Introduction to Introduction to SolitonsSolitons

Institute of Theoretical Physics and AstronomyInstitute of Theoretical Physics and AstronomyVilnius, 2013Vilnius, 2013

University of University of OldenburgOldenburg and BSU Minskand BSU Minsk

YaYa ShnirShnir

KdVKdV inverse scattering problem: examplesinverse scattering problem: examples

One bound state: N=1 N=1 , reflectionless potential: R(kR(k)=0 )=0 ,

FFFF ((((xxxx,,,, tttt)))) ====NNNN∑∑∑∑

nnnn====1111

cccc2222nnnneeee−−−−κκκκnnnnxxxx ++++

1111

2222ππππ

∞∞∞∞∫∫∫∫

−−−−∞∞∞∞

RRRR((((kkkk,,,, tttt))))eeeeiiiikkkkxxxxddddkkkk

ccccnnnn((((tttt)))) ==== ccccnnnn((((0000))))eeee4444κκκκ3333nnnntttt

FFFFFFFF ((((((((xxxxxxxx,,,,,,,, tttttttt)))))))) ======== cccccccc22222222((((((((00000000))))))))eeeeeeee−−−−−−−−κκκκκκκκxxxxxxxx++++++++88888888κκκκκκκκ33333333tttttttt

Discrete spectrum

ψψψψxxxxxxxx ====

((((λλλλ++++

2222

ccccoooosssshhhh2222 xxxx

))))ψψψψ ==== 0000Sturm-Liouville equation (l=1):SturmSturm--LiouvilleLiouville equation equation (l=1):

Single normal discrete mode:

cccc((((0000)))) ====√√√√2222The scattering data cccc((((tttt)))) ====

√√√√2222eeee4444tttt

ψψψψ1111((((xxxx)))) ====1111√√√√2222

1111ccccoooosssshhhhxxxx →→→→ 1111√√√√

22222222 eeee−−−−xxxx aaaassss xxxx→→→→∞∞∞∞

FFFF ((((xxxx,,,, tttt)))) ==== 2222eeee8888tttt−−−−xxxx

GLM equation:GLM equation:GLM equation: KKKK((((xxxx,,,, yyyy;;;; tttt)))) ++++ 2222eeee8888tttt−−−−((((xxxx++++yyyy)))) ++++ 2222

∞∞∞∞∫∫∫∫

xxxx

KKKK((((xxxx,,,, zzzz;;;; tttt))))eeee8888tttt−−−−((((yyyy++++zzzz))))ddddzzzz ==== 0000

Ansatz: KKKK((((xxxx,,,, yyyy,,,, tttt)))) ==== MMMM((((xxxx,,,, tttt))))eeee−−−−yyyy MMMM((((xxxx,,,, tttt)))) ++++ 2222eeee8888tttt−−−−xxxx++++ 2222MMMM((((xxxx,,,, tttt))))eeee8888tttt∞∞∞∞∫∫∫∫

xxxx

eeee−−−−2222zzzzddddzzzz ==== 0000

==== 11112222eeee−−−−2222xxxx

MMMM((((xxxx,,,, tttt)))) ==== −−−− 2222eeee8888tttt−−−−xxxx

1111 ++++ eeee8888tttt−−−−2222xxxx

One-soliton solution propagating to the right

uuuu((((xxxx,,,, tttt)))) ==== 2222∂∂∂∂

∂∂∂∂xxxxKKKK((((xxxx,,,, xxxx,,,, tttt)))) ==== 2222 sssseeeecccchhhh2222((((xxxx−−−− 4444tttt))))

uuuu((((xxxx)))) ==== −−−− llll((((llll ++++ 1111))))

ccccoooosssshhhh2222((((xxxx))));;;; λλλλ ==== −−−−κκκκ2222,,,, κκκκ ==== κκκκ1111 ==== 1111

KdVKdV inverse scattering problem: examplesinverse scattering problem: examples

Two bound states: N=2 N=2 , reflectionless potential: R(kR(k)=0 )=0 ,

ψψψψxxxxxxxx ++++

((((λλλλ++++

6666

ccccoooosssshhhh2222 xxxx

))))ψψψψ ==== 0000 Two normal

discrete modes:

llll((((llll ++++ 1111)))) ==== 6666;;;; λλλλ ==== −−−−kkkk2222,,,, κκκκ1111 ==== 1111;;;; κκκκ2222 ==== 2222

ψψψψ1111 ====√√√√

33332222ttttaaaannnnhhhh xxxxccccoooosssshhhh xxxx

→→→→√√√√6666eeee−−−−xxxx

ψψψψ2222 ====√√√√33332222

1111ccccoooosssshhhh2222 xxxx

→→→→ 2222√√√√3333eeee−−−−xxxx

cccc1111((((0000)))) ====

√√√√6666 ====⇒⇒⇒⇒ cccc1111((((tttt)))) ====

√√√√6666eeee4444tttt

cccc2222((((0000)))) ==== 2222√√√√3333 ====⇒⇒⇒⇒ cccc2222((((tttt)))) ==== 2222

√√√√3333eeee33332222tttt

FFFF ((((xxxx,,,, tttt)))) ====2222∑∑∑∑

nnnn====1111

cccc2222nnnneeee−−−−κκκκnnnnxxxx ==== 6666eeee8888tttt−−−−xxxx ++++ 11112222eeee66664444tttt−−−−2222xxxx

GLM equation:GLM equation:GLM equation:

KKKK((((xxxx,,,, yyyy;;;; tttt))))++++6666eeee8888tttt−−−−((((xxxx++++yyyy))))++++11112222eeee66664444tttt−−−−2222((((xxxx−−−−yyyy))))++++

∞∞∞∞∫∫∫∫

xxxx

KKKK((((xxxx,,,, zzzz;;;; tttt))))6666eeee8888tttt−−−−((((yyyy++++zzzz)))) ++++11112222eeee66664444tttt−−−−2222((((yyyy++++zzzz))))

ddddzzzz ==== 0000

Ansatz: KKKK((((xxxx,,,, yyyy,,,, tttt)))) ==== MMMM1111((((xxxx,,,, tttt))))eeee−−−−yyyy ++++MMMM2222((((xxxx,,,, tttt))))eeee

−−−−2222yyyy Collecting the coefficients at andeeee−−−−yyyy eeee−−−−2222yyyy

MMMM1111 ++++ 6666eeee8888tttt−−−−xxxx ++++ 6666eeee8888ttttMMMM1111

∞∞∞∞∫∫∫∫

xxxxeeee−−−−2222zzzzddddzzzz ++++MMMM2222

∞∞∞∞∫∫∫∫

xxxxeeee−−−−3333zzzzddddzzzz

==== 0000

MMMM2222 ++++ 11112222eeee66664444tttt−−−−2222xxxx ++++ 11112222eeee66664444ttttMMMM1111

∞∞∞∞∫∫∫∫

xxxx

eeee−−−−3333zzzzddddzzzz ++++ MMMM2222

∞∞∞∞∫∫∫∫

xxxx

eeee−−−−4444zzzzddddzzzz

==== 0000

N N bound states, reflectionless potential: R(kR(k)=0)=0 FFFF ((((xxxx,,,, tttt)))) ====NNNN∑∑∑∑

nnnn====1111

cccc2222nnnn((((tttt))))eeee−−−−κκκκnnnnxxxx

The ansatz for the solution of the GLM equation: KKKK((((xxxx,,,, yyyy,,,, tttt)))) ====NNNN∑∑∑∑

nnnn====1111

MMMMnnnn((((xxxx,,,, tttt))))eeee−−−−κκκκnnnnyyyy

KKKKKKKK((((((((xxxxxxxx,,,,,,,, xxxxxxxx,,,,,,,, tttttttt)))))))) ======== MMMMMMMM11111111((((((((xxxxxxxx,,,,,,,, tttttttt))))))))eeeeeeee−−−−−−−−xxxxxxxx ++++++++ MMMMMMMM22222222((((((((xxxxxxxx,,,,,,,, tttttttt))))))))eeeeeeee

−−−−−−−−22222222xxxxxxxx2-soliton solution of the GLM equation:

MMMM1111((((xxxx,,,, tttt)))) ====6666((((eeee77772222tttt−−−−5555xxxx −−−− eeee8888tttt−−−−xxxx))))

1111 ++++ 3333eeee8888tttt−−−−2222xxxx ++++ 3333eeee66664444tttt−−−−4444xxxx ++++ eeee77772222tttt−−−−6666xxxx

