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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 |
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), 477-500
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), 121-137
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 (1) (2013), 247-289.
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) 013501.
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 (5) (2013), 1155-1188.
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, Providence, 2005. |
[7] |
D. Bambusi, Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry, Comm. Math. Phys. 320 (2013), 499-542.
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), 669-705.
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), 1111-1142. |
[10] |
V. S. Buslaev and G. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations, Amer.Math.Soc.Transl., 164 (1995), 75-98.
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), 419-475.
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), 1557-1570.
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, arXiv:1511.06731, (2015) |
[14] |
S. Cuccagna, Stabilization of solution to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145. erratum ibid. 58 (2005), 147.
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), 51-87.
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), 279-331.
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), 443-490.
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), 267-322.
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) 123-181. |
[20] |
I. S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Vth edition, Academic Press, 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), 3691-3747.
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), 1063-1079. |
[23] |
D. Noja and A. Posilicano, Wave equations with concentrated nonlinearities, J. Phys. A: Math. Gen., 38 (2005), 5011-5022.
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), 297-320. |
[25] |
A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146.
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), 376-390.
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), 9-74.
doi: 10.1007/s002220050303. |
[28] |
A. Soffer and M. Weinstein, Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys., 16 (2004), 977-1071.
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), 213905.
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), 153-216.
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),1629-1673.
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), 41-79.
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), 477-500
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), 121-137
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 (1) (2013), 247-289.
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) 013501.
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 (5) (2013), 1155-1188.
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, Providence, 2005. |
[7] |
D. Bambusi, Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry, Comm. Math. Phys. 320 (2013), 499-542.
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), 669-705.
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), 1111-1142. |
[10] |
V. S. Buslaev and G. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations, Amer.Math.Soc.Transl., 164 (1995), 75-98.
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), 419-475.
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), 1557-1570.
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, arXiv:1511.06731, (2015) |
[14] |
S. Cuccagna, Stabilization of solution to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145. erratum ibid. 58 (2005), 147.
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), 51-87.
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), 279-331.
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), 443-490.
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), 267-322.
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) 123-181. |
[20] |
I. S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Vth edition, Academic Press, 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), 3691-3747.
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), 1063-1079. |
[23] |
D. Noja and A. Posilicano, Wave equations with concentrated nonlinearities, J. Phys. A: Math. Gen., 38 (2005), 5011-5022.
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), 297-320. |
[25] |
A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146.
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), 376-390.
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), 9-74.
doi: 10.1007/s002220050303. |
[28] |
A. Soffer and M. Weinstein, Selection of the ground state for nonlinear Schrödinger equations, Rev. Math. Phys., 16 (2004), 977-1071.
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), 213905.
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), 153-216.
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),1629-1673.
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), 41-79.
doi: 10.1007/978-3-319-19935-1_2. |
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