Gausson (physics)

Gausson is a soliton which is the solution of the Schrödinger equation with the logarithmic nonlinearity which describes a quantum particle in the possible nonlinear quantum mechanics but still preserving the dimensional homogeneity of the equation i.e. the property that the product of the independent solutions in one dimension remains the solution in two dimensions etc. despite of the nonlinearity or that the nonlinearity alone cannot cause the quantum entanglement between dimensions without the interaction and therefore it still can be solved by the separation of variables.^[1][2]

Let the nonlinear Schrödinger equation in one dimension will be given by $(\hbar = 1)$:

$i{\partial \psi \over \partial t} = -\frac{1}{ 2} \frac{\partial^2 \psi}{ \partial x^2}- a \ln |\psi|^2\psi$

Let assume the Galilean invariance i.e.

$\frac{}{}\psi(x,t)=e^{-i E t}\psi(x-k t)$

Substituting

$\frac{}{}y=x-k t$

The first equation can be written as

$-\frac{1}{ 2} \left( {\frac{\partial \psi}{ \partial y}+ik}\right)^2 -a \ln |\psi|^2 \psi=\left( E + \frac{k^2}{2} \right) \psi$

Substituting additionally

$\frac{}{}\Psi(y)=e^{-iky}\psi(y)$

and assuming

$\Psi(y)=N e^{-\omega y^2/2}$

we get the normal Schrödinger equation for the quantum harmonic oscillator:

$-\frac{1}{ 2} {\frac{\partial^2 \Psi}{ \partial y^2}} + a \omega y^2\Psi=\left( E + \frac{k^2}{2} +N^2 a \right) \Psi$

The solution is therefore the normal ground state of the harmonic oscillator if only $(a>0)$

$\frac{}{} a \omega=\omega^2/2$

or

$\frac{}{} \omega=2 a$

The full solitonic solution is therefore given by

$\frac{}{}\psi(x,t)=e^{-i E t} e^{ik{(x-kt)}}e^{-a ({x-kt})^2}$

where

$\frac{}{}E=a(1-N^2) - k^2/2$

This solution describes the soliton moving with the constant velocity and not changing the shape (modulus) of the Gaussian function.

References

1. Bialynicki-Birula, Iwo; Mycielski, Jerzy (1979). "Gaussons: Solitons of the Logarithmic Schrödinger Equation" (PDF). Physica Scripta 20 (13): 539. Bibcode:1979PhyS...20..539B. doi:10.1088/0031-8949/20/3-4/033.
2. Gāhler, R.; Klein, A. G.; Zeilinger, A. (1981). "Neutron optical tests of nonlinear wave mechanics". Physical Review A 23 (4): 1611. Bibcode:1981PhRvA..23.1611G. doi:10.1103/PhysRevA.23.1611.