apde.2008.1.267.  Google Scholar [19] Z. Gang and M. Weinstein, Equipartition of energy in Nonlinear Schrödinger-Gross-Pitaevskii equations,, AMRX, 2 (2011), 123.   Google Scholar [20] I. S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products,, Vth edition, (1994).   Google Scholar [21] E. Kirr and Ö. Mizrak, Asymptotic stability of ground states in 3d nonlinear Schrödinger equation including subcritical cases,, J. Funct. Anal., 257 (2009), 3691.  doi: 10.1016/j.jfa.2009.08.010.  Google Scholar [22] A. I. Komech, E. A. Kopylova and D. Stuart, On asymptotic stability of solitary waves for Schrödinger equation coupled to nonlinear oscillator, II,, Comm. Pure Appl. Anal., 202 (2012), 1063.   Google Scholar [23] D. Noja and A. Posilicano, Wave equations with concentrated nonlinearities,, J. Phys. A: Math. Gen., 38 (2005), 5011.  doi: 10.1088/0305-4470/38/22/022.  Google Scholar [24] I.M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions,, Commun. Math. Phys., 2 (1993), 297.   Google Scholar [25] A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations,, Comm. Math. Phys., 133 (1990), 119.  doi: 10.1007/BF02096557.  Google Scholar [26] A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations II, the case of anisotropic potentials and data,, J. Diff. Eq., 98 (1992), 376.  doi: 10.1016/0022-0396(92)90098-8.  Google Scholar [27] A. Soffer and M. Weinstein, Resonances, radiation damping, and instability of Hamiltonian nonlinear waves,, Invent. Math., 136 (1999), 9.  doi: 10.1007/s002220050303.  Google Scholar [28] A. Soffer and M. Weinstein, Selection of the ground state for nonlinear Schrödinger equations,, Rev. Math. Phys., 16 (2004), 977.  doi: 10.1142/S0129055X04002175.  Google Scholar [29] A. Soffer and M. Weinstein, Theory of nonlinear dispersive waves and selection of the ground state,, Phys. Rev. Lett., 95 (2005).  doi: 10.1103/PhysRevLett.95.213905.  Google Scholar [30] T. P. Tsai and H. T. Yau, Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions,, Comm. Pure. Appl. Math, 55 (2002), 153.  doi: 10.1002/cpa.3012.  Google Scholar [31] T. P. Tsai and H. T. Yau, Relaxation of excited states in nonlinear Schrödinger equations,, Int. Math. Res. Not., 31 (2002), 1629.  doi: 10.1155/S1073792802201063.  Google Scholar [32] M. Weinstein, Localized states and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii equation,, Frontiers in Applied Dynamical Systems, 3 (2015), 41.  doi: 10.1007/978-3-319-19935-1_2.  Google Scholar

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##### References:
 [1] R. Adami, G. Dell'Antonio, R. Figari and A. Teta, The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity,, Ann. I. H. Poincaré, 20 (2003), 477.  doi: 10.1016/S0294-1449(02)00022-7.  Google Scholar [2] R. Adami, G. Dell'Antonio, R. Figari and A. Teta, Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity,, Ann. I. H. Poincaré, 21 (2004), 121.  doi: 10.1016/j.anihpc.2003.01.002.  Google Scholar [3] R. Adami, D. Noja, Stability and Symmetry-Breaking Bifurcation for the Ground States of a NLS with a $\delta'$ Interaction,, Commun. Math. Phys., 318 (2013), 247.  doi: 10.1007/s00220-012-1597-6.  Google Scholar [4] R. Adami, D. Noja and C. Ortoleva, Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4772490.  Google Scholar [5] R. Adami, D. Noja, N. Visciglia, Constrained energy minimization and ground states for NLS with point defects,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1155.  doi: 10.3934/dcdsb.2013.18.1155.  Google Scholar [6] S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, Solvable Models in Quantum Mechanics,, American Mathematical Society, (2005).   Google Scholar [7] D. Bambusi, Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry,, Comm. Math. Phys. 320 (2013), 320 (2013), 499.  doi: 10.1007/s00220-013-1684-3.  Google Scholar [8] V. S. Buslaev, A. I. Komech, A.E. Kopylova and D. Stuart, On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator,, Comm. PDE, 33 (2008), 669.  doi: 10.1080/03605300801970937.  Google Scholar [9] V. S. Buslaev and G. Perelman, Scattering for the nonlinear Schrödinger equation: states close to a soliton,, St.Petersbourg Math J., 4 (1993), 1111.   