January  2020, 40(1): 635-682. doi: 10.3934/dcds.2020026

Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

Université de Lille, CNRS, Inria, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France

Received  July 2019 Revised  July 2019 Published  October 2019

We consider a nonlocal family of Gross–Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.

Citation: André de Laire, Pierre Mennuni. Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 635-682. doi: 10.3934/dcds.2020026
References:
[1]

M. Abid, C. Huepe, S. Metens, C. Nore, C. Pham, L. Tuckerman and M. Brachet, Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence, Fluid Dynamics Research, 33 (2003), 509–544. Collection of Papers written by Regional Editors. doi: 10.1016/j.fluiddyn.2003.09.001.

[2]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.

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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.

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C. Audiard, Small energy traveling waves for the Euler-Korteweg system, Nonlinearity, 30 (2017), 3362-3399.  doi: 10.1088/1361-6544/aa7cc2.

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C. BeckerS. StellmerP. Soltan-PanahiS. DörscherM. BaumertE.-M. RichterJ. KronjägerK. Bongs and K. Sengstock, Oscillations and interactions of dark and dark–bright solitons in Bose-Einstein condensates, Nature Physics, 4 (2008), 496-501.  doi: 10.1038/nphys962.

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J. Bellazzini and L. Jeanjean, On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48 (2016), 2028-2058.  doi: 10.1137/15M1015959.

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N. G. Berloff, Quantum vortices, travelling coherent structures and superfluid turbulence, In Stationary and Time Dependent Gross-Pitaevskii Equations, volume 473 of Contemp. Math., pages 27–54. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09223.

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N. G. Berloff and P. H. Roberts, Motions in a Bose condensate Ⅵ. Vortices in a nonlocal model, J. Phys. A, 32 (1999), 5611-5625.  doi: 10.1088/0305-4470/32/30/308.

[10]

F. Béthuel, P. Gravejat and J.-C. Saut, Existence and properties of travelling waves for the Gross-Pitaevskii equation, In Stationary and Time Dependent Gross-Pitaevskii Equations, volume 473 of Contemp. Math., pages 55–103. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09224.

[11]

F. BéthuelP. Gravejat and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation. Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.  doi: 10.1007/s00220-008-0614-2.

[12]

F. BéthuelP. GravejatJ.-C. Saut and D. Smets, Orbital stability of the black soliton for the Gross-Pitaevskii equation, Indiana Univ. Math. J., 57 (2008), 2611-2642.  doi: 10.1512/iumj.2008.57.3632.

[13]

F. BéthuelP. GravejatJ.-C. Saut and D. Smets, On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation. Ⅰ, Int. Math. Res. Not. IMRN, 14 (2009), 2700-2748.  doi: 10.1093/imrn/rnp031.

[14]

F. Bethuel, P. Gravejat and D. Smets, Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1327–1381. doi: 10.24033/asens.2271.

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F. BethuelG. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. (JEMS), 6 (2004), 17-94. 

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F. Béthuel and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation Ⅰ, Ann. Inst. H. Poincaré Phys. Théor., 70 (1999), 147-238. 

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M. Bogdan, A. Kovalev and A. Kosevich, Stability criterion in imperfect Bose gas, Fiz. Nizk. Temp., 15 (1989), 511–514. In Russian.

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N. N. Bogoliubov, On the theory of superfluidity, J. Phys. USSR, 11 (1947), 23–32. Reprinted in: D. Pines, The Many-Body Problem (W. A. Benjamin, New York), 1961,292–301.

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

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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.

[21]

T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003.

[22]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[23]

D. Chiron, Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one, Nonlinearity, 25 (2012), 813-850.  doi: 10.1088/0951-7715/25/3/813.

[24]

D. Chiron, Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one, Anal. PDE, 6 (2013), 1327-1420.  doi: 10.2140/apde.2013.6.1327.

[25]

D. Chiron and M. Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, Arch. Ration. Mech. Anal., 226 (2017), 143-242.  doi: 10.1007/s00205-017-1131-2.

[26]

D. Chiron and F. Rousset, The KdV/KP-Ⅰ limit of the nonlinear Schrödinger equation, SIAM J. Math. Anal., 42 (2010), 64-96.  doi: 10.1137/080738994.

[27]

A. de Laire, Non-existence for travelling waves with small energy for the Gross-Pitaevskii equation in dimension $N\geq 3$, C. R. Math. Acad. Sci. Paris, 347 (2009), 375-380.  doi: 10.1016/j.crma.2009.02.006.

