# American Institute of Mathematical Sciences

September  2015, 14(5): 1941-1960. doi: 10.3934/cpaa.2015.14.1941

## Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

 1 The College of Information and Technology, Nanjing University of Chinese Medicine, Nanjing 210046, China

Received  November 2014 Revised  March 2015 Published  June 2015

In this paper, we investigate the dynamics of a boson gas with three-body interactions in $T^2$. We prove that when the particle number $N$ tends to infinity, the BBGKY hierarchy of $k$-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized $N$-body initial datum, we show that this $N\rightarrow\infty$ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.
Citation: Jianjun Yuan. Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1941-1960. doi: 10.3934/cpaa.2015.14.1941
##### References:
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Anal., 260 (2011), 959-997. doi: 10.1016/j.jfa.2010.11.003.  Google Scholar [6] Thomas Chen and Natasa Pavlović, A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies, Proc. Amer. Math. Soc., 141 (2013), 279-293. doi: 10.1090/S0002-9939-2012-11308-5.  Google Scholar [7] T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms, Ann. H. Poincare, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6.  Google Scholar [8] T. Chen and K. Taliaferro, Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy, Commun. PDE, 39 (2014), 1658-1693. doi: 10.1080/03605302.2014.917380.  Google Scholar [9] T. Chen, C. Hainzl, N. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti,, \emph{CPAM}, ().   Google Scholar [10] X. Chen, Second order corrections to mean field evolution for weakly interacting Bosons in the case of three-body interactions, Arch. Rational Mech. Anal., 203 (2012), 455-497. doi: 10.1007/s00205-011-0453-8.  Google Scholar [11] X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl., 98 (2012), 450-478. doi: 10.1016/j.matpur.2012.02.003.  Google Scholar [12] X. Chen, On the rigorous derivation of the 3d cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Rational Mech. Anal., 210 (2013), 365-408. doi: 10.1007/s00205-013-0645-5.  Google Scholar [13] X. Chen and J. Holmer, On the rigorous derivation of the 2d cubic nonlinear Schrödinger equation from 3d quantum many-body dynamics, Arch. Rational Mech. Anal., 210 (2013), 909-954. doi: 10.1007/s00205-013-0667-z.  Google Scholar [14] X. Chen and J. 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Tzirakis, Global well-posedness for a periodic nonlinear Schrodinger equation in 1D and 2D, Discrete and Continuous Dynamical Systems-Series A, 19 (2007), 37-65. doi: 10.3934/dcds.2007.19.37.  Google Scholar [20] A. Elgart, L. Erdos, B. Schlein and H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Rat. Mech. Anal., 179 (2006), 265-283. doi: 10.1007/s00205-005-0388-z.  Google Scholar [21] A. Elgart and B. Schlein, Mean field dynamics of Boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545. doi: 10.1002/cpa.20134.  Google Scholar [22] L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614. doi: 10.1007/s00222-006-0022-1.  Google Scholar [23] L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370. doi: 10.4007/annals.2010.172.291.  Google Scholar [24] L. Erdős, B. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., 22 (2009), 1099-1156. doi: 10.1090/S0894-0347-09-00635-3.  Google Scholar [25] P. Gressman, V. Sohinger and G. Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy, J. Funct. Anal., 266 (2014), 4705-4764. doi: 10.1016/j.jfa.2014.02.006.  Google Scholar [26] M. G. Grillakis and M. Machedon, Pair excitations and the mean .eld approximation of interacting Bosons, I, Commun. Math. Phys., 324 (2013), 601-636. doi: 10.1007/s00220-013-1818-7.  Google Scholar [27] M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. I, Commun. Math. Phys., 294 (2010), 273-301. doi: 10.1007/s00220-009-0933-y.  Google Scholar [28] M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. II, Adv. Math., 228 (2011) 1788-1815. doi: 10.1016/j.aim.2011.06.028.  Google Scholar [29] Y. Hong, K. Taliaferro and Z. Xie, Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity,, 26pp, ().   Google Scholar [30] Kay Kirkpatrick, Benjamin Schlein and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, American Journal of Mathematics, 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004.  Google Scholar [31] S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.  Google Scholar [32] M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010.  Google Scholar [33] E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett., 88 (2002), 170409, 1-4. Google Scholar [34] E. H. Lieb and R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31. doi: 10.1007/s002200100533.  Google Scholar [35] E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and Its Condensation, 34 (2005), Oberwolfach Seminars Series, Birkhäuser.  Google Scholar [36] I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys., 291 (2009), 31-61. doi: 10.1007/s00220-009-0867-4.  Google Scholar [37] L. Pitaevskii, Vortex lines in an imperfect Bose-gas, Sov. phys. JETP, 13 (1961), 451-454. Google Scholar [38] P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys., 97 (2011), 151-164. doi: 10.1007/s11005-011-0470-4.  Google Scholar [39] H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569.  Google Scholar [40] V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbbT^3$ from the dynamics of many-body quantum systems,, \emph{Ann. I.H.Poincar\'e-AN}., ().  doi: 10.1016/j.anihpc.2014.09.005.  Google Scholar