MMMM2222((((xxxx,,,, tttt)))) ==== −−−− 11112222((((eeee66664444tttt−−−−2222xxxx ++++ eeee77772222tttt−−−−4444xxxx))))

1111 ++++ 3333eeee8888tttt−−−−2222xxxx ++++ 3333eeee66664444tttt−−−−4444xxxx ++++ eeee77772222tttt−−−−6666xxxx

Most general case: NMost general case: N--solitonsoliton solutionsolution

uuuu2222 ==== 2222∂∂∂∂

∂∂∂∂xxxx

((((MMMM1111eeee

−−−−xxxx ++++MMMM2222eeee−−−−2222xxxx)))) ==== −−−−111122223333 ++++ 4444 ccccoooosssshhhh((((2222xxxx−−−− 8888tttt)))) ++++ ccccoooosssshhhh((((4444xxxx −−−− 66664444tttt))))

[[[[3333 ccccoooosssshhhh((((xxxx−−−− 22228888tttt)))) ++++ ccccoooosssshhhh((((3333xxxx −−−− 33336666tttt))))]]]]2222

KdVKdV inverse scattering problem: Ninverse scattering problem: N--solitonsoliton solutionsolution

The solution of the GLM equation is given by uuuuNNNN ==== 2222∂∂∂∂2222

∂∂∂∂xxxx2222llllnnnn ddddeeeettttAAAA((((xxxx,,,, tttt))))

Here the matrix A is defined as NNNN ××××NNNN

Asymptotically, as this solution of the KdV equation represents a superposition of N single-soliton solutions propagating to the right and ordered in space by their speeds (amplitudes):

tttt→→→→ ±±±±∞∞∞∞

AAAAmmmmnnnn ==== δδδδmmmmnnnn ++++cccc2222nnnn((((0000))))

κκκκnnnn ++++ κκκκmmmmeeee−−−−((((κκκκnnnn−−−−κκκκmmmm))))xxxx++++8888κκκκ

3333nnnntttt

uuuuuuuuNNNNNNNN ((((((((xxxxxxxx,,,,,,,, tttttttt)))))))) ∼∼∼∼∼∼∼∼NNNNNNNN∑∑∑∑∑∑∑∑

nnnnnnnn========11111111

22222222κκκκκκκκ22222222nnnnnnnn sssssssseeeeeeeecccccccchhhhhhhh22222222 [[[[[[[[κκκκκκκκnnnnnnnn((((((((xxxxxxxx −−−−−−−− 44444444κκκκκκκκ22222222nnnnnnnntttttttt −−−−−−−− xxxxxxxx±±±±±±±±nnnnnnnn ))))))))]]]]]]]]

The position of the n-th soliton is given by:

xxxx±±±±nnnn ====1111

2222κκκκnnnnllllnnnn

cccc2222nnnn((((0000))))

2222κκκκnnnn±±±± 1111

2222κκκκnnnn

nnnn−−−−1111∑∑∑∑

mmmm====1111

llllnnnn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣κκκκnnnn −−−− κκκκmmmm

κκκκnnnn ++++ κκκκmmmm

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣−−−−NNNN∑∑∑∑

mmmm====nnnn++++1111

llllnnnn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣κκκκnnnn −−−− κκκκmmmm

κκκκnnnn ++++ κκκκmmmm

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

The N-soliton solution is characterised by 2N parameters: κκκκ1111 .... .... .... κκκκNNNN ,,,, cccc1111((((0000)))),,,, .... .... .... ccccNNNN ((((0000))))

The evolution is isospectral, i.e. - the solitons preserve their amplitudes (and velocities) in the interactions; the only change they undergo is an additional phase shift due to collisions.

κκκκnnnn ==== ccccoooonnnnsssstttt

δδδδnnnn ==== xxxx++++nnnn −−−−xxxx−−−−nnnn

KdVKdV solitonssolitons: 2: 2--soliton solutionsoliton solution

uuuu2222 ==== 2222∂∂∂∂2222

∂∂∂∂xxxx2222llllnnnnAAAA((((xxxx,,,, tttt))))AAAA((((xxxx,,,, tttt)))) ==== 1111++++eeeexxxxpppp

κκκκ1111((((xxxx−−−−xxxx1111))))−−−− 4444κκκκ33331111tttt

++++eeeexxxxpppp

κκκκ2222((((xxxx−−−−xxxx2222))))−−−− 4444κκκκ33332222tttt

++++

((((κκκκ1111−−−−κκκκ2222κκκκ1111++++κκκκ2222

))))2222eeeexxxxpppp

κκκκ1111((((xxxx−−−−xxxx1111))))−−−− 4444κκκκ33331111tttt++++κκκκ2222((((xxxx−−−−xxxx2222))))−−−− 4444κκκκ33332222tttt

Asymptotically, as tttt→→→→ ±±±±∞∞∞∞uuuu2222((((xxxx,,,, tttt))))∼∼∼∼2222κκκκ22221111 sssseeeecccchhhh

2222[[[[κκκκ1111((((xxxx−−−−4444κκκκ22221111tttt−−−−xxxx1111))))++++2222κκκκ22222222 sssseeeecccchhhh2222[[[[κκκκ2222((((xxxx−−−−4444κκκκ22222222tttt−−−−xxxx2222))))]]]]

For a two-soliton collision the outcome is the phase shift

δδδδ1111 ==== 2222xxxx1111 ====1111

κκκκ1111llllnnnn

((((κκκκ1111 ++++ κκκκ2222κκκκ1111 −−−− κκκκ2222

)))),,,, δδδδ2222 ==== 2222xxxx2222 ==== −−−−

1111

κκκκ2222llllnnnn

((((κκκκ1111 ++++ κκκκ2222κκκκ1111 −−−− κκκκ2222

))))((((κκκκ1111 >>>> κκκκ2222))))

As a result of the interaction, the taller soliton gets an additional shift forward by the distance dd11 while the shorter soliton is shifted backwards by the distance --dd22.

Fast and slow solitons interaction for the KdV Equation

0 10 20 30 40

x

0

2

4

6

8

10

t

0

5

10

u

KdVKdV solitonssolitons: 2: 2--soliton collisionsoliton collision

Fast and slow solitons interaction for the KdV Equation

0

10

20

30

40

x

0

2

4

6

8

10

t

0

5

10

u

0

10

20

30x

κκκκ1111 ==== 1111;;;; κκκκ2222 ==== 2222

KdVKdV solitonssolitons: : HirotaHirota methodmethod

Another way around:Another way around: Let us apply a different approach to find the 2Let us apply a different approach to find the 2--soliton solution soliton solution

Substitute into the KdVSubstitute into the Substitute into the KdVKdVStep I: Step I: uuuu ==== vvvvxxxx

Potential KdV equation:Potential Potential KdVKdV equation:equation:1-soliton KdV

Step II: Step II: Hirota substitutionHirota substitution

uuuutttt ++++ 6666uuuuuuuuxxxx ++++ uuuuxxxxxxxxxxxx ==== 0000

∂∂∂∂

∂∂∂∂xxxx

((((vvvvtttt ++++ 3333vvvv2222xxxx ++++ vvvvxxxxxxxxxxxx

))))==== 0000

uuuuuuuu((((((((θθθθθθθθ)))))))) ========vvvvvvvv

22222222sssssssseeeeeeeecccccccchhhhhhhh2222

2222

((((((((√√√√√√√√vvvvvvvvθθθθθθθθ

22222222

))))))))

vvvv((((θθθθ)))) ====√√√√vvvv

((((ttttaaaannnnhhhh

[[[[√√√√vvvvθθθθ

2222++++ 1111

]]]]))))

====√√√√vvvv

((((2222eeee√√√√

vvvvθθθθ

eeee√√√√

vvvvθθθθ ++++ 1111

))))

==== 2222ηηηηxxxx

ηηηη;;;; ηηηη ==== 1111 ++++ eeee

√√√√vvvvθθθθ

vvvvvvvvtttttttt ++++++++ 33333333vvvvvvvv22222222xxxxxxxx ++++++++ vvvvvvvvxxxxxxxxxxxxxxxxxxxxxxxx ======== 00000000vvvv ==== 2222ηηηηxxxx////ηηηη ==== 2222∂∂∂∂xxxx llllnnnn ηηηη