[28]

A. de Laire, Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition at infinity, Comm. Partial Differential Equations, 35 (2010), 2021-2058.  doi: 10.1080/03605302.2010.497200.

[29]

A. de Laire, Nonexistence of traveling waves for a nonlocal Gross-Pitaevskii equation, Indiana Univ. Math. J., 61 (2012), 1451-1484.  doi: 10.1512/iumj.2012.61.4707.

[30]

A. de Laire and P. Gravejat, Stability in the energy space for chains of solitons of the Landau-Lifshitz equation, J. Differential Equations, 258 (2015), 1-80.  doi: 10.1016/j.jde.2014.09.003.

[31]

A. de Laire and P. Gravejat, The Sine-Gordon regime of the Landau-Lifshitz equation with a strong easy-plane anisotropy, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1885-1945.  doi: 10.1016/j.anihpc.2018.03.005.

[32]

J. DenschlagJ. E. SimsarianD. L. FederC. W. ClarkL. A. CollinsJ. CubizollesL. DengE. W. HagleyK. Helmerson and W. P. Reinhardt, Generating solitons by phase engineering of a Bose-Einstein condensate, Science, 287 (2000), 97-101.  doi: 10.1126/science.287.5450.97.

[33]

C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Comm. Partial Differential Equations, 33 (2008), 729-771.  doi: 10.1080/03605300802031614.

[34]

P. Gérard, The Gross-Pitaevskii equation in the energy space, Stationary and Time Dependent Gross-Pitaevskii Equations, 129–148, Contemp. Math., 473, Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09226.

[35]

P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004.

[36]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[37]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.  doi: 10.1007/BF01214768.

[38]

L. Grafakos, Classical Fourier Analysis, volume 249 of Graduate Texts in Mathematics, Springer, New York, second edition, 2008.

[39]

P. Gravejat and D. Smets, Asymptotic stability of the black soliton for the Gross-Pitaevskii equation, Proc. Lond. Math. Soc. (3), 111 (2015), 305-353.  doi: 10.1112/plms/pdv025.

[40]

E. P. Gross, Hydrodynamics of a superfluid condensate, Journal of Mathematical Physics, 4 (1963), 195-207.  doi: 10.1063/1.1703944.

[41]

S. GustafsonK. Nakanishi and T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré, 8 (2007), 1303-1331.  doi: 10.1007/s00023-007-0336-6.

[42]

S. GustafsonK. Nakanishi and T.-P. Tsai, Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math., 11 (2009), 657-707.  doi: 10.1142/S0219199709003491.

[43]

Y. V. Kartashov and L. Torner, Gray spatial solitons in nonlocal nonlinear media, Opt. Lett., 32 (2007), 946-948.  doi: 10.1364/OL.32.000946.

[44]

R. KillipT. OhO. Pocovnicu and M. Vişan, Global well-posedness of the Gross-Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions, Math. Res. Lett., 19 (2012), 969-986.  doi: 10.4310/MRL.2012.v19.n5.a1.

[45]

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein and T. Pfau, The physics of dipolar bosonic quantum gases, Reports on Progress in Physics, 72 (2009), 126401. doi: 10.1088/0034-4885/72/12/126401.

[46]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.

[47]

Z. Lin, Stability and instability of traveling solitonic bubbles, Adv. Differential Equations, 7 (2002), 897-918. 

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P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

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O. Lopes and M. Mariş, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal., 254 (2008), 535-592.  doi: 10.1016/j.jfa.2007.10.004.

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Y. Luo and A. Stylianou, Ground states for a nonlocal cubic-quartic Gross-Pitaevskii equation, Preprint, http://arXiv.org/abs/1806.00697.

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M. Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, Ann. of Math. (2), 178 (2013), 107-182.  doi: 10.4007/annals.2013.178.1.2.

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M. Mariş, On some minimization problems in RN, In New Trends in Differential Equations, Control Theory and Optimization, 215–230. World Sci. Publ., Hackensack, NJ, 2016.

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M. Mariş, Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, SIAM J. Math. Anal., 40 (2008), 1076-1103.  doi: 10.1137/070711189.

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show all references

References:
[1]

M. Abid, C. Huepe, S. Metens, C. Nore, C. Pham, L. Tuckerman and M. Brachet, Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence, Fluid Dynamics Research, 33 (2003), 509–544. Collection of Papers written by Regional Editors. doi: 10.1016/j.fluiddyn.2003.09.001.

[2]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 of National Bureau of Standards Applied Mathematics Series, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.

