August  2021, 26(8): 4493-4513. doi: 10.3934/dcdsb.2020297

Scattering and strong instability of the standing waves for dipolar quantum gases

School of Mathematical Science, and V.C. & V.R. Key Lab of Sichuan Province, Sichuan Normal University, Chengdu 610068, China

* Corresponding author: Juan Huang

Received  June 2020 Revised  August 2020 Published  August 2021 Early access  October 2020

This paper concerns the nonlinear Schrödinger equation which describes the dipolar quantum gases. When the energy plus mass is lower than the mass of the ground state, we find we can use the kinetic energy and mass of the initial data to divide the subspace into two parts. If the initial data are in one of the parts, the solutions exist globally. Moreover, by using the Kening-Merle roadmap method, we find that these solutions will scatter. If initial data are in the other part, the solutions will collapse. And hence, the standing waves are strong unstable.

Citation: Juan Huang. Scattering and strong instability of the standing waves for dipolar quantum gases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4493-4513. doi: 10.3934/dcdsb.2020297
References:
[1]

P. Antonelli and C. Sparber, Existence of solitary waves in dipolar quantum gases, Phys. D, 240 (2011), 426-431.  doi: 10.1016/j.physd.2010.10.004.

[2]

W. BaoY. Cai and H. Wang, Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), 7874-7892.  doi: 10.1016/j.jcp.2010.07.001.

[3]

J. Bellazzini and L. Forcella, Asymptotic dynamic for dipolar quantum gases below the ground state energy threshold, J. Funct. Anal., 277 (2019), 1958-1998.  doi: 10.1016/j.jfa.2019.04.005.

[4]

J. Bourgain, Scattering in the energy space and below for 3D NLS, J. Anal. Math., 75 (1998), 267–297. doi: 10.1007/BF02788703.

[5]

R. Carles and H. Hajaiej, Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases, Bull. Lond. Math. Soc., 47 (2015), 509-518.  doi: 10.1112/blms/bdv024.

[6]

R. CarlesP. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590.  doi: 10.1088/0951-7715/21/11/006.

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.  doi: 10.1002/cpa.20029.

[9]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[10]

M. S. EllioJ. J. Valentini and D. W. Chandler, Subkelvin cooling NO molecules via "billiard-like" collisions with argon, Science, 302 (2003), 1940-1943.  doi: 10.1126/science.1090679.

[11]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1063/1.523491.

[12]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl., 64 (1985), 363-401. 

[13]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.

[14]

J. Huang and J. Zhang, Exact value of cross-constrain problem and strong instability of standing waves in trapped dipolar quantum gases, Appl. Math. Lett., 70 (2017), 32-38.  doi: 10.1016/j.aml.2017.03.002.

[15]

S. IbrahimN. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.  doi: 10.2140/apde.2011.4.405.

[16]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[17]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[18]

L. Ma and P. Cao, The threshold for the focusing Gross-Pitaevskii equation with trapped dipolar quantum gases, J. Math. Anal. Appl., 381 (2011), 240-246.  doi: 10.1016/j.jmaa.2011.02.031.

[19]

L. Ma and J. Wang, Sharp threshold of the Gross-Pitaevskii equation with trapped dipolar quantum gases, Canad. Math. Bull., 56 (2013), 378-387.  doi: 10.4153/CMB-2011-181-2.

[20]

J. Rauch, Partial Differential Equations, Graduate Texts in Mathematics, 128, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.

[21]

M. Vengalattore, S. R. Leslie, J. Guzman and D. M. Stamper-Kurn, Spontaneously modulated spin textures in a dipolar spinor Bose-Einstein condensate, Phys. Rev. Lett., 100 (2008), 170403. doi: 10.1103/PhysRevLett.100.170403.

[22]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1982/83), 567-576.  doi: 10.1007/BF01208265.

[23]

S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61 (2000). doi: 10.1103/PhysRevA.61.041604.

show all references

References:
[1]

P. Antonelli and C. Sparber, Existence of solitary waves in dipolar quantum gases, Phys. D, 240 (2011), 426-431.  doi: 10.1016/j.physd.2010.10.004.

[2]

W. BaoY. Cai and H. Wang, Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), 7874-7892.  doi: 10.1016/j.jcp.2010.07.001.

[3]

J. Bellazzini and L. Forcella, Asymptotic dynamic for dipolar quantum gases below the ground state energy threshold, J. Funct. Anal., 277 (2019), 1958-1998.  doi: 10.1016/j.jfa.2019.04.005.

[4]

J. Bourgain, Scattering in the energy space and below for 3D NLS, J. Anal. Math., 75 (1998), 267–297. doi: 10.1007/BF02788703.

[5]

R. Carles and H. Hajaiej, Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases, Bull. Lond. Math. Soc., 47 (2015), 509-518.  doi: 10.1112/blms/bdv024.

[6]

R. CarlesP. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590.  doi: 10.1088/0951-7715/21/11/006.

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[8]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.  doi: 10.1002/cpa.20029.

[9]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[10]

M. S. EllioJ. J. Valentini and D. W. Chandler, Subkelvin cooling NO molecules via "billiard-like" collisions with argon, Science, 302 (2003), 1940-1943.  doi: 10.1126/science.1090679.

[11]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1063/1.523491.

[12]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl., 64 (1985), 363-401. 

[13]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.

[14]

J. Huang and J. Zhang, Exact value of cross-constrain problem and strong instability of standing waves in trapped dipolar quantum gases, Appl. Math. Lett., 70 (2017), 32-38.  doi: 10.1016/j.aml.2017.03.002.

[15]

S. IbrahimN. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.  doi: 10.2140/apde.2011.4.405.

[16]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[17]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[18]

L. Ma and P. Cao, The threshold for the focusing Gross-Pitaevskii equation with trapped dipolar quantum gases, J. Math. Anal. Appl., 381 (2011), 240-246.  doi: 10.1016/j.jmaa.2011.02.031.

[19]

L. Ma and J. Wang, Sharp threshold of the Gross-Pitaevskii equation with trapped dipolar quantum gases, Canad. Math. Bull., 56 (2013), 378-387.  doi: 10.4153/CMB-2011-181-2.

[20]

J. Rauch, Partial Differential Equations, Graduate Texts in Mathematics, 128, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.

[21]

M. Vengalattore, S. R. Leslie, J. Guzman and D. M. Stamper-Kurn, Spontaneously modulated spin textures in a dipolar spinor Bose-Einstein condensate, Phys. Rev. Lett., 100 (2008), 170403. doi: 10.1103/PhysRevLett.100.170403.

[22]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1982/83), 567-576.  doi: 10.1007/BF01208265.

[23]

S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61 (2000). doi: 10.1103/PhysRevA.61.041604.

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