ηηηηxxxxttttηηηη −−−− ηηηηxxxxηηηηtttt ++++ ηηηηxxxxxxxxxxxxxxxxηηηη −−−− 4444ηηηηxxxxηηηηxxxxxxxxxxxx ++++ 3333ηηηη2222xxxxxxxx ==== 0000 Hirota form of KdV (bilinear equation)HirotaHirota form of form of KdVKdV (bilinear equation)(bilinear equation)

Hirota’s D-operatorHirota’s D-operator DDDDDDDDnnnnnnnnxxxxxxxxffffffff ········ gggggggg ≡≡≡≡≡≡≡≡ ((((((((∂∂∂∂∂∂∂∂xxxxxxxx11111111−−−−−−−−∂∂∂∂∂∂∂∂xxxxxxxx22222222))))))))

nnnnnnnnffffffff((((((((xxxxxxxx11111111))))))))gggggggg((((((((xxxxxxxx22222222))))))))

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣xxxxxxxx11111111========xxxxxxxx22222222========xxxxxxxx

≡≡≡≡≡≡≡≡ ∂∂∂∂∂∂∂∂nnnnnnnnyyyyyyyy ffffffff((((((((xxxxxxxx++++++++ yyyyyyyy))))))))gggggggg((((((((xxxxxxxx−−−−−−−− yyyyyyyy))))))))

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣yyyyyyyy========00000000

DDDDxxxxDDDDttttηηηη ···· ηηηη ==== 2222((((ηηηηxxxxttttηηηη −−−− ηηηηxxxxηηηηtttt))))DDDD4444

xxxxηηηη ···· ηηηη ==== 2222((((ηηηηxxxxxxxxxxxxxxxxηηηη −−−− 4444ηηηηxxxxηηηηxxxxxxxxxxxx ++++3333ηηηη2222xxxxxxxx))))

((((DDDD4444xxxx ++++DDDDxxxxDDDDtttt))))ηηηη ···· ηηηη ==== 0000

HirotaHirota ’’ ss bilinear operatorbilinear operator

DDDDnnnnxxxxffff ···· gggg ≡≡≡≡ ((((∂∂∂∂xxxx1111−−−−∂∂∂∂xxxx2222))))

nnnnffff ((((xxxx1111))))gggg((((xxxx2222))))

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣xxxx1111====xxxx2222====xxxx

Note:Note: the operator D acts on a product of 2 functions similar to the usual Leibnizrule, except for a crucial sign difference:

DDDDxxxxffff ···· gggg ==== ffffxxxxgggg −−−− ffffggggxxxx

DDDDxxxxDDDDttttffff ···· gggg ==== ffffggggxxxxtttt −−−− ffffxxxxggggtttt −−−− ffffttttggggxxxx ++++ ffffggggxxxxtttt

How to construct the How to construct the solitonsoliton solutions of solutions of KdVKdV ??

Trivial solution: Trivial solution: ηηηη ==== 1111

Almost Perturbatively!Almost Almost PerturbativelyPerturbatively !!

1 1 ––solitonsoliton solution: solution:

22––solitonsoliton solution: solution: ηηηη ==== 1111 ++++ eeeeθθθθ1111 ++++ eeeeθθθθ2222 ++++ aaaaeeeeθθθθ1111++++θθθθ2222

κκκκ ====√√√√vvvv

ηηηη ==== 1111 ++++ eeeeθθθθ ;;;; θθθθ ==== κκκκxxxx −−−− κκκκ3333tttt ++++ δδδδ0000

θθθθθθθθ11111111 ======== κκκκκκκκ11111111xxxxxxxx −−−−−−−− κκκκκκκκ3333333311111111tttttttt ++++++++ δδδδδδδδ((((((((00000000))))))))11111111 ;;;;;;;;

θθθθθθθθ22222222 ======== κκκκκκκκ22222222xxxxxxxx −−−−−−−− κκκκκκκκ3333333322222222tttttttt ++++++++ δδδδδδδδ((((((((00000000))))))))22222222aaaa====

((((κκκκ1111−−−−κκκκ2222κκκκ1111++++κκκκ2222

))))2222

NN––solitonsoliton solution: solution:

DDDD2222xxxxffff ···· gggg ==== ffffxxxxxxxxgggg −−−− 2222ffffxxxxggggxxxx ++++ ggggxxxxxxxxffff

ηηηη ==== 1111 ++++∞∞∞∞∑∑∑∑

nnnn====1111

ǫǫǫǫnnnnηηηηnnnn((((xxxx,,,, tttt)))) - expansion in powers of e

DDDDxxxx

((((DDDDtttt ++++DDDD3333

xxxx

))))ηηηη ···· ηηηη ==== 0000

- 1-soliton - 2-soliton ηηηη1111 ==== eeeeθθθθ1111 ηηηη1111 ==== eeeeθθθθ1111 ++++ eeeeθθθθ2222ηηηη1111 ====

NNNN∑∑∑∑

iiii====1111

eeeeθθθθiiii - N-soliton

Nonlinear Nonlinear SchrSchröödinger Equationdinger Equation

Non-linear Schrödinger equation:NonNon --linear linear SchrSchr öödinger dinger equation:equation: iiiiiiiiψψψψψψψψtttttttt ++++++++ ψψψψψψψψxxxxxxxxxxxxxxxx ++++++++ 22222222σσσσσσσσ||||||||ψψψψψψψψ||||||||22222222ψψψψψψψψ ======== 00000000 σσσσ ==== ±±±±1111

Hamiltonian: HHHH ====1111

2222||||ψψψψxxxx||||2222 ++++

σσσσ

2222||||ψψψψ4444||||Symmetries:

Translational invariance:Translational invariance: tttt→→→→ tttt,,,, xxxx→→→→ xxxx++++ δδδδxxxx,,,, ψψψψ →→→→ ψψψψ

Time invariance:Time invariance: tttt→→→→ tttt++++ δδδδtttt,,,, xxxx→→→→ xxxx,,,, ψψψψ →→→→ ψψψψ

Scale invariance:Scale invariance: tttt→→→→ aaaa2222tttt,,,, xxxx→→→→ aaaaxxxx,,,, ψψψψ →→→→ ψψψψ////aaaa

Galilean invariance:Galilean invariance: tttt →→→→ tttt,,,, xxxx →→→→ xxxx −−−− cccctttt ψψψψ →→→→ ψψψψeeeeiiiicccc((((xxxx−−−− cccc2222tttt))))////2222

One-parameter Lie group of symmetries

Lagrangian: LLLL ==== iiii((((ψψψψψψψψ∗∗∗∗tttt −−−− ψψψψttttψψψψ∗∗∗∗)))) ++++

1111

2222||||ψψψψxxxx||||2222 −−−−

σσσσ

2222||||ψψψψ||||4444

NSE NSE SolitonsSolitons

Ansatz for the soliton solution: ψψψψ((((xxxx,,,, tttt)))) ==== uuuu((((xxxx))))eeeeiiiiφφφφ((((tttt)))) −−−−uuuuφφφφtttt ++++ uuuuxxxxxxxx ++++2222σσσσuuuu3333 ==== 0000

ddddφφφφ

ddddtttt====

uuuuxxxxxxxx

uuuu++++ 2222σσσσuuuu2222 ==== CCCC ==== ccccoooonnnnsssstttt ====⇒⇒⇒⇒ φφφφ ==== CCCCtttt

uuuuxxxxxxxx ==== −−−−2222σσσσuuuu3333 ++++ CCCCuuuu X integrating factoruuuuxxxx ((((uuuuxxxx))))2222 ==== −−−−σσσσuuuu4444 ++++ CCCCuuuu2222 ++++ CCCC0000

Shape of the solitary waves depends on the sign of s

iiiiiiiiψψψψψψψψtttttttt ++++++++ ψψψψψψψψxxxxxxxxxxxxxxxx ++++++++ 22222222σσσσσσσσ||||||||ψψψψψψψψ ||||||||22222222ψψψψψψψψ ======== 00000000

Bright Bright SolitonSoliton :: ss=1 (Focusing NLS) ((((uuuuxxxx))))2222 ==== −−−−uuuu4444 ++++ CCCCuuuu2222 ++++ CCCC0000

Boundary conditions: uuuu ==== uuuu′′′′ ==== 0000,,,, aaaassss xxxx→→→→ ±±±±∞∞∞∞∫∫∫∫

dddduuuu

uuuu√√√√CCCC −−−− uuuu2222

====

∫∫∫∫ddddxxxx

Simplest solution (C=1):

