Nonlinear Phenomena Soliton & Tsunami - Saha Institute of Nuclear ...

Nonlinear Phenomena Soliton & Tsunami
Anjan Kundu
Theory Division
Saha Institute of Nuclear Physics
February 21, 2005

NONLINEAR PHENOMENA SOLITON & TSUNAMI 1
Traditional Tsunami Theory
• A plane wave of long wave length
Evolution Equation for water elevation U (x,t) (x → - direction from sea to
shore):
1
v 2U(x, t) tt = U(x, t) xx ,
∂U
∂x ≡ U x, velocity v = √ gH
• Near shore Tsunami effects : i) H ↓↓ v, (slowing down of waves)
ii) due to conservation of energy Amplitude A ↑ (increase in amplitude)
➥ Therefore everything seems to be well-described
IS IT REALLY !
Few Questions about Tsunami05
➥ Tsunami traveled: from epicenter of (Near Banda Aceh, Northern Sumatra,
Indonesia): −→ to Port Elizabeth, South Africa
8000 km distance!
• How can a linear plane wave travel such a long distance without
dispersion ➥Not a single but a group of waves hit the shores.

NONLINEAR PHENOMENA SOLITON & TSUNAMI 2
• The individual waves have different intensities. Why
• At Banda Aceh the 2nd wave was the most intense, while at
Nagapattanam it was the 1st one:
Why switching of intensities
• At Banda Aceh the interval between the 1st & 2nd waves was 10 min.,
while at Nagapattanam it was 40 min.:
How wave-length can change
Can the traditional linear wave theory Answer them
Possibly, NO !!!
So, we have to look for some other explanation.
⋄ Let us make together a time travel of 30 years near the same spot.
Andaman Sea: October 1976
Northern Sumatra Indonesia !! ( Science, 1980, vol 208, 451 , (by A. R.
Osborn ( Research Physicist at Exon Prod. research Com., Texas) & T Burch
Oceanographer, Massachusets)

NONLINEAR PHENOMENA SOLITON & TSUNAMI 3
Appolo-Soyuz Space-Photo

NONLINEAR PHENOMENA SOLITON & TSUNAMI 4
Experiment in the open sea : Internal waves

NONLINEAR PHENOMENA SOLITON & TSUNAMI 5
Experimental data on single Soliton

NONLINEAR PHENOMENA SOLITON & TSUNAMI 6
Single Soliton!

NONLINEAR PHENOMENA SOLITON & TSUNAMI 7
To Understand this let us make a space-time-travel →
August 1834: A canal near Edinburg, UK
Where a British Naval Engineer was riding on a horse back.

NONLINEAR PHENOMENA SOLITON & TSUNAMI 8
John Scott Russel :
(British Naval Engineer)

NONLINEAR PHENOMENA SOLITON & TSUNAMI 9
”I was observing the motion of a boat which was rapidly drawn along by
a pair of horses, when the boat suddenly stopped-not so the mass of
water... it accumulated round the prow of the vessel in a state of violent
agitation.
Then suddenly ...rolled forward with a great velocity, assuming the form of a
large solitary elevation, a rounded, smooth and well-defined heap of water.,
which continued its course along the channel apparently without change of
form and or diminution of speed.
I followed it on horse-back...thirty feet long and a foot to a foot and a half in
height..and after a chase of 1 or 2 miles I lost it.
Such in the month of August 1934 was my first chance interview with that
singular and beautiful phenomenon”

NONLINEAR PHENOMENA SOLITON & TSUNAMI 10
Mathematical formulation of this Nonlnear Phenomena came in 1895:
Korteveg & de Vries (KdV)
(Derived from Navier-Stokes eqn)

NONLINEAR PHENOMENA SOLITON & TSUNAMI 10
Mathematical formulation of this Nonlnear Phenomena came in 1895:
Korteveg & de Vries (KdV)
(Derived from Navier-Stokes eqn)
∂u
∂t = ∂3 u
∂x + u∂u
3 ∂x
Where the the ’repulsive’ dispersive term finely balances the ’attractive’
nonlinear interaction term. Hence emerges
☞ Stable localised solution
☞ Preserves its shape and velocity during collition with other such objects!!
(Behaves like ’particles’ under interaction)
☞ This nonlinear nonperturbative exact slution is called Soliton

NONLINEAR PHENOMENA SOLITON & TSUNAMI 11
Further advancement:
Nonlinear Integrable Systems
☞ Allow exact nonperturbative solutions
☞ Infinite number of conserved quantities
☞ Apart from the KdV many other nonlinear equations belong to this class
☞ Theory was laid by stalwarts like Liouville, Poincaré, Painleveé and
Kovalewskaya

NONLINEAR PHENOMENA SOLITON & TSUNAMI 12
Sophia Kovalewskaya :
(Russian Mathematician)

