October  2022, 42(10): 4937-4964. doi: 10.3934/dcds.2022081

On multi-solitons for coupled Lowest Landau Level equations

Université Lorraine, CNRS, IECL, F-54000 Nancy, France

Received  November 2021 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: The author is supported by the grants "BEKAM" ANR-15-CE40-0001 and "ISDEEC" ANR-16-CE40-0013

We consider a coupled system of nonlinear Lowest Landau Level equations. We first show the existence of multi-solitons with an exponentially localised error term in space, and then we prove a uniqueness result. We also show a long time stability result of the sum of traveling waves having all the same speed, under the condition that they are localised far away enough from each other. Finally, we observe that these multi-solitons provide examples of dynamics for the linear Schrödinger equation with harmonic potential perturbed by a time-dependent potential.

Citation: Laurent Thomann. On multi-solitons for coupled Lowest Landau Level equations. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4937-4964. doi: 10.3934/dcds.2022081
References:
[1]

A. AftalionX. Blanc and J. Dalibard, Vortex patterns in a fast rotating Bose-Einstein condensate, Physical Review A, 71 (2005), 023611.  doi: 10.1103/PhysRevA.71.023611.

[2]

A. AftalionX. Blanc and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates, J. Functional Anal., 241 (2006), 661-702.  doi: 10.1016/j.jfa.2006.04.027.

[3]

A. BiasiP. BizonB. Craps and O. Evnin, Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates, Phys. Rev. A, 96 (2017), 053615.  doi: 10.1103/PhysRevA.96.053615.

[4]

E. A. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal., 97 (1991), 231-249.  doi: 10.1016/0022-1236(91)90022-W.

[5]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[6]

R. Côte and S. Le Coz, High-speed excited multi-solitons in nonlinear Schröinger equations, J. Math. Pures Appl., 96 (2011), 135-166.  doi: 10.1016/j.matpur.2011.03.004.

[7]

R. CôteY. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.  doi: 10.4171/RMI/636.

[8]

M. De Clerck and O. Evnin, Time-periodic quantum states of weakly interacting bosons in a harmonic trap, Phys. Lett. A, 384 (2020), 126930, 11 pp. doi: 10.1016/j.physleta.2020.126930.

[9]

F. DelebecqueS. Le Coz and R. M. Weishäupl, Multi-speed solitary waves of nonlinear Schrödinger systems: Theoretical and numerical analysis, Commun. Math. Sci., 14 (2016), 1599-1624.  doi: 10.4310/CMS.2016.v14.n6.a7.

[10]

E. FaouP. Germain and Z. Hani, The weakly nonlinear large box limit of the 2D cubic NLS, J. Amer. Math. Soc., 29 (2016), 915-982.  doi: 10.1090/jams/845.

[11]

E. Faou and P. Raphaël, On weakly turbulent solutions to the perturbed linear harmonic oscillator, Am. J. Math., to appear.

[12]

G. Ferriere, Existence of multi-solitons for the focusing logarithmic non-linear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 841-875.  doi: 10.1016/j.anihpc.2020.09.002.

[13]

G. Ferriere, The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition, Discrete Contin. Dyn. Syst., 40 (2020), 6247-6274.  doi: 10.3934/dcds.2020277.

[14]

P. GérardP. Germain and L. Thomann, On the cubic lowest Landau level equation, Arch. Ration. Mech. Anal., 231 (2019), 1073-1128.  doi: 10.1007/s00205-018-1295-4.

[15]

P. GermainZ. Hani and L. Thomann, On the continuous resonant equation for NLS. Ⅰ. Deterministic analysis, J. Math. Pures Appl., 105 (2016), 131-163.  doi: 10.1016/j.matpur.2015.10.002.

[16]

P. GermainZ. Hani and L. Thomann, On the continuous resonant equation for NLS. Ⅱ. Statistical study, Anal. & PDE., 8 (2015), 1733-1756.  doi: 10.2140/apde.2015.8.1733.

[17]

T. L. Ho, Bose-Einstein condensates with large number of vortices, Physical Review letters, 87 (2001), 060403.  doi: 10.1103/PhysRevLett.87.060403.

[18]

I. Ianni and S. Le Coz, Multi-speed solitary wave solutions for nonlinear Schröinger systems, J. Lond. Math. Soc., 89 (2014), 623-639.  doi: 10.1112/jlms/jdt083.

[19]

J. KriegerY. Martel and P. Raphaël, Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math., 62 (2009), 1501-1550.  doi: 10.1002/cpa.20292.

[20]

S. Le Coz and T.-P. Tsai, Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations, Proceedings of the Sixth International Congress of Chinese Mathematicians. Vol. I, 43–56, Adv. Lect. Math. (ALM), 36, Int. Press, Somerville, MA, 2017.

[21]

Y. Martel, Interaction of solitons from the PDE point of view, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. III. Invited lectures, 2439–2466, World Sci. Publ., Hackensack, NJ, 2018.

[22]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.

[23]

Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation, Ann. Sci. Éc. Norm. Supér., 51 (2018), 701-737.  doi: 10.24033/asens.2364.

[24]

E. Mueller and T.-L. Ho, Two-component Bose-Einstein condensates with a large number of vortices, Phys. Rev. Lett., 88 (2002), 180403.  doi: 10.1103/PhysRevLett.88.180403.

[25]

F. Nier, Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics, Rev. Math. Phys., 19 (2007), 101-130.  doi: 10.1142/S0129055X07002900.