Using Galilean and scale symmetry:

uuuu ==== sssseeeecccchhhh xxxx;;;; φφφφ ==== tttt ====⇒⇒⇒⇒ ψψψψ ==== sssseeeecccchhhh xxxxeeeeiiiitttt −−−− llllnnnn((((1111++++

√√√√1111−−−−uuuu2222

uuuu

))))==== xxxx

ψψψψ ==== AAAA sssseeeecccchhhh AAAA((((xxxx−−−− cccctttt))))eeeeiiii((((cccc2222xxxx++++((((AAAA2222−−−− cccc2222

4444))))tttt)))) Two-parameter family

of bright solitons

Homework: Consider C=-1

c=2 c=5 c=10 c=20

Instability of the bright Instability of the bright solitonsoliton :: for sufficiently large values of c the envelope has spatial oscillations of the same period as the carrier wave

Re y

Dark Dark SolitonSoliton :: ss=-1 (Defocusing NLS) ((((((((uuuuuuuuxxxxxxxx))))))))22222222 ======== uuuuuuuu44444444 ++++++++ CCCCCCCCuuuuuuuu22222222 ++++++++ CCCCCCCC00000000

Simplest solution (C=-1, C0=¼):

∫∫∫∫dddduuuu

uuuu2222 −−−− 11112222

====

∫∫∫∫ddddxxxx uuuu====

1111√√√√2222ttttaaaannnnhhhh

xxxx√√√√2222;;;; φφφφ====−−−−tttt

ψψψψ ====1111√√√√2222ttttaaaannnnhhhh

xxxx√√√√2222eeee−−−−iiiitttt Homework: Consider C=1

Using Galilean and scale symmetry:

Two-parameter familyof dark solitons

ψψψψ ====AAAA√√√√2222ttttaaaannnnhhhh

AAAA((((xxxx −−−− cccctttt))))√√√√2222

eeeeiiii(((( cccc2222xxxx((((AAAA2222++++ cccc2222

4444))))tttt))))

Re y

c=1 c=3 c=10 c=20

Focusing Focusing NSE:BreathersNSE:Breathers

Euler hydrodynamical equationsshallow water surface waves

deep water surface waves

KdV equationKdVKdV equationequation

NLS equationNLSNLS equationequation

Freak (rogue) wave: Freak (rogue) wave: a single wave or a very short wave group with a signicantlylarger steepness than the surrounding waves – BreatherBreather solution of the NLS equation

Note:Note: While for a bright soliton there is always a reference frame where theenvelope is stationary, this is not so for breathers (“dynamical solitons")||||ψψψψ||||

ψψψψ ====ccccoooossss((((ΩΩΩΩtttt−−−− 2222iiiikkkk))))−−−− ccccoooosssshhhh((((kkkk)))) ccccoooosssshhhh((((ppppxxxx))))

ccccoooossss((((ΩΩΩΩtttt))))−−−− ccccoooosssshhhh((((kkkk)))) ccccoooosssshhhh((((ppppxxxx))))eeee2222iiiitttt;;;; ΩΩΩΩ ==== 2222 ssssiiiinnnnhhhh((((2222kkkk)))),,,, pppp ==== 2222 ssssiiiinnnnhhhh kkkk

t=0 t=0.5 t=1

||||ψψψψ||||Limit of zero breathing period :kkkkkkkk →→→→→→→→ 00000000

Peregrine breather (1983)

ψψψψ ====

[[[[1111−−−− 4444((((1111 ++++ 4444iiiitttt))))

1111 ++++ 4444xxxx2222 ++++ 11116666tttt2222

]]]]eeee2222iiiitttt

xt

ZSZS--AKNS techniqueAKNS technique

The ZSZS--AKNS schemeAKNS scheme is a generalisation of the Sturm-Liouville equation

Zakharov and Shabat - Ablowitz, Kaup, Newell, and Segur

Consider the 2x2 spectral linear problem

Note:Note: If we recoverthe linear Sturm-Liouville equation:

ψψψψ((((1111))))xxxx ==== −−−−iiiiλλλλψψψψ((((1111)))) ++++ qqqq((((xxxx,,,, tttt))))ψψψψ((((2222))))

ψψψψ((((2222))))xxxx ==== iiiiλλλλψψψψ((((2222)))) ++++ pppp((((xxxx,,,, tttt))))ψψψψ((((1111))))pppp((((xxxx,,,, tttt)))) ==== 0000

((((ψψψψ((((2222))))))))xxxxxxxx ++++ ((((λλλλ2222 ++++ qqqq))))ψψψψ((((2222)))) ==== 0000 Time evolution:Time evolution:ψψψψ((((1111))))tttt ==== AAAA((((xxxx))))ψψψψ((((1111)))) ++++ BBBB((((xxxx))))ψψψψ((((2222))))

ψψψψ((((2222))))tttt ==== CCCC((((xxxx))))ψψψψ((((1111)))) ++++DDDD((((xxxx))))ψψψψ((((2222))))

((((ψψψψ((((aaaa))))xxxx ))))tttt ==== ((((ψψψψ((((aaaa))))tttt ))))xxxx

DDDD ==== −−−−AAAA;;;; AAAAxxxx ==== qqqqCCCC −−−− ppppBBBB

BBBBxxxx ++++ 2222iiiiλλλλBBBB ==== qqqqtttt −−−− 2222AAAAqqqq;;;;

CCCCxxxx −−−− 2222iiiiλλλλCCCC ==== pppptttt ++++ 2222AAAArrrr

Various solutions of this system yield different integrable PDE

We take σσσσσσσσ ======== ±±±±±±±±11111111

AAAA ==== −−−−2222iiiiλλλλ2222 ++++ iiiiσσσσqqqqqqqq∗∗∗∗;;;;

BBBB ==== 2222qqqqλλλλ++++ iiiiqqqqxxxx;;;;

CCCC ==== −−−−2222σσσσqqqq∗∗∗∗λλλλ++++ iiiiσσσσqqqq∗∗∗∗xxxx

provided that satisfies NLS:qqqq((((xxxx,,,, tttt))))

iiiiqqqqtttt ++++ qqqqxxxxxxxx ++++ 2222σσσσ||||qqqq||||2222qqqq ==== 0000

ψψψψ((((1111))))tttt ==== ((((−−−−2222iiiiλλλλ2222 ++++ iiiiσσσσ||||qqqq||||2222))))ψψψψ((((1111)))) ++++ ((((2222qqqqλλλλ++++ iiiiqqqqxxxx))))ψψψψ

((((2222))))

ψψψψ((((2222))))tttt ==== ((((−−−−2222σσσσqqqq∗∗∗∗λλλλ++++ iiiiσσσσqqqq∗∗∗∗xxxx))))ψψψψ

((((1111)))) ++++ ((((2222iiiiλλλλ2222 −−−− iiiiσσσσ||||qqqq||||2222))))ψψψψ((((2222))))

pppppppp ======== −−−−−−−−σσσσσσσσqqqqqqqq∗∗∗∗∗∗∗∗

NLS direct scattering problemNLS direct scattering problem

Focusing NLS:Focusing NLS: ss=1 Spectral problem:Spectral problem:Spectral problem:

ψψψψ((((1111))))xxxx ==== −−−−iiiiλλλλψψψψ((((1111)))) ++++ qqqq((((xxxx,,,, tttt))))ψψψψ((((2222))))

ψψψψ((((2222))))xxxx ==== iiiiλλλλψψψψ((((2222)))) −−−− qqqq∗∗∗∗((((xxxx,,,, tttt))))ψψψψ((((1111))))qqqq((((xxxx,,,, tttt))))→→→→ 0000 aaaassss xxxx→→→→ ±±±±∞∞∞∞

Complex conjugation: if is an eigenvector, so is ((((((((ψψψψψψψψ((((((((11111111)))))))) ,,,,,,,, ψψψψψψψψ((((((((22222222)))))))))))))))) ((((((((ψψψψψψψψ((((((((22222222))))))))∗∗∗∗∗∗∗∗ ,,,,,,,,−−−−−−−−ψψψψψψψψ((((((((11111111))))))))∗∗∗∗∗∗∗∗))))))))