NONLINEAR PHENOMENA SOLITON & TSUNAMI 13
Triumph of Integrable Systems
Over the years it was understood that our world is interacting & hence
Nonlinear
☞ Many natural phenomena are described by it:
starting from ocean waves to atmospheric vortices,
from nerve signal propagation to plasma soliton,
from cosmic objects to knot theory

NONLINEAR PHENOMENA SOLITON & TSUNAMI 14
Knots & Links possess Solitonic features

NONLINEAR PHENOMENA SOLITON & TSUNAMI 15
Universality of Nonlinear Integrable System
Apart from KdV a host of other systems found to show the same featires:
Integrability & Soliton solution
•Modified KdV
u t = u xxx + u 2 u x , u = u(x, t), u t ≡ ∂u
∂t
• Nonlinear Schrödinger equation (NLS)
• Sine-Gordon equation
• Liouville equation
iψ t = ψ xx + 2(ψ † ψ)ψ = 0, ψ = ψ(x, t)
u tt − u xx = m 2 sin u, u ≡ u(x, t)
u tt − u xx = e ηu u ≡ u(x, t) etc.

NONLINEAR PHENOMENA SOLITON & TSUNAMI 16
☞ All these models share common properties & solved using Inverse
Scattering method
People involved:Kruskal-Zabuztki-Ablowitz-Zakharov-Novikov (1960-’85),
etc. Next breakthrough: Quantum Integrable Systems
☞ Eigenvalue problem could be solved exactly and nonperturbatively for a
family of Interacting quantum many-body problems:
• Isotropic Heisenburg spin chain( Hans Bethe, 1931
H =
N∑
(˜σ n˜σ n+1 , H|m >= E m |m >
n
• Anisotropic spin chains, spin ladder models
• Jayenes-Cummings type model
• Electron models like: Hubbard model & t-J model, etc ☞ All models are
solved using Quantum Inverse Scattering method
People involved:Bethe-Baxter-Faddeev-Sklyanin-Takhtajan-Jimbo
(1979-’90) etc.

NONLINEAR PHENOMENA SOLITON & TSUNAMI 17
How to find the Soliton solution
KdV equation
Inverse Scattering Theory
• The Idea:
Nonlinear equation ⇒ Linear Scattering system
⇒ Evolution of Scattering data: (reflection (r(k)) & transmission t(i) coeff.)
⇒
Mapping back to soln. of Nonlinear eqn.
(r ∼ ’radiative waves’, t(i) ∼ ’Solitons’)

NONLINEAR PHENOMENA SOLITON & TSUNAMI 18
Solitons! + radiative ripples
☞ We can find exact analytic solutions!

NONLINEAR PHENOMENA SOLITON & TSUNAMI 19
1-Soliton solution:
u(x, t) = Asech 2 ( x − vt
2
), A ∼ v, ∆ ∼ √
∆ v

NONLINEAR PHENOMENA SOLITON & TSUNAMI 20
2-Soliton solution :
u(x, t) = A v 2cosech 2 (ξ 2 ) + v 1 sech 2 (ξ 1 )
( √ v 2 coth(ξ 2 ) − √ v 1 tanh(ξ 1 )) 2, ξ i ≡ x − v it
∆ i

NONLINEAR PHENOMENA SOLITON & TSUNAMI 21
☞ Hence we could answer & explain through Nonlinear Phenomena related
KdV Solitons two mysterious observational facts anout Tsunami’05, which
the well-accepted traditional linear theory could not answer !
We have to see now whether the important near shore Tsunami effects can
also be explained by KdV soliton.
For constant sea-depth H the KdV equation:
u t = √ Hu x + H 5 2 uxxx + + 1 √
H
uu x
Therefore, the Soliton solution is of the same form, only
Now the Soliton velocity V = v √ H
☞ Therefore near-shore H ↓, hence V ↓, i.e. slowing down of waves ∼ exactly
like in standard Tsunami model
☞To verify the last near-shore feature (i.e. Tsunami effect = increase in
amplitute) for KdV solitons we do some calculation (qualitative) for
propagation of soliton with different near-shore sea-depth profile H(x)

NONLINEAR PHENOMENA SOLITON & TSUNAMI 22
And the result:
Shallow sea-depth profile

NONLINEAR PHENOMENA SOLITON & TSUNAMI 23
Breaking wave: less devastation !!!
Stiff sea-depth profile:

NONLINEAR PHENOMENA SOLITON & TSUNAMI 24
Concluding Remarks
• KdV soliton can explain apparentlyNot Only the traditional near-shore
Tsunami effects as in the traditional linear theory,
• But Also can explain possibly some curious observed facts on varying
Tsunami wave lengths & intensities, which
CAN NOT BE ANSWERED BY LINEAR THEORY!
• Therefore for conclusive evidence we have to collect similar relevant datas
on varying Tsunami wave lengths & intensities from all other Tsunami events
(if these are available !!)