1. Solitons in relativistic mean field models
David Augaitis Fogaça
and
Fernando S. Navarra

2. Introduction and Motivation
Hydrodynamics gives a good description of nuclear matter
in many situations
Fast particles traversing nuclear matter may have supersonic motion :
ex. 1: hard parton (jet starter) in hot and dense matter at RHIC
ex. 2: fast proton hitting and entering a heavy nucleus
They may generate shockwaves and ,under certain conditions
Korteweg de-Vries Solitons !
G.N. Fowler, S.Raha, N. Stelte, R.M. Weiner, PLB (1982)
Experimental signature: nuclear transparency at low energies
in p-A collisions
Very difficult to observe: outgoing p (in p-A collisions)
coincides with the beam

3. Supersonic flow in hot and dense matter at RHIC
J. Casalderrey-Solana, E.V. Shuryak, D. Teaney, hep-ph/

4. Supersonic flow in hot and dense matter at RHIC
Away side jet measured by STAR
shockwave ?
(maybe a soliton
in the middle?)
If the shockwave is real, then there may be a soliton too !
To be checked in the future...

5. Conditions for soliton formation
Euler equation
Continuity equation
Using:
we have:
EOS
The combination of Euler and Continuity equation gives
the KdV equation which has a solitonic solution. If
then

6. Equation of state
The search for the EOS is given by using a phenomenological description of the nuclear matter: Quantum Hadrodynamics (QHD)
QHD + non-linear terms + derivative coupling
Mean field approx. and keeping the spatial derivatives

7. Derivation of the KdV equation
The combination of Euler and continuity equations using dimensionless variables:
In the - space, given by the “stretched coordinates” :
with the following expansion:
Gives two equations that contain power series in  up to  !

8. with the analytical solution:
Since the coefficients in these series are independent we get a set of equations, which, when combined, lead to the KdV equation for :
with the analytical solution:

9. speed of sound is not << 1 relativistic corrections ?
Numerical results
Soliton width:
speed of sound is not << relativistic corrections ?

10. Relativistic Fluid Dynamics
where
Thermodynamical relations:

11. Ansatz for the energy density:
can also be found
in the following works:
Following the same procedure we find the KdV equation:
Non-relativistic limit : >> speed of sound

12. The analytical solution:
With the condition for :
and of course:

13. Conclusions and Perspectives
The nucleus can be treated as a perfect fluid, we have investigated the possibility of formation and propagation of KdV solitons in nuclear matter. We have updated the early estimates of the soliton properties, using more reliable equations of state.
The KdV equation may be found in Relativistic Fluid Dynamics, and also depends on energy density.
Future numerical approach.
Study systems at finite temperatures.
Thank you!