Asymptotically ((((ψψψψ((((1111)))),,,, ψψψψ((((2222))))))))→→→→ ((((kkkk1111eeee−−−−iiiiλλλλxxxx,,,, kkkk2222eeee

iiiiλλλλxxxx)))) aaaassss xxxx→→→→ ±±±±∞∞∞∞

Consider the set

as a basis for solutions of the spectral problem ((((ψψψψ((((2222))))++++∗∗∗∗,,,,−−−−ψψψψ((((1111))))++++ ∗∗∗∗))))→→→→((((0000,,,,−−−−eeeeiiiiλλλλxxxx)))) aaaassss xxxx→→→→++++∞∞∞∞

((((ψψψψ((((1111))))++++ ,,,, ψψψψ

((((2222))))++++ )))) →→→→ ((((eeee−−−−iiiiλλλλxxxx,,,,0000)))) aaaassss xxxx→→→→++++∞∞∞∞

((((((((ψψψψψψψψ((((((((11111111))))))))++++++++ ,,,,,,,, ψψψψψψψψ

((((((((22222222))))))))++++++++ )))))))),,,,,,,, ((((((((ψψψψψψψψ

((((((((22222222))))))))++++++++

∗∗∗∗∗∗∗∗ ,,,,,,,,−−−−−−−−ψψψψψψψψ((((((((11111111))))))))++++++++

∗∗∗∗∗∗∗∗))))))))

Scattering problem((((ψψψψ

((((1111))))−−−− ,,,, ψψψψ

((((2222))))−−−− )))) →→→→ ((((eeee−−−−iiiiλλλλxxxx,,,, 0000)))) aaaassss xxxx→→→→−−−−∞∞∞∞

((((ψψψψ((((2222))))−−−−

∗∗∗∗,,,,−−−−ψψψψ((((1111))))−−−− ∗∗∗∗))))→→→→ ((((0000,,,,−−−−eeeeiiiiλλλλxxxx)))) aaaassss xxxx→→→→−−−−∞∞∞∞

ψψψψ((((1111))))−−−− ==== aaaa((((λλλλ))))ψψψψ

((((1111))))++++ ++++ bbbb((((λλλλ))))ψψψψ

((((2222))))++++

∗∗∗∗

ψψψψ((((2222))))−−−− ==== aaaa((((λλλλ))))ψψψψ

((((2222))))++++ −−−− bbbb((((xxxx))))ψψψψ

((((1111))))++++

∗∗∗∗Scattering data: aaaaaaaa((((((((λλλλλλλλ)))))))),,,,,,,, bbbbbbbb((((((((λλλλλλλλ))))))))

aaaa((((λλλλ))))→→→→ 1111,,,, bbbb((((λλλλ))))→→→→ 0000 aaaassss ||||λλλλ|||| →→→→∞∞∞∞

Note:Note: Discrete spectrum is introduced as those values of ll, for which eigenvectors decay both for and for (“bound states”) The discretespectrum coincides with the zeros of the function in the upper half-plane.

xxxx→→→→ ++++∞∞∞∞ xxxx→→→→ −−−−∞∞∞∞aaaa((((λλλλ))))

Jost solutions

NLS inverse scattering problemNLS inverse scattering problem

Recall: Recall: The inverse scattering problem – reconstruct the function from the scattering data,

qqqq((((xxxx,,,, tttt))))aaaaaaaa((((((((λλλλλλλλ)))))))),,,,,,,, bbbbbbbb((((((((λλλλλλλλ))))))))

ψψψψ((((1111))))++++ ==== eeee−−−−iiiiλλλλxxxx ++++

∞∞∞∞∫∫∫∫

xxxx

KKKK1111((((xxxx,,,, yyyy))))eeee−−−−iiiiλλλλyyyyddddyyyy

ψψψψ((((2222))))++++ ====

∞∞∞∞∫∫∫∫

xxxx

KKKK2222((((xxxx,,,, yyyy))))eeee−−−−iiiiλλλλyyyyddddyyyy

KKKK1111((((xxxx,,,, yyyy)))) ==== KKKK2222((((xxxx,,,, yyyy)))) ==== 0000 aaaassss yyyy <<<< xxxx

ψψψψ((((1111))))xxxx ==== −−−−iiiiλλλλψψψψ((((1111)))) ++++ qqqq((((xxxx,,,, tttt))))ψψψψ((((2222))))

ψψψψ((((2222))))xxxx ==== iiiiλλλλψψψψ((((2222)))) −−−− qqqq∗∗∗∗((((xxxx,,,, tttt))))ψψψψ((((1111))))

Spectral problem:

The boundary problem:subject to

∂∂∂∂KKKK1111

∂∂∂∂xxxx++++ ∂∂∂∂KKKK1111

∂∂∂∂yyyy==== qqqqKKKK2222

∂∂∂∂KKKK2222

∂∂∂∂xxxx−−−− ∂∂∂∂KKKK2222

∂∂∂∂yyyy==== −−−−qqqq∗∗∗∗KKKK1111KKKK1111((((xxxx,,,, yyyy))))→→→→ 0000,,,,

KKKK2222((((xxxx,,,, yyyy))))→→→→ 0000 aaaassss yyyy →→→→ 0000;;;;KKKK2222((((xxxx,,,, xxxx)))) ====

11112222qqqq∗∗∗∗

if we can find from our knowledge of the scattering data, we can reconstruct from

KKKK2222((((xxxx,,,, xxxx))))qqqq((((xxxx))))

qqqq((((xxxx)))) ==== 2222KKKK∗∗∗∗2222 ((((xxxx,,,, xxxx))))

Consider representation with restrictions on the kernels of the Jost functions

NLS inverse scattering problemNLS inverse scattering problemAnalogue of the Gelfand-Levitan-Marchenko equation:

KKKK∗∗∗∗2222 ((((xxxx,,,, yyyy)))) ≡≡≡≡ KKKK((((xxxx,,,, yyyy)))),,,, KKKK((((xxxx,,,, yyyy)))) −−−− FFFF ∗∗∗∗((((xxxx++++ yyyy)))) ++++

∞∞∞∞∫∫∫∫

xxxx

GGGG((((xxxx,,,, yyyy,,,, zzzz))))KKKK((((xxxx,,,, zzzz))))ddddzzzz ==== 0000

where

GGGG((((xxxx,,,, yyyy,,,, zzzz)))) ====

∞∞∞∞∫∫∫∫

xxxx

FFFF ((((yyyy++++ ssss))))FFFF ∗∗∗∗((((ssss++++ zzzz))))ddddssss;;;; FFFF ((((xxxx)))) ==== −−−−iiiiNNNN∑∑∑∑

nnnn====1111

ccccnnnneeeeiiiiλλλλnnnnxxxx ++++

1111

2222ππππ

∞∞∞∞∫∫∫∫

−−−−∞∞∞∞

bbbb((((λλλλ))))

aaaa((((λλλλ))))eeeeiiiiλλλλxxxxddddλλλλ

Discrete spectrum data Reflection coefficient

λλλλnnnn are the N zeros of in the upper half plane; aaaa((((λλλλ))))

One bound state: a single zero of at and reflecteonless potential:aaaa((((λλλλ)))) λλλλ ==== λλλλ1111bbbb((((λλλλ))))aaaa((((λλλλ))))

==== 0000

FFFF ((((xxxx)))) ==== −−−−iiiicccc1111eeeeiiiiλλλλ1111xxxx GGGG((((xxxx,,,, yyyy,,,, zzzz)))) ====iiii||||cccc1111||||2222

λλλλ1111 −−−− λλλλ∗∗∗∗1111eeeeiiiiλλλλ1111((((xxxx++++yyyy))))eeee−−−−iiiiλλλλ∗∗∗∗1111((((xxxx++++zzzz))))

KKKK((((xxxx,,,, yyyy)))) −−−− iiiicccc∗∗∗∗1111eeee−−−−iiiiλλλλ∗∗∗∗1111((((xxxx++++yyyy)))) −−−− iiii||||cccc1111||||2222

((((λλλλ1111 −−−− λλλλ∗∗∗∗1111))))

∞∞∞∞∫∫∫∫

xxxx

eeeeiiiiλλλλ1111((((xxxx++++zzzz))))eeee−−−−iiiiλλλλ∗∗∗∗1111((((xxxx++++yyyy))))KKKK((((xxxx,,,, zzzz))))ddddzzzz ==== 0000

IS data equation:IS data equation:

Ansatz: KKKK((((xxxx,,,, yyyy)))) ==== MMMM((((xxxx))))eeee−−−−iiiiλλλλ∗∗∗∗1111yyyy

λλλλ1111 ==== αααα ++++ iiiiββββ

MMMM((((xxxx)))) ====iiiicccc∗∗∗∗1111((((λλλλ1111 −−−− λλλλ∗∗∗∗1111))))

