Universal dynamics for the defocusing logarithmic Schrödinger equation. (English) Zbl 1394.35467

Summary: We consider the Schrödinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time, and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in working in hydrodynamical variables to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker-Planck operator.


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
35Q31 Euler equations
35Q84 Fokker-Planck equations
Full Text: DOI arXiv Euclid


[1] M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Stand. Appl. Math. Ser. 55, U.S. Government, Washington, D.C., 1964. · Zbl 0171.38503
[2] T. Alazard and R. Carles, Loss of regularity for supercritical nonlinear Schrödinger equations, Math. Ann. 343 (2009), 397-420. · Zbl 1161.35047 · doi:10.1007/s00208-008-0276-6
[3] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Panor. Synthèses 10, Soc. Math. France, Paris, 2000. · Zbl 0982.46026
[4] P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys. 287 (2009), 657-686. · Zbl 1177.82127 · doi:10.1007/s00220-008-0632-0
[5] P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Ration. Mech. Anal. 203 (2012), 499-527. · Zbl 1290.76165 · doi:10.1007/s00205-011-0454-7
[6] A. H. Ardila, Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations 2016 (2016), no. 335. · Zbl 1358.35163
[7] A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations 26 (2001), 43-100. · Zbl 0982.35113 · doi:10.1081/PDE-100002246
[8] J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys. 25 (1984), 3270-3273. · Zbl 0554.35123 · doi:10.1063/1.526074
[9] I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics 100 (1976), 62-93.
[10] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Int. Math. Res. Not. IMRN 1996, no. 6, 277-304. · Zbl 0934.35166 · doi:10.1155/S1073792896000207
[11] H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev, and D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E 68 (2003), no. 036697.
[12] R. Carles, R. Danchin, and J.-C. Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity 25 (2012), 2843-2873. · Zbl 1251.35142
[13] R. Carles and L. Miller, Semiclassical nonlinear Schrödinger equations with potential and focusing initial data, Osaka J. Math. 41 (2004), 693-725. · Zbl 1155.35090
[14] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7 (1983), 1127-1140. · Zbl 0529.35068 · doi:10.1016/0362-546X(83)90022-6
[15] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Amer. Math. Soc., Providence, 2003. · Zbl 1055.35003
[16] T. Cazenave and A. Haraux, Équations d’évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (6) 2 (1980), 21-51. · Zbl 0411.35051
[17] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), 39-113. · Zbl 1197.35265 · doi:10.1007/s00222-010-0242-2
[18] M. Combescure and D. Robert, Coherent States and Applications in Mathematical Physics, Theoret. Math. Phys., Springer, Dordrecht, 2012. · Zbl 1243.81004
[19] P. d’Avenia, E. Montefusco, and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math. 16 (2014), no. 1350032. · Zbl 1292.35259
[20] L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and B.G.K. equations, Arch. Ration. Mech. Anal. 110 (1990), 73-91. · Zbl 0705.76070 · doi:10.1007/BF00375163
[21] E. Feireisl and H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal. 150 (1999), 77-96. · Zbl 0946.35058
[22] T. Gallay and G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations 150 (1998), 42-97. · Zbl 0913.35086 · doi:10.1006/jdeq.1998.3459
[23] T. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \({\mathbf{R}}^{2}\), Arch. Ration. Mech. Anal. 163 (2002), 209-258. · Zbl 1042.37058
[24] T. Gallay and C. E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation, Comm. Math. Phys. 255 (2005), 97-129. · Zbl 1139.35084 · doi:10.1007/s00220-004-1254-9
[25] T. Gallay and C. E. Wayne, Long-time asymptotics of the Navier-Stokes equation in \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\), ZAMM Z. Angew. Math. Mech. 86 (2006), 256-267. · Zbl 1094.35089
[26] I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptot. Anal. 14 (1997), 97-116. · Zbl 0877.76087
[27] P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE 5 (2012), 1139-1155. · Zbl 1268.35013 · doi:10.2140/apde.2012.5.1139