[3]

J. Albert, Positivity properties and uniqueness of solitary wave solutions of the intermediate long-wave equation, In Evolution Equations (Baton Rouge, LA, 1992), volume 168 of Lecture Notes in Pure and Appl. Math., pages 11–20. Dekker, New York, 1995.

[4]

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.

[5]

C. Audiard, Small energy traveling waves for the Euler-Korteweg system, Nonlinearity, 30 (2017), 3362-3399.  doi: 10.1088/1361-6544/aa7cc2.

[6]

C. BeckerS. StellmerP. Soltan-PanahiS. DörscherM. BaumertE.-M. RichterJ. KronjägerK. Bongs and K. Sengstock, Oscillations and interactions of dark and dark–bright solitons in Bose-Einstein condensates, Nature Physics, 4 (2008), 496-501.  doi: 10.1038/nphys962.

[7]

J. Bellazzini and L. Jeanjean, On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48 (2016), 2028-2058.  doi: 10.1137/15M1015959.

[8]

N. G. Berloff, Quantum vortices, travelling coherent structures and superfluid turbulence, In Stationary and Time Dependent Gross-Pitaevskii Equations, volume 473 of Contemp. Math., pages 27–54. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09223.

[9]

N. G. Berloff and P. H. Roberts, Motions in a Bose condensate Ⅵ. Vortices in a nonlocal model, J. Phys. A, 32 (1999), 5611-5625.  doi: 10.1088/0305-4470/32/30/308.

[10]

F. Béthuel, P. Gravejat and J.-C. Saut, Existence and properties of travelling waves for the Gross-Pitaevskii equation, In Stationary and Time Dependent Gross-Pitaevskii Equations, volume 473 of Contemp. Math., pages 55–103. Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09224.

[11]

F. BéthuelP. Gravejat and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation. Ⅱ, Comm. Math. Phys., 285 (2009), 567-651.  doi: 10.1007/s00220-008-0614-2.

[12]

F. BéthuelP. GravejatJ.-C. Saut and D. Smets, Orbital stability of the black soliton for the Gross-Pitaevskii equation, Indiana Univ. Math. J., 57 (2008), 2611-2642.  doi: 10.1512/iumj.2008.57.3632.

[13]

F. BéthuelP. GravejatJ.-C. Saut and D. Smets, On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation. Ⅰ, Int. Math. Res. Not. IMRN, 14 (2009), 2700-2748.  doi: 10.1093/imrn/rnp031.

[14]

F. Bethuel, P. Gravejat and D. Smets, Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1327–1381. doi: 10.24033/asens.2271.

[15]

F. BethuelG. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. (JEMS), 6 (2004), 17-94. 

[16]

F. Béthuel and J.-C. Saut, Travelling waves for the Gross-Pitaevskii equation Ⅰ, Ann. Inst. H. Poincaré Phys. Théor., 70 (1999), 147-238. 

[17]

M. Bogdan, A. Kovalev and A. Kosevich, Stability criterion in imperfect Bose gas, Fiz. Nizk. Temp., 15 (1989), 511–514. In Russian.

[18]

N. N. Bogoliubov, On the theory of superfluidity, J. Phys. USSR, 11 (1947), 23–32. Reprinted in: D. Pines, The Many-Body Problem (W. A. Benjamin, New York), 1961,292–301.

[19]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[20]

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.

[21]

T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003.

[22]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[23]

D. Chiron, Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one, Nonlinearity, 25 (2012), 813-850.  doi: 10.1088/0951-7715/25/3/813.

[24]

D. Chiron, Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one, Anal. PDE, 6 (2013), 1327-1420.  doi: 10.2140/apde.2013.6.1327.

[25]

D. Chiron and M. Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, Arch. Ration. Mech. Anal., 226 (2017), 143-242.  doi: 10.1007/s00205-017-1131-2.

[26]

D. Chiron and F. Rousset, The KdV/KP-Ⅰ limit of the nonlinear Schrödinger equation, SIAM J. Math. Anal., 42 (2010), 64-96.  doi: 10.1137/080738994.

[27]

A. de Laire, Non-existence for travelling waves with small energy for the Gross-Pitaevskii equation in dimension $N\geq 3$, C. R. Math. Acad. Sci. Paris, 347 (2009), 375-380.  doi: 10.1016/j.crma.2009.02.006.

[28]

A. de Laire, Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition at infinity, Comm. Partial Differential Equations, 35 (2010), 2021-2058.  doi: 10.1080/03605302.2010.497200.