Google Scholar [10] V. S. Buslaev and G. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations,, Amer.Math.Soc.Transl., 164 (1995), 75.  doi: 10.1090/trans2/164/04.  Google Scholar [11] V. S. Buslaev and C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equation,, Ann. I. H. Poincaré, 20 (2003), 419.  doi: 10.1016/S0294-1449(02)00018-5.  Google Scholar [12] C.Cacciapuoti, D.Finco, D.Noja and A.Teta, The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit,, Lett. Math. Phys., 104 (2014), 1557.  doi: 10.1007/s11005-014-0725-y.  Google Scholar [13] C. Cacciapuoti, D.Finco, D.Noja and A.Teta, The point-like limit for a NLS Equation with concentrated nonlinearity in dimension three,, , (2015).   Google Scholar [14] S. Cuccagna, Stabilization of solution to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 1110.  doi: 10.1002/cpa.1018.  Google Scholar [15] S. Cuccagna and T. Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations,, Comm. Math. Phys., 284 (2008), 51.  doi: 10.1007/s00220-008-0605-3.  Google Scholar [16] S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states,, Comm. Math. Phys., 305 (2011), 279.  doi: 10.1007/s00220-011-1265-2.  Google Scholar [17] Z. Gang and I. M. Sigal, Relaxation of solitons in nonlinear Schrödinger equations with potentials,, Adv. Math., 216 (2007), 443.  doi: 10.1016/j.aim.2007.04.018.  Google Scholar [18] Z. Gang and M. Weinstein, Dynamics of nonlinear Schrödinger-Gross-Pitaevskii equations; mass transfer in systems with solitons and degenerates neutral modes,, Analysis & PDE, 1 (2008), 267.  doi: 10.2140/apde.2008.1.267.  Google Scholar [19] Z. Gang and M. Weinstein, Equipartition of energy in Nonlinear Schrödinger-Gross-Pitaevskii equations,, AMRX, 2 (2011), 123.   Google Scholar [20] I. S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products,, Vth edition, (1994).   Google Scholar [21] E. Kirr and Ö. Mizrak, Asymptotic stability of ground states in 3d nonlinear Schrödinger equation including subcritical cases,, J. Funct. Anal., 257 (2009), 3691.  doi: 10.1016/j.jfa.2009.08.010.  Google Scholar [22] A. I. Komech, E. A. Kopylova and D. Stuart, On asymptotic stability of solitary waves for Schrödinger equation coupled to nonlinear oscillator, II,, Comm. Pure Appl. Anal., 202 (2012), 1063.   Google Scholar [23] D. Noja and A. Posilicano, Wave equations with concentrated nonlinearities,, J. Phys. A: Math. Gen., 38 (2005), 5011.  doi: 10.1088/0305-4470/38/22/022.  Google Scholar [24] I.M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions,, Commun. Math. Phys., 2 (1993), 297.   Google Scholar [25] A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations,, Comm. Math. Phys., 133 (1990), 119.  doi: 10.1007/BF02096557.  Google Scholar [26] A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations II, the case of anisotropic potentials and data,, J. Diff. Eq., 98 (1992), 376.  doi: 10.1016/0022-0396(92)90098-8.  Google Scholar [27] A. Soffer and M. Weinstein, Resonances, radiation damping, and instability of Hamiltonian nonlinear waves,, Invent. Math., 136 (1999), 9.  doi: 10.1007/s002220050303.  Google Scholar [28] A. Soffer and M. Weinstein, Selection of the ground state for nonlinear Schrödinger equations,, Rev. Math. Phys., 16 (2004), 977.  doi: 10.1142/S0129055X04002175.  Google Scholar [29] A. Soffer and M. Weinstein, Theory of nonlinear dispersive waves and selection of the ground state,, Phys. Rev. Lett., 95 (2005).  doi: 10.1103/PhysRevLett.95.213905.  Google Scholar [30] T. P. Tsai and H. T. Yau, Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions,, Comm. Pure. Appl. Math, 55 (2002), 153.  doi: 10.1002/cpa.3012.  Google Scholar [31] T. P. Tsai and H. T. Yau, Relaxation of excited states in nonlinear Schrödinger equations,, Int. Math. Res. Not., 31 (2002), 1629.  doi: 10.1155/S1073792802201063.  Google Scholar [32] M. Weinstein, Localized states and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii equation,, Frontiers in Applied Dynamical Systems, 3 (2015), 41.  doi: 10.1007/978-3-319-19935-1_2.  Google Scholar
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