Logarithmic Schrödinger equation

In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mechanics,[1][2][3] quantum optics,[4] nuclear physics,[5][6] transport and diffusion phenomena,[7][8] open quantum systems and information theory,[9][10] [11][12][13][14] effective quantum gravity and physical vacuum models[15][16][17][18] and theory of superfluidity and Bose–Einstein condensation.[19][20] Its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) was first proposed by Gerald Rosen.[21] It is an example of an integrable model.

The equation

The logarithmic Schrödinger equation is the partial differential equation. In mathematics and mathematical physics one often uses its dimensionless form:

for the complex-valued function ψ = ψ(x, t) of the particles position vector x = (x, y, z) at time t, and

is the Laplacian of ψ in Cartesian coordinates. The logarithmic term has been shown indispensable in determining the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures.[22] In spite of the logarithmic term, it has been shown in the case of central potentials, that even for non-zero angular momentum, the LogSE retains certain symmetries similar to those found in its linear counterpart, making it potentially applicable to atomic and nuclear systems.[23]

The relativistic version of this equation can be obtained by replacing the derivative operator with the D'Alembertian, similarly to the Klein–Gordon equation. Soliton-like solutions known as Gaussons figure prominently as analytical solutions to this equation for a number of cases.

See also

References
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  2. Białynicki-Birula, Iwo; Mycielski, Jerzy (1975). "Uncertainty relations for information entropy in wave mechanics". Communications in Mathematical Physics. 44 (2): 129–132. Bibcode:1975CMaPh..44..129B. doi:10.1007/BF01608825. ISSN 0010-3616.
  3. Bialynicki-Birula, Iwo; Mycielski, Jerzy (1979). "Gaussons: Solitons of the Logarithmic Schrödinger Equation". Physica Scripta. 20 (3–4): 539–544. Bibcode:1979PhyS...20..539B. doi:10.1088/0031-8949/20/3-4/033. ISSN 0031-8949.
  4. Buljan, H.; Šiber, A.; Soljačić, M.; Schwartz, T.; Segev, M.; Christodoulides, D. N. (2003). "Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media". Physical Review E. 68 (3). doi:10.1103/PhysRevE.68.036607. ISSN 1063-651X.
  5. Hefter, Ernst F. (1985). "Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics". Physical Review A. 32 (2): 1201–1204. doi:10.1103/PhysRevA.32.1201. ISSN 0556-2791. PMID 9896178.
  6. Kartavenko, V. G.; Gridnev, K. A.; Greiner, W. (1998). "Nonlinear Effects in Nuclear Cluster Problem". International Journal of Modern Physics E. 07 (2): 287–299. arXiv:nucl-th/9907015. doi:10.1142/S0218301398000129. ISSN 0218-3013.
  7. Martino, S. De; Falanga, M; Godano, C; Lauro, G (2003). "Logarithmic Schrödinger-like equation as a model for magma transport". Europhysics Letters (EPL). 63 (3): 472–475. doi:10.1209/epl/i2003-00547-6. ISSN 0295-5075.
  8. Hansson, T.; Anderson, D.; Lisak, M. (2009). "Propagation of partially coherent solitons in saturable logarithmic media: A comparative analysis". Physical Review A. 80 (3). doi:10.1103/PhysRevA.80.033819. ISSN 1050-2947.
  9. Yasue, Kunio (1978). "Quantum mechanics of nonconservative systems". Annals of Physics. 114 (1–2): 479–496. doi:10.1016/0003-4916(78)90279-8. ISSN 0003-4916.
  10. Lemos, Nivaldo A. (1980). "Dissipative forces and the algebra of operators in stochastic quantum mechanics". Physics Letters A. 78 (3): 239–241. doi:10.1016/0375-9601(80)90080-8. ISSN 0375-9601.
  11. Brasher, James D. (1991). "Nonlinear wave mechanics, information theory, and thermodynamics". International Journal of Theoretical Physics. 30 (7): 979–984. doi:10.1007/BF00673990. ISSN 0020-7748.
  12. Schuch, Dieter (1997). "Nonunitary connection between explicitly time-dependent and nonlinear approaches for the description of dissipative quantum systems". Physical Review A. 55 (2): 935–940. doi:10.1103/PhysRevA.55.935. ISSN 1050-2947.
  13. M. P. Davidson, Nuov. Cim. B 116 (2001) 1291.
  14. López, José L. (2004). "Nonlinear Ginzburg-Landau-type approach to quantum dissipation". Physical Review E. 69 (2). doi:10.1103/PhysRevE.69.026110. ISSN 1539-3755.
  15. Zloshchastiev, K. G. (2010). "Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences". Gravitation and Cosmology. 16 (4): 288–297. arXiv:0906.4282. doi:10.1134/S0202289310040067. ISSN 0202-2893.
  16. Zloshchastiev, Konstantin G. (2011). "Vacuum Cherenkov effect in logarithmic nonlinear quantum theory". Physics Letters A. 375 (24): 2305–2308. arXiv:1003.0657. Bibcode:2011PhLA..375.2305Z. doi:10.1016/j.physleta.2011.05.012. ISSN 0375-9601.
  17. Zloshchastiev, K.G. (2011). "Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory". Acta Physica Polonica B. 42 (2): 261–292. arXiv:0912.4139. Bibcode:2011AcPPB..42..261Z. doi:10.5506/APhysPolB.42.261. ISSN 0587-4254.
  18. Scott, T.C.; Zhang, Xiangdong; Mann, Robert; Fee, G.J. (2016). "Canonical reduction for dilatonic gravity in 3 + 1 dimensions". Physical Review D. 93 (8): 084017. arXiv:1605.03431. Bibcode:2016PhRvD..93h4017S. doi:10.1103/PhysRevD.93.084017.
  19. Avdeenkov, Alexander V; Zloshchastiev, Konstantin G (2011). "Quantum Bose liquids with logarithmic nonlinearity: self-sustainability and emergence of spatial extent". Journal of Physics B: Atomic, Molecular and Optical Physics. 44 (19): 195303. arXiv:1108.0847. Bibcode:2011JPhB...44s5303A. doi:10.1088/0953-4075/44/19/195303. ISSN 0953-4075.
  20. Zloshchastiev, Konstantin G. (2019). "Temperature-driven dynamics of quantum liquids: Logarithmic nonlinearity, phase structure and rising force". Int. J. Mod. Phys. B. 33 (17): 1950184. arXiv:2001.04688. doi:10.1142/S0217979219501844.
  21. Rosen, Gerald (1969). "Dilatation Covariance and Exact Solutions in Local Relativistic Field Theories". Physical Review. 183 (5): 1186–1188. doi:10.1103/PhysRev.183.1186. ISSN 0031-899X.
  22. Scott, T. C.; Zloshchastiev, K. G. (2019). "Resolving the puzzle of sound propagation in liquid helium at low temperatures". Low Temperature Physics. 45 (12): 1231–1236. arXiv:2006.08981. doi:10.1063/10.0000200.
  23. Shertzer, J.; Scott, T.C. (2020). "Solution of the 3D logarithmic Schrödinger equation with a central potential,". J. Phys. Commun. 4 (6): 065004. doi:10.1088/2399-6528/ab941d.
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