[29]

A. de Laire, Nonexistence of traveling waves for a nonlocal Gross-Pitaevskii equation, Indiana Univ. Math. J., 61 (2012), 1451-1484.  doi: 10.1512/iumj.2012.61.4707.

[30]

A. de Laire and P. Gravejat, Stability in the energy space for chains of solitons of the Landau-Lifshitz equation, J. Differential Equations, 258 (2015), 1-80.  doi: 10.1016/j.jde.2014.09.003.

[31]

A. de Laire and P. Gravejat, The Sine-Gordon regime of the Landau-Lifshitz equation with a strong easy-plane anisotropy, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1885-1945.  doi: 10.1016/j.anihpc.2018.03.005.

[32]

J. DenschlagJ. E. SimsarianD. L. FederC. W. ClarkL. A. CollinsJ. CubizollesL. DengE. W. HagleyK. Helmerson and W. P. Reinhardt, Generating solitons by phase engineering of a Bose-Einstein condensate, Science, 287 (2000), 97-101.  doi: 10.1126/science.287.5450.97.

[33]

C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Comm. Partial Differential Equations, 33 (2008), 729-771.  doi: 10.1080/03605300802031614.

[34]

P. Gérard, The Gross-Pitaevskii equation in the energy space, Stationary and Time Dependent Gross-Pitaevskii Equations, 129–148, Contemp. Math., 473, Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09226.

[35]

P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004.

[36]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[37]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.  doi: 10.1007/BF01214768.

[38]

L. Grafakos, Classical Fourier Analysis, volume 249 of Graduate Texts in Mathematics, Springer, New York, second edition, 2008.

[39]

P. Gravejat and D. Smets, Asymptotic stability of the black soliton for the Gross-Pitaevskii equation, Proc. Lond. Math. Soc. (3), 111 (2015), 305-353.  doi: 10.1112/plms/pdv025.

[40]

E. P. Gross, Hydrodynamics of a superfluid condensate, Journal of Mathematical Physics, 4 (1963), 195-207.  doi: 10.1063/1.1703944.

[41]

S. GustafsonK. Nakanishi and T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré, 8 (2007), 1303-1331.  doi: 10.1007/s00023-007-0336-6.

[42]

S. GustafsonK. Nakanishi and T.-P. Tsai, Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math., 11 (2009), 657-707.  doi: 10.1142/S0219199709003491.

[43]

Y. V. Kartashov and L. Torner, Gray spatial solitons in nonlocal nonlinear media, Opt. Lett., 32 (2007), 946-948.  doi: 10.1364/OL.32.000946.

[44]

R. KillipT. OhO. Pocovnicu and M. Vişan, Global well-posedness of the Gross-Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions, Math. Res. Lett., 19 (2012), 969-986.  doi: 10.4310/MRL.2012.v19.n5.a1.

[45]

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein and T. Pfau, The physics of dipolar bosonic quantum gases, Reports on Progress in Physics, 72 (2009), 126401. doi: 10.1088/0034-4885/72/12/126401.

[46]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.

[47]

Z. Lin, Stability and instability of traveling solitonic bubbles, Adv. Differential Equations, 7 (2002), 897-918. 

[48]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[49]

O. Lopes and M. Mariş, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal., 254 (2008), 535-592.  doi: 10.1016/j.jfa.2007.10.004.

[50]

Y. Luo and A. Stylianou, Ground states for a nonlocal cubic-quartic Gross-Pitaevskii equation, Preprint, http://arXiv.org/abs/1806.00697.

[51]

M. Mariş, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, Ann. of Math. (2), 178 (2013), 107-182.  doi: 10.4007/annals.2013.178.1.2.

[52]

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Figure 1.  Curve $ E_ \text{min} $ and solitons in the case $ {\cal W} = \delta_0 $
Figure 2.  Curve $ E_ \text{min} $ and solitons for the potential in (7.1), with $ \alpha = 0.05 $ and $ \beta = 0.15 $
Figure 3.  Curve $ E_ \text{min} $ and solitons for the potential in (7.2), with $ \alpha = 0.8 $
Figure 4.  Curve $ E_ \text{min} $ and solitons for the potential in (7.3), with $ \sigma = 10 $
Figure 5.  Dispersion curve associated with potential (7.4), with $ a = -36 $, $ b = 2687 $, $ c = 30 $. Here $ \xi_m\sim 0.33 $ and $ \xi_r\sim 0.53 $
Figure 6.  Curves $ E_ \text{min} $ and solitons for the potential in (7.4), with $ a = -36 $, $ b = 2687 $, $ c = 30 $
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