1

Soliton dynamics in Soliton dynamics in a trapped a trapped

condensatecondensateLev PitaevskiiLev Pitaevskii

Kapitza Unstitute for Kapitza Unstitute for Physical Problems;Physical Problems;

University of TrentoUniversity of Trento

Nizhniy Novgorod, July Nizhniy Novgorod, July 20072007

2

Equation for the condensate Equation for the condensate wave functionwave function

2

2

22

,,

(1961) skiiL.P.Pitaev (1961), Gross E.P.

length scattering theis ,04

2

txtxn

aam

g

xUgmt

i

3

““Grey” soliton in an uniform Grey” soliton in an uniform condensatecondensate

(1971) Tsuzuki T.

,,,

tahn

2

2min22

c

v

n

n

m

gncvcuvttX

tXxmu

c

u

c

vintXx

4

Burger et al., 1999

5

Solitons in BECSolitons in BEC

Engels and Atherton, 2007

6

Energy of a solitonEnergy of a soliton

potentialchemicaltheis

)/(

3

4

3

4

2

2

32/322

mc

vmEE

ug

mvcm

gE

ss

s

7

"Number of atoms" in a "Number of atoms" in a solitonsoliton

s

s

s

EN

ug

dxnN22

8

Landau dynamic of Landau dynamic of elementary excitationselementary excitations

constnH

nHn

rp

rpp

,

,,

9

Weakly inhomogeneous Weakly inhomogeneous condensatecondensate

222

2/322

,

3

4

onconservati Energy

,;,,

),(

soliton ofcenter theof coordinate

uXcdt

dXconstu

constEvXcmg

dt

dXv

t

XvtXtvXv

Xcc

X

s

10

Local density approximationLocal density approximation

2004 ,Pitaevskii L. Konotop, V.

,2

/

22**

222

20

2

0

0

0

ucmEmm

ucmxUdt

dXm

mxUcXc

xUx

11

Harmonic trapHarmonic trap

. arbitrary an for alidActually v

2000. Anglin, J. Bush, T.

:th soliton wi aFor

2/,2/22

v

cv

xmxU xx

12

Density perturbationDensity perturbation

mul

Xxt

Xxmuc

unxn

/ :lengthSoliton

.given at on dependnot Does

ch

1)(

220

2

0

13

14

3/1

0

1

11

21

21

or if validis theory The

0

where,

pointsbetween oscillatesSoliton

x

x

Rc

u

lXR

XnXv

uXcX

15

Simple physical Simple physical interpretationinterpretation

sI

sI

ss

sss

Nmm

dv

dEm

dX

dUN

dt

dv

dv

dE

vdX

dU

d

dE

dt

dvv

dv

dE

dt

dE

/

02:mass"Inertial"

2

:equation"Newton "

02

*

2

2

2

16

BEC condensateBEC condensate

mmm

mN

vcg

mm

vmEE

Is

I

ss

2,2

4

/:BECIn

*2/

2/122

2

17

Adiabatic change of the trap Adiabatic change of the trap frequencyfrequency

~,2

2

22

1

/2

1,

sds

s

xs

s

EEE

I

T

v

dx

dE

dI

vpE

constpdxtEI

18

Effective energyEffective energy

2/1

1

3/22/1*1

3/2*

3/13/1220

*

~ oscillator usualan For

~/~

nsoscillatio of Amplitude

~

~~,

X

EX

E

EuucmE s

19

Snake instability of a flat Snake instability of a flat solitonsoliton

1988 Turitsyn, S. and Kuznetsov E. 1975, ; ZakharovV.

1970; li,Petviashvi V. and Kadomtsev B.

3/:BEC

,

2

1

2

11 :energy Surface

cos,,,}(,

:soliton Deformed

2/122

2/1

22

2

22

2

2

00

kvci

km

EikE

dt

dm

kyX

U

dy

dEU

kytytyttXytX

I

ssI

ss

20

Modified EquationsModified Equations

(2002) al.,et Salasnich L.

t confinemen radialin weak Solitons1/2,

.0point near the gas Bose 1D 2.

0

2*

2*2

22

g

g

xUgxmt

i

21

(2005) ,Pitaevskii L. Konotop, V.

frequency trap theof change Adiabatic

7.1,2/1

3.2,2

2

:0at energy Effective

2

2

*

*

2**

E

mm

mm

XUv

mE

v

equations. modifiedin solitons Slow