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Asymptotic stability of the black soliton for the Gross-Pitaevskii equation. (English) Zbl 1326.35346

This paper considers finite-energy solutions of the Gross-Pitaevskii (GP) equation \[ i\partial_t \Psi+\partial_{xx}\Psi+\Psi(1-|\Psi|^2)=0, \] where \(\Psi\) is a complex-valued function of \((x,t)\in \mathbb{R}^2\) satisfying the boundary condition \(|\Psi(x,t)|\to 1\) as \(|x|\to +\infty.\) Solitons in this paper are travelling wave solutions of the form \(\Psi(x,t)=U_c(x-ct).\) There exist finite-energy non-constant solitons of speed \(c\) if and only if \(|c|<\sqrt{2},\) which are uniquely given by the formula \[ U_c(x)=\sqrt{\dfrac{2-c^2}{2}}\text{tanh}\left(\dfrac{\sqrt{2-c^2}}{2}x \right)+i\dfrac{c}{\sqrt{2}}, \] up to the invariances of the problem, that is, multiplication by a constant of modulus one and translation. The particular solution when the speed \(c=0\) is called the black soliton of the GP equation.
This paper studies the nonlinear stability of the black soliton. The first result of this paper concerns the orbital stability of the black soliton in the energy space. The proof relies on a variational approach, in the spirit of the work by M. I. Weinstein [Commun. Pure Appl. Math. 39, 51–67 (1986; Zbl 0594.35005)] and M. Grillakis et al. [J. Funct. Anal. 74, 160–197 (1987; Zbl 0656.35122)]. The main ingredient in the proof is to establish the coercivity of the energy functional. The second result concerns the asymptotic stability of the black soliton. The proof of the asymptotic stability uses ideas and techniques developed by Y. Martel and F. Merle [Math. Ann. 341, No. 2, 391–427 (2008; Zbl 1153.35068)] for the generalized Korteweg-de Vries equation.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C08 Soliton solutions
35C07 Traveling wave solutions
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[1] Béthuel, Existence and properties of travelling waves for the Gross–Pitaevskii equation, in: Stationary and time dependent Gross–Pitaevskii equations pp 55– (2008)
[2] DOI: 10.1007/s00220-008-0614-2 · Zbl 1190.35196
[3] DOI: 10.1512/iumj.2008.57.3632 · Zbl 1171.35012
[4] DOI: 10.5802/aif.2838 · Zbl 1337.35131
[5] Béthuel, Asymptotic stability in the energy space for dark solitons of the Gross–Pitaevskii equation, Ann. Sci. École Norm. Sup. (2015)
[6] Béthuel, Vortex rings for the Gross–Pitaevskii equation, J. Eur. Math. Soc. 6 pp 17– (2004) · Zbl 1091.35085
[7] Béthuel, Travelling waves for the Gross–Pitaevskii equation I, Ann. Inst. H. Poincaré Phys. Théor. 70 pp 147– (1999)
[8] Buslaev, Scattering for the nonlinear Schrödinger equation: states close to a soliton, St. Petersburg Math. J. 4 pp 1111– (1993)
[9] Buslaev, On the stability of solitary waves for nonlinear Schrödinger equations, in: Nonlinear evolution equations pp 75– (1995) · Zbl 0841.35108
[10] DOI: 10.1016/S0294-1449(02)00018-5 · Zbl 1028.35139
[11] DOI: 10.1007/BF01403504 · Zbl 0513.35007
[12] DOI: 10.1088/0951-7715/25/3/813 · Zbl 1278.35226
[13] Chiron D. , Maris M. , ’Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. II’, Preprint, 2012, http://arxiv.org/abs/1203.1912 .
[14] DOI: 10.1002/cpa.1018 · Zbl 1031.35129
[15] DOI: 10.1142/S0129055X03001849 · Zbl 1084.35089
[16] Cuccagna S. , Jenkins R. , ’On asymptotic stability of N-solitons of the Gross–Pitaevskii equation’, Preprint, 2014, http://arxiv.org/abs/1410.6887 .
[17] Dunford, Linear operators. Part II. Spectral theory. Self-adjoint operators in Hilbert space (1963) · Zbl 0128.34803
[18] DOI: 10.3934/dcds.2005.13.583 · Zbl 1083.35019
[19] Escauriaza, Hardy’s uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. 10 pp 883– (2008) · Zbl 1158.35018
[20] DOI: 10.1016/j.aim.2007.04.018 · Zbl 1126.35065
[21] Gérard, The Gross–Pitaevskii equation in the energy space, in: Stationary and time dependent Gross–Pitaevskii equations pp 129– (2008)
[22] DOI: 10.1016/j.matpur.2008.09.009 · Zbl 1232.35152
[23] DOI: 10.1016/0022-1236(87)90044-9 · Zbl 0656.35122
[24] DOI: 10.1016/0022-1236(90)90016-E · Zbl 0711.58013
[25] DOI: 10.1063/1.1703944
[26] Kenig, Asymptotic stability of solitons for the Benjamin–Ono equation, Rev. Mat. Iberoamericana 25 pp 909– (2009) · Zbl 1247.35133
[27] DOI: 10.1016/S0370-1573(97)00073-2
[28] Lin, Stability and instability of traveling solitonic bubbles, Adv. Differential Equations 7 pp 897– (2002) · Zbl 1033.35117
[29] DOI: 10.4007/annals.2013.178.1.2 · Zbl 1315.35207
[30] DOI: 10.1137/050637510 · Zbl 1126.35055
[31] DOI: 10.1016/S0021-7824(00)00159-8 · Zbl 0963.37058
[32] DOI: 10.1007/s002050100138 · Zbl 0981.35073
[33] DOI: 10.1088/0951-7715/18/1/004 · Zbl 1064.35171
[34] DOI: 10.1007/s00208-007-0194-z · Zbl 1153.35068
[35] Martel, Refined asymptotics around solitons for the gKdV equations with a general nonlinearity, Discrete Contin. Dynam. Syst. 20 pp 177– (2008)
[36] DOI: 10.1007/s00220-002-0723-2 · Zbl 1017.35098
[37] DOI: 10.1215/S0012-7094-06-13331-8 · Zbl 1099.35134
[38] DOI: 10.1137/S0036141098346827 · Zbl 0981.35066
[39] DOI: 10.1007/BF02101705 · Zbl 0805.35117
[40] Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13 pp 451– (1961)
[41] Soffer, Multichannel nonlinear scattering theory for nonintegrable equations, in: Integrable systems and applications pp 312– (1989) · Zbl 0712.35074
[42] DOI: 10.1007/BF02096557 · Zbl 0721.35082
[43] DOI: 10.1016/0022-0396(92)90098-8 · Zbl 0795.35073
[44] DOI: 10.1023/A:1021179311172 · Zbl 1080.35060
[45] Vartanian, Long-time asymptotics of solutions to the Cauchy problem for the defocusing non-linear Schrödinger equation with finite-density initial data. I. Solitonless sector, in: Recent developments in integrable systems and Riemann–Hilbert problems pp 91– (2003) · Zbl 1061.35140
[46] DOI: 10.1137/0516034 · Zbl 0583.35028
[47] DOI: 10.1002/cpa.3160390103 · Zbl 0594.35005
[48] Zakharov, Interaction between solitons in a stable medium, Sov. Phys. JETP 37 pp 823– (1973)
[49] Zhidkov, Korteweg–De Vries and nonlinear Schrödinger equations: qualitative theory (2001)
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