Solitons and shock waves in Bose-Einstein condensates A.M. Kamchatnov*, A. Gammal**, R.A. Kraenkel*** *Institute of Spectroscopy RAS, Troitsk, Russia **Universidade de São Paulo, São Paulo, Brazil ***Instituto de Física Teórica, São Paulo, Brazil

Gross-Pitaevskii equation Dynamics of a dilute condensate is described by the Gross-Pitaevskii equation where

is the atom-atom scattering length, is number of atoms in the trap.

Cigar-shaped trap or

If then transverse motion is “frozen” and the condensate wave function can be factorized where is a harmonic oscillator ground state function of transverse motion:

The axial motion is described by the equation where

Disc-shaped trap

Quasi-one-dimensional expansion Hydrodynamic-like variables are introduced by where is density of condensate and is its velocity.

In Thomas-Fermi approximation the stationary state is described by the distributions where is axial half-length of the condensate.

After turning off the axial potential the condensate expands in self-similar way:

Analytical solution is given by where has an order of magnitude of the sound velocity in the initial state: is the density of the condensate.

Shock wave in Bose-Einstein condensate Let the initial state have the density distribution

A formal hydrodynamic solution has wave breaking points: Taking into account of dispersion effects leads to generation of oscillations in the regions of transitions from high density to low density gas.

Numerical solution of 2D Gross- Pitaevskii equation

Density profiles at y=0

Analytical theory of shocks The region of oscillations is presented as a modulated periodic wave: where

The parameterschange slowly along the shock. Their evolution is described by the Whitham modulational equations

Solution of Whitham equations has the form where functionsare determined by the Initial conditions. This solution defines implicitly as functions of

Substitution of into periodic solution gives profile of dissipationless shock wave:

Formation of dark solitons Let an initial profile of density have a “hole”

After wave breaking two shocks are formed which develop eventually into two soliton trains:

Analytical form of each emerging soliton is parameterized by an “eigenvalue” wherecan be found with the use of the generalized Bohr-Sommerfeld quantization rule

Formation of solitons in BEC with attractive interaction Solitons are formed due to modulational instability. If initial distribution of density has sharp fronts, then Whitham analytical theory can be developed.

Results of 3D numerics

1D cross sections of density distributions

Whitham theory

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