[26]

V. Schwinte and L. Thomann, Growth of Sobolev norms for coupled Lowest Landau Level equations, Pure Appl. Anal., 3 (2021), 189-222.  doi: 10.2140/paa.2021.3.189.

[27]

L. Thomann, Growth of Sobolev norms for linear Schrödinger operators, Ann. Henri Lebesgue, 4 (2021), 1595-1618.  doi: 10.5802/ahl.111.

[28]

K. Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, 263. Springer, New York, 2012. x+344 pp. doi: 10.1007/978-1-4419-8801-0.

show all references

References:
[1]

A. AftalionX. Blanc and J. Dalibard, Vortex patterns in a fast rotating Bose-Einstein condensate, Physical Review A, 71 (2005), 023611.  doi: 10.1103/PhysRevA.71.023611.

[2]

A. AftalionX. Blanc and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates, J. Functional Anal., 241 (2006), 661-702.  doi: 10.1016/j.jfa.2006.04.027.

[3]

A. BiasiP. BizonB. Craps and O. Evnin, Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates, Phys. Rev. A, 96 (2017), 053615.  doi: 10.1103/PhysRevA.96.053615.

[4]

E. A. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal., 97 (1991), 231-249.  doi: 10.1016/0022-1236(91)90022-W.

[5]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[6]

R. Côte and S. Le Coz, High-speed excited multi-solitons in nonlinear Schröinger equations, J. Math. Pures Appl., 96 (2011), 135-166.  doi: 10.1016/j.matpur.2011.03.004.

[7]

R. CôteY. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.  doi: 10.4171/RMI/636.

[8]

M. De Clerck and O. Evnin, Time-periodic quantum states of weakly interacting bosons in a harmonic trap, Phys. Lett. A, 384 (2020), 126930, 11 pp. doi: 10.1016/j.physleta.2020.126930.

[9]

F. DelebecqueS. Le Coz and R. M. Weishäupl, Multi-speed solitary waves of nonlinear Schrödinger systems: Theoretical and numerical analysis, Commun. Math. Sci., 14 (2016), 1599-1624.  doi: 10.4310/CMS.2016.v14.n6.a7.

[10]

E. FaouP. Germain and Z. Hani, The weakly nonlinear large box limit of the 2D cubic NLS, J. Amer. Math. Soc., 29 (2016), 915-982.  doi: 10.1090/jams/845.

[11]

E. Faou and P. Raphaël, On weakly turbulent solutions to the perturbed linear harmonic oscillator, Am. J. Math., to appear.

[12]

G. Ferriere, Existence of multi-solitons for the focusing logarithmic non-linear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 841-875.  doi: 10.1016/j.anihpc.2020.09.002.

[13]

G. Ferriere, The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition, Discrete Contin. Dyn. Syst., 40 (2020), 6247-6274.  doi: 10.3934/dcds.2020277.

[14]

P. GérardP. Germain and L. Thomann, On the cubic lowest Landau level equation, Arch. Ration. Mech. Anal., 231 (2019), 1073-1128.  doi: 10.1007/s00205-018-1295-4.

[15]

P. GermainZ. Hani and L. Thomann, On the continuous resonant equation for NLS. Ⅰ. Deterministic analysis, J. Math. Pures Appl., 105 (2016), 131-163.  doi: 10.1016/j.matpur.2015.10.002.

[16]

P. GermainZ. Hani and L. Thomann, On the continuous resonant equation for NLS. Ⅱ. Statistical study, Anal. & PDE., 8 (2015), 1733-1756.  doi: 10.2140/apde.2015.8.1733.

[17]

T. L. Ho, Bose-Einstein condensates with large number of vortices, Physical Review letters, 87 (2001), 060403.  doi: 10.1103/PhysRevLett.87.060403.

[18]

I. Ianni and S. Le Coz, Multi-speed solitary wave solutions for nonlinear Schröinger systems, J. Lond. Math. Soc., 89 (2014), 623-639.  doi: 10.1112/jlms/jdt083.

[19]

J. KriegerY. Martel and P. Raphaël, Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math., 62 (2009), 1501-1550.  doi: 10.1002/cpa.20292.

[20]

S. Le Coz and T.-P. Tsai, Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations, Proceedings of the Sixth International Congress of Chinese Mathematicians. Vol. I, 43–56, Adv. Lect. Math. (ALM), 36, Int. Press, Somerville, MA, 2017.

[21]

Y. Martel, Interaction of solitons from the PDE point of view, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. III. Invited lectures, 2439–2466, World Sci. Publ., Hackensack, NJ, 2018.

[22]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.

[23]

Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation, Ann. Sci. Éc. Norm. Supér., 51 (2018), 701-737.  doi: 10.24033/asens.2364.

[24]

E. Mueller and T.-L. Ho, Two-component Bose-Einstein condensates with a large number of vortices, Phys. Rev. Lett., 88 (2002), 180403.  doi: 10.1103/PhysRevLett.88.180403.

[25]

F. Nier, Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics, Rev. Math. Phys., 19 (2007), 101-130.  doi: 10.1142/S0129055X07002900.

[26]

V. Schwinte and L. Thomann, Growth of Sobolev norms for coupled Lowest Landau Level equations, Pure Appl. Anal., 3 (2021), 189-222.  doi: 10.2140/paa.2021.3.189.

[27]

L. Thomann, Growth of Sobolev norms for linear Schrödinger operators, Ann. Henri Lebesgue, 4 (2021), 1595-1618.  doi: 10.5802/ahl.111.

[28]

K. Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, 263. Springer, New York, 2012. x+344 pp. doi: 10.1007/978-1-4419-8801-0.

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