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 & Continuous Dynamical Systems - A, 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.   Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

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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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