[28] P. Gérard and S. Grellier, An explicit formula for the cubic Szegö equation, Trans. Amer. Math. Soc. 367, no. 4 (2015), 2979-2995. · Zbl 1318.37024 · doi:10.1090/S0002-9947-2014-06310-1
[29] P. Gérard and S. Grellier, The cubic Szegő equation and Hankel operators, Astérisque 389, Soc. Math. France, Paris, 2017. · Zbl 1410.37001
[30] J. Ginibre, “An introduction to nonlinear Schrödinger equations” in Nonlinear Waves (Sapporo, 1995), GAKUTO Internat. Ser. Math. Sci. Appl. 10, Gakkōtosho, Tokyo, 1997, 85-133. · Zbl 0891.35146
[31] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations, III: Gevrey spaces and low dimensions, J. Differential Equations 175 (2001), 415-501. · Zbl 0991.35057 · doi:10.1006/jdeq.2000.3969
[32] M. Guardia, E. Haus, and M. Procesi, Growth of Sobolev norms for the analytic NLS on \(\mathbb{T}^{2}\), Adv. Math. 301 (2016), 615-692. · Zbl 1353.35260 · doi:10.1016/j.aim.2016.06.018
[33] M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS) 17 (2015), 71-149. · Zbl 1311.35284 · doi:10.4171/JEMS/499
[34] P. Guerrero, J. L. López, and J. Montejo-Gámez, A wavefunction description of quantum Fokker-Planck dissipation: Derivation and numerical approximation of transient dynamics, J. Phys. A 47 (2014), no. 035303. · Zbl 1293.82018
[35] P. Guerrero, J. L. López, and J. Nieto, Global \(H^{1}\) solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal. Real World Appl. 11 (2010), 79-87. · Zbl 1180.81071
[36] G. A. Hagedorn, Semiclassical quantum mechanics, I: The \(ℏ→0\) limit for coherent states, Comm. Math. Phys. 71 (1980), 77-93.
[37] Z. Hani, B. Pausader, N. Tzvetkov, and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi 3 (2015), no. e4. · Zbl 1326.35348 · doi:10.1017/fmp.2015.5
[38] N. Hayashi, T. Mizumachi, and P. I. Naumkin, Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D, Differential Integral Equations 16 (2003), 159-179. · Zbl 1031.35134
[39] N. Hayashi and P. I. Naumkin, Domain and range of the modified wave operator for Schrödinger equations with a critical nonlinearity, Comm. Math. Phys. 267 (2006), 477-492. · Zbl 1113.81121 · doi:10.1007/s00220-006-0057-6
[40] N. Hayashi and P. I. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations, Int. Math. Res. Not. IMRN 2015 (2015), 5604-5643. · Zbl 1330.35401 · doi:10.1093/imrn/rnu102
[41] E. F. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. A 32 (1985), 1201-1204.
[42] E. S. Hernández and B. Remaud, General properties of Gausson-conserving descriptions of quantal damped motion, Phys. A 105 (1981), 130-146.
[43] W. Krolikowski, D. Edmundson, and O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E 61 (2000), 3122-3126.
[44] L. Landau and E. Lifschitz, Physique théorique (“Landau-Lifchitz”), III: Mécanique quantique, Théorie non relativiste, Éditions Mir, Moscow, 1967.
[45] T. Li and D. Wang, Blowup phenomena of solutions to the Euler equations for compressible fluid flow, J. Differential Equations 221 (2006), 91-101. · Zbl 1083.76051 · doi:10.1016/j.jde.2004.12.004
[46] J. L. López and J. Montejo-Gámez, A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics, Phys. D 238 (2009), 622-644. · Zbl 1160.37430
[47] E. Madelung, Quanten Theorie in hydrodynamischer Form, Zeit. F. Physik 40 (1927), 322.
[48] S. de Martino, M. Falanga, C. Godano, and G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Lett. 63 (2003), 472-475.
[49] K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space, Commun. Pure Appl. Anal. 1 (2002), 237-252. · Zbl 1013.35056 · doi:10.3934/cpaa.2002.1.237
[50] K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space, II, Ann. Henri Poincaré 3 (2002), 503-535. · Zbl 1040.35054 · doi:10.1007/s00023-002-8626-5
[51] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (1991), 479-493. · Zbl 0742.35043 · doi:10.1007/BF02101876
[52] O. Pocovnicu, Explicit formula for the solution of the Szegö equation on the real line and applications, Discrete Contin. Dyn. Syst. 31 (2011), 607-649. · Zbl 1235.35263 · doi:10.3934/dcds.2011.31.607
[53] J. Rauch, Partial Differential Equations, Grad. Texts in Math. 128, Springer, New York, 1991. · Zbl 0742.35001
[54] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, Amer. Math. Soc., Providence, 2003. · Zbl 1106.90001
[55] H. Xu, Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z. 286 (2017), 443-489. · Zbl 1367.35159 · doi:10.1007/s00209-016-1768-9
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