2222eeee−−−−iiiiλλλλ∗∗∗∗1111xxxx

((((λλλλ1111 −−−− λλλλ∗∗∗∗1111))))2222 −−−− ||||cccc1111||||2222eeee2222iiii((((λλλλ1111−−−−λλλλ∗∗∗∗1111 ))))xxxx

ccccnnnn ==== bbbb((((λλλλnnnn))))aaaa((((λλλλnnnn))))

qqqq((((xxxx)))) ==== 2222MMMM((((xxxx))))eeee−−−−iiiiλλλλ∗∗∗∗1111xxxx ==== iiiicccc∗∗∗∗1111eeee−−−−2222iiiiααααxxxx 2222ββββ

||||cccc1111||||sssseeeecccchhhh ((((2222ββββxxxx−−−− llllnnnn

||||cccc1111||||2222ββββ

))))

Time evolution of the scattering dataTime evolution of the scattering data

Time evolution:Time evolution:ψψψψ((((1111))))tttt ==== ((((−−−−2222iiiiλλλλ2222 ++++ iiiiqqqqqqqq∗∗∗∗))))ψψψψ((((1111)))) ++++ ((((2222λλλλqqqq ++++ iiiiqqqqxxxx))))ψψψψ

((((2222))))

ψψψψ((((2222))))tttt ==== ((((−−−−2222λλλλqqqq∗∗∗∗ ++++ iiiiqqqq∗∗∗∗xxxx))))ψψψψ

((((1111)))) ++++ ((((2222iiiiλλλλ2222 −−−− iiiiqqqqqqqq∗∗∗∗))))ψψψψ((((2222))))

Asymptotically qqqq →→→→ 0000;;;; ψψψψ((((1111))))−−−− ,,,, ψψψψ((((2222))))−−−− →→→→ aaaa((((λλλλ,,,, tttt))))eeee−−−−iiiiλλλλxxxx,,,,−−−−bbbb((((λλλλ,,,, tttt))))eeeeiiiiλλλλxxxx aaaassss xxxx→→→→ ++++∞∞∞∞

aaaatttt ==== −−−−2222iiiiλλλλ2222aaaa;;;; bbbbtttt ==== 2222iiiiλλλλ2222bbbb aaaa((((λλλλ,,,, tttt)))) ==== eeee−−−−2222iiiiλλλλ2222ttttaaaa((((λλλλ,,,, 0000))));;;; bbbb((((λλλλ,,,, tttt)))) ==== eeee2222iiiiλλλλ

2222ttttbbbb((((λλλλ,,,, 0000))))

Note:Note: The zeros of (i,e. the discrete spectrum) are independent of time, aaaa((((λλλλ,,,, tttt))))

ccccnnnn((((tttt)))) ==== eeee4444iiiiλλλλ2222ttttccccnnnn((((0000))))

This yields the single bright NLS soliton:

qqqq((((xxxx)))) ==== iiiicccc∗∗∗∗1111((((0000))))eeeeiiii((((−−−−2222ααααxxxx++++4444((((ββββ2222−−−−αααα2222))))tttt 2222ββββ

||||cccc1111((((0000))))||||sssseeeecccchhhh ((((2222ββββ((((xxxx++++ 4444ααααtttt)))) −−−− llllnnnn

||||cccc1111((((0000))))||||2222ββββ

))))

Another example:Another example: The Lax pair for the Boussinesq-type equation

ψψψψtttt ==== AAAAψψψψ ≡≡≡≡ ((((∂∂∂∂2222xxxxxxxx ++++2222

3333uuuu))))ψψψψ

˙uuuu ==== 2222vvvv′′′′ −−−− uuuu′′′′′′′′;;;;

˙vvvv ==== vvvv′′′′′′′′ −−−− 22223333uuuu′′′′′′′′′′′′ −−−− 2222

3333uuuuuuuu′′′′

uuuutttttttt ==== −−−−1111

3333uuuuxxxxxxxxxxxx −−−−

4444

3333((((uuuuuuuuxxxx))))xxxx

Boussinesq-type equations describe waves which can propagate both to the right, and to the left (“the two-way long-wave equations”).

Apropos: Apropos: BoussinesqBoussinesqequationequation

uuuutttttttt −−−− uuuuxxxxxxxx ++++ 3333((((uuuu2222))))xxxxxxxx ++++ uuuuxxxxxxxxxxxxxxxx ==== 0000

uuuuuuuu ≡≡≡≡≡≡≡≡ uuuuuuuu((((((((θθθθθθθθ)))))))) wwwwwwwwhhhhhhhheeeeeeeerrrrrrrreeeeeeee θθθθθθθθ ======== xxxxxxxx −−−−−−−− vvvvvvvvttttttttTravelling wave solution has the form

uuuuuuuu((((((((xxxxxxxx,,,,,,,, tttttttt)))))))) ======== 22222222aaaaaaaa22222222sssssssseeeeeeeecccccccchhhhhhhh22222222((((((((aaaaaaaa((((((((xxxxxxxx −−−−−−−− vvvvvvvvtttttttt))))))))))))))));;;;;;;; vvvvvvvv ======== ±±±±±±±±√√√√√√√√11111111 −−−−−−−− 44444444aaaaaaaa22222222

LLLLψψψψ ≡≡≡≡ ((((−−−−∂∂∂∂2222xxxxxxxx −−−− uuuu))))ψψψψ ==== λλλλψψψψ ψψψψtttt ==== AAAAψψψψ ≡≡≡≡ ((((−−−−4444∂∂∂∂3333xxxxxxxxxxxx −−−− 6666uuuu∂∂∂∂xxxx −−−− 3333uuuuxxxx))))ψψψψ

Recall:Recall: The Lax pair for the KdV equation:

LLLLψψψψ ≡≡≡≡ ((((−−−−∂∂∂∂3333xxxxxxxxxxxx ++++ uuuu∂∂∂∂xxxx ++++ vvvv))))ψψψψ ==== λλλλψψψψ

dddd

ddddtttt((((−−−−∂∂∂∂3333 ++++ uuuu∂∂∂∂ ++++ vvvv)))) ==== [[[[AAAA,,,,LLLL]]]] ==== ((((2222vvvv′′′′ −−−− uuuu′′′′′′′′))))∂∂∂∂ ++++ vvvv′′′′′′′′ −−−− 2222

3333uuuu′′′′′′′′′′′′ −−−− 2222

3333uuuuuuuu′′′′

FermiFermi--PastaPasta--UlamUlam systemsystem

E Fermi, J Pasta, and S Ulam (1955): numerical study of the dynamics of an anharmonic chain of particles connected to their nearest neighbours by weakly nonlinear springs

MANIACMANIAC --11(Mathematical Analyzer Numerical Integrator And Computer)

a

- displacement of the n-thparticle from the equilibriumuuuuuuuunnnnnnnn

Weak non-linearitymmmmmmmm ¨uuuuuuuunnnnnnnn ======== ffffffff ((((((((uuuuuuuunnnnnnnn++++++++11111111 −−−−−−−− uuuuuuuunnnnnnnn)))))))) −−−−−−−− ffffffff ((((((((uuuuuuuunnnnnnnn −−−−−−−− uuuuuuuunnnnnnnn−−−−−−−−11111111))))))));;;;;;;; nnnnnnnn ======== 11111111,,,,,,,, 22222222 ........ ........ ........ NNNNNNNN ======== 6666666644444444

ffff((((∆∆∆∆uuuu)))) ==== kkkk∆∆∆∆uuuu++++ αααα((((∆∆∆∆uuuu))))2222

mmmm ¨uuuunnnn ==== kkkk((((uuuunnnn++++1111 −−−− 2222uuuunnnn ++++ uuuunnnn−−−−1111)))) ++++ αααα[[[[((((uuuunnnn++++1111 −−−− uuuunnnn))))

2222 −−−− ((((uuuunnnn −−−− uuuunnnn−−−−1111))))2222]]]]

A general solution of the linearized system(a=0) is given by the expansion in the normal modes:

uuuukkkknnnn((((tttt)))) ==== AAAAkkkk ssssiiiinnnn

((((kkkkππππnnnnaaaa

NNNN ++++ 1111

))))ccccoooossss((((ωωωωkkkktttt++++ δδδδkkkk)))))))) ωωωωkkkk ==== 2222

√√√√kkkk

mmmmssssiiiinnnn

((((kkkkππππaaaa

2222((((NNNN ++++ 1111))))

))))

FermiFermi--PastaPasta--UlamUlam…… + Mary + Mary TsingouTsingou

All numerical simulations of the Fermi-Pasta-Ulamproblem were performed by Mary Tsingou

Note:Note: There is no energy transfer between the modes in the linear approximation. In the nonlinear chain (a≠0) modes become coupled. It was expected that if all the initial energy was put into a few lowest modes, the nonlinear coupling would yield equal distribution of the energy among the normal modes.