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##### References:
 [1] R. Adami, C. Bardos, F. Golse and A. Teta, Towards a rigorous derivation of the cubic nonlinear Schrodinger equation in dimension one, Asymptot. Anal., 40 (2004), 93-108.  Google Scholar [2] R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys., 127 (2007), 1193-1220. doi: 10.1007/s10955-006-9271-z.  Google Scholar [3] E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J., 59 (1989), 337-357. doi: 10.1215/S0012-7094-89-05915-2.  Google Scholar [4] Thomas Chen and Natasa Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discr. Contin. Dyn. Syst. A, 27 (2010), 715-739. doi: 10.3934/dcds.2010.27.715.  Google Scholar [5] Thomas Chen and Natasa Pavlović, The quintic NLS as the mean field limit of a Boson gas with three-body interactions, J. Funct. Anal., 260 (2011), 959-997. doi: 10.1016/j.jfa.2010.11.003.  Google Scholar [6] Thomas Chen and Natasa Pavlović, A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies, Proc. Amer. Math. Soc., 141 (2013), 279-293. doi: 10.1090/S0002-9939-2012-11308-5.  Google Scholar [7] T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms, Ann. H. Poincare, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6.  Google Scholar [8] T. Chen and K. Taliaferro, Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy, Commun. PDE, 39 (2014), 1658-1693. doi: 10.1080/03605302.2014.917380.  Google Scholar [9] T. Chen, C. Hainzl, N. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti,, \emph{CPAM}, ().   Google Scholar [10] X. Chen, Second order corrections to mean field evolution for weakly interacting Bosons in the case of three-body interactions, Arch. Rational Mech. Anal., 203 (2012), 455-497. doi: 10.1007/s00205-011-0453-8.  Google Scholar [11] X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl., 98 (2012), 450-478. doi: 10.1016/j.matpur.2012.02.003.  Google Scholar [12] X. Chen, On the rigorous derivation of the 3d cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Rational Mech. Anal., 210 (2013), 365-408. doi: 10.1007/s00205-013-0645-5.  Google Scholar [13] X. Chen and J. Holmer, On the rigorous derivation of the 2d cubic nonlinear Schrödinger equation from 3d quantum many-body dynamics, Arch. Rational Mech. Anal., 210 (2013), 909-954. doi: 10.1007/s00205-013-0667-z.  Google Scholar [14] X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation,, 41pp, ().   Google Scholar [15] X. Chen and J. Holmer, Focusing quantum many-body dynamics II: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation from 3D,, 48pp, ().   Google Scholar [16] X. Chen and J. Holmer, On the Klainerman-Machedon onjecture of the quantum BBGKY hierarchy with self-interaction,, preprint, ().   Google Scholar [17] X. Chen and J. Holmer, Correlation structures, many-body scattering processes and the derivation of the Gross-Pitaevskii hierarchy,, 48pp, ().   Google Scholar [18] X. Chen and P. Smith, On the unconditional uniqueness of solutions to the infinite radial Chern- Simons-Schrödinger hierarchy, Analysis and PDE, 7 (2014), 1683-1712. doi: 10.2140/apde.2014.7.1683.  Google Scholar [19] D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis, Global well-posedness for a periodic nonlinear Schrodinger equation in 1D and 2D, Discrete and Continuous Dynamical Systems-Series A, 19 (2007), 37-65. doi: 10.3934/dcds.2007.19.37.  Google Scholar [20] A. Elgart, L. Erdos, B. Schlein and H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Rat. Mech. Anal., 179 (2006), 265-283. doi: 10.1007/s00205-005-0388-z.  Google Scholar [21] A. Elgart and B. Schlein, Mean field dynamics of Boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545. doi: 10.1002/cpa.20134.  Google Scholar [22] L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614. doi: 10.1007/s00222-006-0022-1.  Google Scholar [23] L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370. doi: 10.4007/annals.2010.172.291.  Google Scholar [24] L. Erdős, B. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., 22 (2009), 1099-1156. doi: 10.1090/S0894-0347-09-00635-3.  Google Scholar [25] P. Gressman, V. Sohinger and G. Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy, J. Funct. Anal., 266 (2014), 4705-4764. doi: 10.1016/j.jfa.2014.02.006.  Google Scholar [26] M. G. Grillakis and M. Machedon, Pair excitations and the mean .eld approximation of interacting Bosons, I, Commun. Math. Phys., 324 (2013), 601-636. doi: 10.1007/s00220-013-1818-7.  Google Scholar [27] M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. I, Commun. Math. Phys., 294 (2010), 273-301. doi: 10.1007/s00220-009-0933-y.  Google Scholar [28] M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. II, Adv. Math., 228 (2011) 1788-1815. doi: 10.1016/j.aim.2011.06.028.  Google Scholar [29] Y. Hong, K. Taliaferro and Z. Xie, Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity,, 26pp, ().   Google Scholar [30] Kay Kirkpatrick, Benjamin Schlein and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, American Journal of Mathematics, 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004.  Google Scholar [31] S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.  Google Scholar [32] M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010.  Google Scholar [33] E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett., 88 (2002), 170409, 1-4. Google Scholar [34] E. H. Lieb and R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31. doi: 10.1007/s002200100533.  Google Scholar [35] E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and Its Condensation, 34 (2005), Oberwolfach Seminars Series, Birkhäuser.  Google Scholar [36] I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys., 291 (2009), 31-61. doi: 10.1007/s00220-009-0867-4.  Google Scholar [37] L. Pitaevskii, Vortex lines in an imperfect Bose-gas, Sov. phys. JETP, 13 (1961), 451-454. Google Scholar [38] P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys., 97 (2011), 151-164. doi: 10.1007/s11005-011-0470-4.  Google Scholar [39] H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569.  Google Scholar [40] V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbbT^3$ from the dynamics of many-body quantum systems,, \emph{Ann. I.H.Poincar\'e-AN}., ().  doi: 10.1016/j.anihpc.2014.09.005.  Google Scholar
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