In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schrödinger equation in presence of a smooth external potential.
A.H.Ardila, Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations335 (2016), 1–9.
2.
A.H.Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Analysis155 (2017), 52–64. doi:10.1016/j.na.2017.01.006.
3.
F.A.Berezin and M.Shubin, The Schrödinger Equation, Kluwer Acad. Publ., Dordrecht, 1991.
4.
I.Bialynicki-Birula and J.Mycielski, Nonlinear wave mechanics, Ann. Phys100 (1976), 62–93. doi:10.1016/0003-4916(76)90057-9.
5.
P.Blanchard and J.Stubbe, Stability of ground states for nonlinear classical field theories, in: Lecture Notes in Physics, Vol. 347, Springer, Heidelberg, 1989, pp. 19–35.
6.
P.Blanchard, J.Stubbe and L.Vázquez, On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri-Poncaré, Phys. Théor.47 (1987), 309–336.
7.
J.Bronski and R.Jerrard, Soliton dynamics in a potential, Math. Res. Letters7 (2000), 329–342. doi:10.4310/MRL.2000.v7.n3.a7.
8.
H.Buljan, A.Siber, M.Soljacic, T.Schwartz, M.Segev and D.N.Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E68 (2003), Article ID 036607. doi:10.1103/PhysRevE.68.036607.
9.
R.Carles and I.Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, 2016, preprint, https://hal.archives-ouvertes.fr/hal-01398526.
10.
T.Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal.7 (1983), 1127–1140. doi:10.1016/0362-546X(83)90022-6.
11.
T.Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003.
12.
T.Cazenave and A.Haraux, Equations d’évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math.2 (1980), 21–51. doi:10.5802/afst.543.
13.
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.
14.
P.d’Avenia, E.Montefusco and M.Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math.16 (2014), Article ID 1350032. doi:10.1142/S0219199713500326.
15.
P.d’Avenia and M.Squassina, Soliton dynamics for the Schrödinger–Newton system, Math. Models Methods Appl. Sci.24 (2014), 553–572. doi:10.1142/S0218202513500590.
16.
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. doi:10.1209/epl/i2003-00547-6.
17.
E.F.Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev.32 (1985), 1201–1204. doi:10.1103/PhysRevA.32.1201.
18.
T.Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966.
19.
S.Keraani, Semiclassical limit for nonlinear Schrödinger equation with potential. II, Asymptotic Anal.47 (2006), 171–186.
20.
S.Le Coz, Standing waves in nonlinear Schrödinger equations, in: Analytical and Numerical Aspects of Partial Differential Equations, Walter de Gruyter, Berlin, 2009, pp. 151–192.
21.
E.Lieb and M.Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.
22.
E.Montefusco, B.Pellacci and M.Squassina, Soliton dynamics for CNLS systems with potentials, Asymptot. Anal.66 (2010), 61–86.
23.
S.Secchi and M.Squassina, Soliton dynamics for fractional Schrödinger equation, Appl. Anal.93 (2014), 1702–1729. doi:10.1080/00036811.2013.844793.
24.
A.Selvitella, Asymptotic evolution for the semiclassical nonlinear Schrödinger equation in presence of electric and magnetic fields, J. Differential Equations245 (2008), 2566–2584. doi:10.1016/j.jde.2008.05.012.
25.
M.A.Shubin, Partial Differential Equations VII: Spectral Theory of Differential Operators, Encyclopaedia of Mathematical Sciences, Vol. 64, Springer-Verlag, Berlin, 1994. doi:10.1007/978-3-662-06719-2_1.
26.
M.Squassina, Soliton dynamics for the nonlinear Schrödinger equation with magnetic field, Manuscripta Math.130 (2009), 461–494. doi:10.1007/s00229-009-0307-y.
27.
M.Weinstein, Modulation stability of ground state of nonlinear Schrödinger equations, SIAM J. Math. Anal.16 (1985), 472–491. doi:10.1137/0516034.
28.
M.Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure. Appl. Math39 (1986), 51–68. doi:10.1002/cpa.3160390103.
29.
J.Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys.51 (2000), 498–503. doi:10.1007/PL00001512.
30.
K.G.Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol.16 (2010), 288–297. doi:10.1134/S0202289310040067.
31.
K.G.Zloshchastiev, Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory, Acta Physica Polonica B42 (2011), 261–292. doi:10.5506/APhysPolB.42.261.
