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.

[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.

[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.

[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.

[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.

[6]

S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, Solvable Models in Quantum Mechanics,, American Mathematical Society, (2005).

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[14]

S. Cuccagna, Stabilization of solution to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 1110. doi: 10.1002/cpa.1018.

[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.

[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.

[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.

[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.

[19]

Z. Gang and M. Weinstein, Equipartition of energy in Nonlinear Schrödinger-Gross-Pitaevskii equations,, AMRX, 2 (2011), 123.

[20]

I. S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products,, Vth edition, (1994).

[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.

[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.

[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.

[24]

I.M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions,, Commun. Math. Phys., 2 (1993), 297.

[25]

A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations,, Comm. Math. Phys., 133 (1990), 119. doi: 10.1007/BF02096557.

[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.

[27]

A. Soffer and M. Weinstein, Resonances, radiation damping, and instability of Hamiltonian nonlinear waves,, Invent. Math., 136 (1999), 9. doi: 10.1007/s002220050303.

[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.

[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.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[6]

S. Albeverio, F. Gesztesy, R. Högh-Krohn and H. Holden, Solvable Models in Quantum Mechanics,, American Mathematical Society, (2005).

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[14]

S. Cuccagna, Stabilization of solution to nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 54 (2001), 1110. doi: 10.1002/cpa.1018.

[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.

[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.

[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.

[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.

[19]

Z. Gang and M. Weinstein, Equipartition of energy in Nonlinear Schrödinger-Gross-Pitaevskii equations,, AMRX, 2 (2011), 123.

[20]

I. S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products,, Vth edition, (1994).

[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.

[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.

[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.

[24]

I.M. Sigal, Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions,, Commun. Math. Phys., 2 (1993), 297.

[25]

A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations,, Comm. Math. Phys., 133 (1990), 119. doi: 10.1007/BF02096557.

[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.

[27]

A. Soffer and M. Weinstein, Resonances, radiation damping, and instability of Hamiltonian nonlinear waves,, Invent. Math., 136 (1999), 9. doi: 10.1007/s002220050303.

[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.

[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.

[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.

[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.

[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.

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