What was observed?What was observed?

RecurrencesRecurrences

However:However: If the energy was initially in the mode of lowest frequency, it returned almost entirely to that mode after interaction with a few other low frequency modes

From FermiFrom Fermi --PastaPasta--UlamUlam to to BoussinesqBoussinesqequationequation

Continuum approximation Continuum approximation (Zabusky and Kruskal (1965)):

uuuunnnn((((tttt)))) ==== uuuu((((xxxxnnnn,,,, tttt)))) ==== uuuu((((nnnnaaaa,,,, tttt))));;;; uuuunnnn±±±±1111((((tttt)))) ==== uuuu((((xxxxnnnn ±±±± aaaa,,,, tttt))))

Gradient expansion:Gradient expansion:

mmmm ¨uuuunnnn ==== kkkk((((uuuunnnn++++1111 −−−− 2222uuuunnnn ++++ uuuunnnn−−−−1111)))) ++++ αααα[[[[((((uuuunnnn++++1111 −−−− uuuunnnn))))

2222 −−−− ((((uuuunnnn −−−− uuuunnnn−−−−1111))))2222]]]]

FPU:FPU:FPU:

uuuunnnn±±±±1111((((tttt)))) ≈≈≈≈ uuuu((((xxxxnnnn,,,, tttt))))±±±± aaaauuuu′′′′((((xxxxnnnn,,,, tttt))))++++aaaa2222

2222uuuu′′′′′′′′((((xxxxnnnn,,,, tttt))))±±±±

aaaa3333

3333!!!!uuuu′′′′′′′′′′′′((((xxxxnnnn,,,, tttt))))++++

aaaa4444

4444!!!!uuuu′′′′′′′′′′′′′′′′((((xxxxnnnn,,,, tttt))))++++ .... .... ....

uuuuuuuutttttttttttttttt −−−−−−−− cccccccc22222222uuuuuuuuxxxxxxxxxxxxxxxx ======== εεεεεεεεcccccccc22222222((((((((uuuuuuuuxxxxxxxxuuuuuuuuxxxxxxxxxxxxxxxx ++++++++ δδδδδδδδ22222222uuuuuuuuxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx))))))))Boussinesq:BoussinesqBoussinesq:: cccc2222==== kkkkaaaa2222

mmmm;;;; εεεε==== 2222ααααaaaa

kkkk;;;; δδδδ2222==== aaaa2222

11112222εεεε

Note:Note: the leading order nonlinear and dispersive contributions in the r.h.s. are balanced at the same order of ε

J. Boussinesq originally derived a system of two first-order (in time) equations for weakly nonlinear surface waves in shallow water (1881)

ηηηηtttt ++++ uuuuxxxx ++++ ((((ηηηηuuuu))))xxxx ==== 0000uuuutttt ++++ ηηηηxxxx ++++ uuuuuuuuxxxx −−−− uuuuxxxxxxxxtttt ==== 0000

u is the horizontal velocity

h is the free surface elevation

From From BoussinesqBoussinesqequation to equation to KdVKdV

uuuuuuuutttttttttttttttt −−−−−−−− cccccccc22222222uuuuuuuuxxxxxxxxxxxxxxxx ======== εεεεεεεεcccccccc22222222((((((((uuuuuuuuxxxxxxxxuuuuuuuuxxxxxxxxxxxxxxxx ++++++++ δδδδδδδδ22222222uuuuuuuuxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx))))))))

Asymptotic Asymptotic miltiplemiltiple --scale expansion: scale expansion:

vvvvtttttttt −−−− cccc2222vvvvxxxxxxxx ==== 2222ccccffffTTTTθθθθ ++++ cccc2222ffffθθθθffffθθθθθθθθ ++++ cccc2222δδδδ2222ffffθθθθθθθθθθθθθθθθ ++++ .... .... ....

uuuu((((xxxx,,,, tttt)))) ==== ffff((((θθθθ,,,, TTTT )))) ++++ εεεεvvvv((((xxxx,,,, tttt))))θθθθ ==== xxxx−−−− cccctttt,,,, TTTT ==== εεεεtttt

Note:Note: The function grows linearly in unless the r.h.s is not zero:vvvv((((xxxx,,,, tttt)))) ¯θθθθ ==== xxxx++++ cccctttt

2222ccccffffTTTTθθθθ ++++ cccc2222ffffθθθθffffθθθθθθθθ ++++ cccc2222δδδδ2222ffffθθθθθθθθθθθθθθθθ ==== 0000

Change of variables:Change of variables: qqqqqqqq ========ffffffffθθθθθθθθ

66666666;;;;;;;; ττττττττ ========

ccccccccTTTTTTTT

22222222

qqqqττττ ++++ 6666qqqqqqqqθθθθ ++++ δδδδ2222qqqqθθθθθθθθθθθθ ==== 0000KdV:KdV:

In 1965 Kruskal and Zabusky numerically studied the dynamics of the KdV equation with sinusoidal initial conditions (for small d =0.022 with periodic boundary conditions)

The appearing solitary waves interact with each other elasticallyThey have called the waves solitonssolitons since they behave like particlesExplanation of the FPU reccurence as property of the system of solitons movingwith different speed. Since the system studied was of finite length, solitons eventuallyreassembled in the (x, t) plane and approximately recreated the initial configuration

The The FrenkelFrenkel--KontorovaKontorova modelmodel

a The original FK model (1938) was proposed to describe dislocations in metals. The atoms are treated as a one-dimensional chain subjected to an external periodic potential produced by the surrounding atoms.2V0

LLLL ==== TTTT −−−− VVVV ====∑∑∑∑

nnnn

mmmm ˙uuuu2222nnnn2222

−−−− αααα

2222

∑∑∑∑

nnnn

((((uuuunnnn++++1111 −−−− uuuunnnn))))2222 −−−− VVVV0000

∑∑∑∑

nnnn

((((1111 −−−− ccccoooossss

2222ππππuuuunnnn

aaaa

))))

Rescaling I:Rescaling I: uuuunnnn →→→→2222ππππuuuunnnn

aaaa;;;; tttt→→→→ 2222ππππ

aaaa

√√√√VVVV0000mmmm

tttt;;;; αααα→→→→ αααα((((aaaa////2222ππππ))))2222

VVVV0000

¨uuuunnnn −−−− αααα((((uuuunnnn++++1111 −−−− 2222uuuunnnn ++++ uuuunnnn−−−−1111)))) ==== −−−− ssssiiiinnnn uuuunnnn

Gradient expansion:Gradient expansion: uuuutttttttt −−−− ααααaaaa2222uuuuxxxxxxxx ==== −−−− ssssiiiinnnn uuuu

Rescaling II:Rescaling II: xxxx→→→→ xxxx

aaaa√√√√αααα uuuuuuuutttttttttttttttt −−−−−−−− uuuuuuuuxxxxxxxxxxxxxxxx ++++++++ ssssssssiiiiiiiinnnnnnnn uuuuuuuu ======== 00000000

This is the sine-Gordon equationThis is the sine-Gordon equation

SineSine--Gordon model: Josephson junctionsGordon model: Josephson junctions

Superconducting transmission line

Capacitance CCCC per unit lengthInductance LLLL per unit lengthCritical current IIII0 per unit length

Josephson phase f= f2- f1Voltage V

∂∂∂∂VVVV

∂∂∂∂xxxx==== −−−−LLLL∂∂∂∂IIII

∂∂∂∂tttt

∂∂∂∂φφφφ

∂∂∂∂tttt====

2222eeeeVVVV

∂∂∂∂IIII

∂∂∂∂xxxx==== −−−−CCCC ∂∂∂∂VVVV

∂∂∂∂tttt−−−− IIII0000 ssssiiiinnnnφφφφ

∂∂∂∂φφφφ

∂∂∂∂xxxx==== −−−−2222eeeeLLLL

IIII

shunt current ∂∂∂∂

∂∂∂∂xxxx

1111

cccc2222∂∂∂∂2222φφφφ

∂∂∂∂tttt2222−−−− ∂∂∂∂2222φφφφ

∂∂∂∂xxxx2222++++

1111

λλλλ2222ssssiiiinnnn φφφφ ==== 0000

cccccccc ======== 11111111////////√√√√√√√√LLLLLLLLCCCCCCCC

Swihart velocity:SwihartSwihart velocity:velocity: Josephson lengthJosephson lengthJosephson length

λλλλ ====√√√√////2222eeeeLLLLIIII0000

sine-Gordon equation!sine-Gordon equation!

