1. Waves and Solitons in Multi-component Self-gravitating Systems
Kinwah Wu
Mullard Space Science Laboratory
University College London
Curtis Saxton (MSSL, UCL)
Ignacio Ferreras (P&A, UCL)

2. Outline
Collective oscillations, non-linear waves and solitons (a brief overview)
Multi-component self-gravitating systems
Two studies: results and some thoughts
- “tsunami” & “quakes” in galaxy clusters
- solitons in self-gravitating sheets and 1D
infinite media

3. Collective phenonmona in multi-component systems
- Coupled oscillators:
different oscillation normal modes and dissipation processes
non-linear mode coupling
resonance
- Two-stream instability
- Landau damping
cf. electron-ion plasma
wave-wave interaction
particle-particle interaction
wave-particle interaction

4. Solitons as non-linear wave packets
Non-linear, non-dispersive waves:
The nonlinearity which leads to wave steeping counteracts the wave dispersion.
Interact with one another so to keep their basic identity -- particle liked
Linear superposition often not applicable
Propagation speed proportional to pulse height
An example soliton pulse profile:
pulse height
propagation speed

5. Situations that give rise to solitons
- fluids with surface tension
- ion acoustic oscillations in plasmas
- etc
Kadomstev-Petviasgvili (KP) Equation
Korteweg-de Vries (KdV) Equation
mathematical methods developed to solve various soliton equations
e.g. Baecklund transformation, inverse scattering method,
Zakharov-Shabat method ……

6. Mutiple solitons
development of “soliton trains” (Zabusky & Kruskal 1965)
kdV equation with a cosine-like function as initial condition
component solitions with larger pulse height travel faster
a train of solitions lined up with the tallest leading the way
=> “big” solitions are more likely to find each other
depending on dimension
- resonance and phase shift
before collision
resonant states: momentum exchange
phase shift
time
after collision

7. Multi-component self-gravitating systems
The fluid description (non-rotational case)
Conservation equations:
Poisson equation:

8. Quakes and Tsunami in Galaxy Clusters

9. Galaxy clusters as “spherical” 2-component coupled oscillators
DM - momentum carrier
Gas - coolant (dissipater)
inflow
gas cooling
Solve to obtain the stationary structure

10. Perturbative analysis
Lagrange perturbation for the DM and gas components:
the perturbed Poisson equation
plus 6 coupled perturbed hydrodynamic equations
Diagonise the matrix
Define the boundary conditions
Numerically integrate the DEs
algebraic functions of hydrodynamic variables

11. Normal mode oscillations (the DM component)
mass inflow rate = 100 M_sun per year
(size ~ 1 Mpc; kT_max ~ 10 keV)
eigen-mode
- oscillation modes depend on BCs
- high order modes not damped
- different stability properties for
even and odd modes in some cases
similar eigen-planes can be generated
for the gas component

12. Quakes and tsunami in galaxy clusters: dark-matter oscillations and gas dissipation
- close proximity between clusters
--> excitation of DM oscillations, ie. cluster quakes
high-order modes are also fast growing
--> oscillations may occur in a wide range of scales
oscillations in DM coupled with oscillations in gas
cooler interior of gas (due to radiative loss)
--> slower sound speeds in the inner cluster region (“cooling flow” core)
--> waves piled up when propagating inward, ie. cluster tsunami
mode cascade
--> inducing turbulence and hence heating of the cluster throughout
cf. “original” cluster tsunami model of Fujita, Suzaki & Wada (2004) and
Fujita, Matsumoto & Wada (2005)
stationary DM provide background potential (ie. no quake), waves in gas
piled up when propagating inward (“self-excited” tsunami only)

13. Solitons in self-gravitating sheets and 1D infinite media

14. Planar systems: two component infinite self-gravitating sheets
Suppose: 1. inertia of one component is unimportant
2. the component is approximately isothermal
3. polytropic EOS
“quasi-1D” approximation

15. Perturbative expansion
constant yet to
be determined
Use two new variables:

16. Formation of solitons KdV equation --> soliton solution
effective sound speed
rescaling variables
KdV equation --> soliton solution
cf. Solitons in single-component self-gravitating systems
(Semekin et al. 2001)

17. Some thoughts and questions:
2 colliding DM solitons
resonant state
Suppose the resonant half life
Q1: Are ridge solitons manifested as filaments in cosmic
sheets?
Q2: Can soliton collisions make globular clusters?

18. Summary Collective and non-linear oscillations, which may not be
present in single component self-gravitating systems,
could be important in multi-component systems.
DM and gas play different roles in exciting and sustaining
oscillations in astrophysical systems.
Galaxy clusters can be considered as couple oscillators
with DM as the mode resonant medium and gas as the
energy dissipater. Gas in clusters can be compression
heated by acoustic coupling with the DM oscillations.
Solitons can be excited in DM/gas sheets and infinite
self-gravitating systems, and they could lead to “bright”
structure formation, provided that certain dynamical
conditions are satisfied.