# 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] quantum optics,[2] nuclear physics,[3][4] transport and diffusion phenomena,[5][6] open quantum systems and information theory,[7][8][9][10][11][12] effective quantum gravity and physical vacuum models[13][14][15] and theory of superfluidity and Bose–Einstein condensation.[16] Its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) was first proposed by G. Rosen.[17] 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:

$i \frac{\partial \psi}{\partial t} + \Delta \psi + \psi \ln |\psi|^2 = 0.$

for the complex-valued function $\psi=\psi (\mathrm{\mathbf{x}},t)$. Here $\Delta\,$ is the Laplacian with respect to the vector $\mathrm{\mathbf{x}}$.

The relativistic version of this equation can be obtained by replacing the derivative operator with the D'Alembertian, similarly to the Klein–Gordon equation.

## References

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7. ^ K. Yasue, Quantum mechanics of nonconservative systems, Annals Phys. 114 (1978) 479.
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12. ^ J. L. Lopez, Phys. Rev. E. 69 (2004) 026110.
13. ^ K. G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol. 16 (2010) 288–297 ArXiv:0906.4282.
14. ^ K. G. Zloshchastiev, Vacuum Cherenkov effect in logarithmic nonlinear quantum theory, Phys. Lett. A 375 (2011) 2305–2308 ArXiv:1003.0657.
15. ^ K. G. Zloshchastiev, Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory, Acta Phys. Polon. B 42 (2011) 261–292 ArXiv:0912.4139.
16. ^ A. V. Avdeenkov and K.G. Zloshchastiev, Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent, J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 195303 ArXiv:1108.0847.
17. ^ G. Rosen, Phys. Rev. 183 (1969) 1186.