Field equation:

SineSine--Gordon model: scalar field theoryGordon model: scalar field theory

LLLL ====1111

2222∂∂∂∂µµµµφφφφ∂∂∂∂

µµµµφφφφ−−−− UUUU ((((φφφφ))));;;; UUUU((((φφφφ)))) ==== 1111−−−− ccccoooossssφφφφUUUU ((((φφφφ))))

Symmetry:

φφφφ→→→→ φφφφ±±±± 2222ππππnnnn,,,, nnnn ∈∈∈∈ ZZZZ

Integration:Integration: X integrating factor

Note:Note: The model is Lorentz-invariant

Non-trivial static solutions: the function f(x) interpolatesbetween φφφφ((((−−−−∞∞∞∞)))) ==== 0000 aaaannnndddd φφφφ((((++++∞∞∞∞)))) ==== 2222ππππ

ddddφφφφddddxxxx

φφφφtttttttt −−−− φφφφxxxxxxxx ++++ ssssiiiinnnnφφφφ ==== 0000

φφφφxxxxxxxx ==== ssssiiiinnnnφφφφ1111

2222

((((ddddφφφφ

ddddxxxx

))))2222==== −−−− ccccoooossssφφφφ ++++ CCCC Boundary condition: φφφφxxxx →→→→ 0000 aaaassss xxxx→→→→ ±±±±∞∞∞∞ CCCC ==== 1111

It looks like an equation of motion of a “particle” in the effective potential

Separating the variables:Separating the variables:

Kink solution:Kink solution: Busted kink: xxxx→→→→ xxxx−−−− vvvvtttt√√√√1111 −−−− vvvv2222

xxxx −−−− xxxx0000 ==== ±±±±∫∫∫∫

ddddφφφφ√√√√

2222((((1111 −−−− ccccoooossss φφφφ))))==== ±±±±

∫∫∫∫ddddφφφφ

2222 ssssiiiinnnn((((φφφφ////2222))))====

∫∫∫∫dddd((((llllnnnn ttttaaaannnn

φφφφ

4444))))

φφφφ ==== ±±±±4444 aaaarrrrccccttttaaaannnn eeeexxxxpppp ((((xxxx−−−− xxxx0000))))

Topological current:

SineSine--Gordon model: scalar field theoryGordon model: scalar field theory

φφφφKKKK

φφφφ ¯KKKK

Kink solution:

Energy density: EEEEEEEE ========44444444

ccccccccoooooooosssssssshhhhhhhh22222222((((((((xxxxxxxx −−−−−−−− xxxxxxxx00000000))))))))Mass of the kink: MMMM ====

∫∫∫∫EEEEddddxxxx ==== 8888

Topological charge: QQQQQQQQ ======== 1111111122222222ππππππππ

∞∞∞∞∞∞∞∞∫∫∫∫∫∫∫∫

−−−−−−−−∞∞∞∞∞∞∞∞ddddddddxxxxxxxx ∂∂∂∂∂∂∂∂φφφφφφφφ

∂∂∂∂∂∂∂∂xxxxxxxx - n-fold cover of [0,2[0,2]]

JJJJµµµµ ====1111

2222ππππεεεεµµµµνννν∂∂∂∂

ννννφφφφ,,,, ∂∂∂∂µµµµJJJJµµµµ ≡≡≡≡ 0000

Note: Note: This is not a Noether current!

Unlike KdV solitary wave the function f(x) does not go to 0 as The amplitude of the SG soliton is independent of its velocity SG soliton is topologicalThe SG model is integrableThe SG model is relativistic-invariant

xxxx→→→→ ±±±±∞∞∞∞

φφφφKKKK ¯KKKK ==== ±±±±4444 aaaarrrrccccttttaaaannnn((((eeee−−−−xxxx++++xxxx0000))))

Note:Note:

For small f(x) 1111−−−− ccccoooossss φφφφ ≈≈≈≈ φφφφ2222

2222++++

φφφφ4444

4444++++ .... .... ....

LLLL ====1111

2222∂∂∂∂µµµµφφφφ∂∂∂∂

µµµµφφφφ −−−− ((((φφφφ2222 −−−− 1111))))2222ff4 4 modelmodel

SineSine--Gordon model is Gordon model is integrableintegrable!!

BBääcklundcklund transformation:transformation: if we have a solution of an integrable system, even a trivial one, there is a transformation which transforms it into a new non-trivial solution.

Example I:Example I: Laplace equation in 2d ∆∆∆∆uuuu((((xxxx,,,, yyyy)))) ==== ((((∂∂∂∂2222xxxx ++++ ∂∂∂∂2222yyyy))))uuuu ==== 0000

Let us take another equation for a new function :vvvv((((xxxx,,,, yyyy))))

∆∆∆∆∆∆∆∆vvvvvvvv((((((((xxxxxxxx,,,,,,,, yyyyyyyy)))))))) ======== ((((((((∂∂∂∂∂∂∂∂22222222xxxxxxxx ++++++++ ∂∂∂∂∂∂∂∂22222222yyyyyyyy ))))))))vvvvvvvv ======== 00000000

Note:Note: the functions and are not independent:uuuu((((xxxx,,,, tttt)))) vvvv((((xxxx,,,, tttt))))

∂∂∂∂∂∂∂∂xxxxxxxxuuuuuuuu ======== ∂∂∂∂∂∂∂∂yyyyyyyyvvvvvvvv;;;;;;;; ∂∂∂∂∂∂∂∂yyyyyyyyuuuuuuuu ======== −−−−−−−−∂∂∂∂∂∂∂∂xxxxxxxxvvvvvvvv

Indeed , so sum of these two

equations yields the original Laplace equation.

∂∂∂∂xxxx((((∂∂∂∂xxxxuuuu)))) ==== ∂∂∂∂xxxx((((∂∂∂∂yyyyvvvv)))),,,, ∂∂∂∂yyyy((((∂∂∂∂yyyyuuuu)))) ==== −−−−∂∂∂∂yyyy((((∂∂∂∂xxxxvvvv))))

Now we take the trivial solution vvvvvvvv((x,yx,y) = ) = xyxy and plug it into the Bäcklund transformation:

Bäcklund transformationBäcklund transformation

uuuuuuuuxxxxxxxx ======== xxxxxxxx;;;;;;;; uuuuuuuuyyyyyyyy ======== −−−−−−−−yyyyyyyy iiiiiiii........eeeeeeee........ uuuuuuuu ======== 1111111122222222

((((((((xxxxxxxx22222222 −−−−−−−− yyyyyyyy22222222

))))))))

BBääcklundcklund transformation for the sinetransformation for the sine--Gordon modelGordon modelLight cone coordinates:Light cone coordinates:

Then the SG equation becomes

xxxx±±±± ==== 11112222((((xxxx±±±± tttt)))) ∂∂∂∂xxxx ==== 1111

2222∂∂∂∂++++ ++++ 1111

2222∂∂∂∂−−−−;;;; ∂∂∂∂tttt ====

11112222∂∂∂∂++++ −−−− 1111

2222∂∂∂∂−−−−

∂∂∂∂2222tttt −−−− ∂∂∂∂2222xxxx ==== −−−−∂∂∂∂−−−−∂∂∂∂++++ ∂∂∂∂−−−−∂∂∂∂++++φφφφ ==== ssssiiiinnnnφφφφ

Consider the pair of equations:Consider the pair of equations:

∂∂∂∂++++ψψψψ ==== ∂∂∂∂++++φφφφ−−−− 2222λλλλ ssssiiiinnnn

((((φφφφ++++ ψψψψ2222

)))),,,, ∂∂∂∂−−−−ψψψψ ==== −−−−∂∂∂∂−−−−φφφφ++++

2222λλλλssssiiiinnnn

((((φφφφ −−−− ψψψψ2222

))))

∂∂∂∂−−−−∂∂∂∂++++ψψψψ ==== ∂∂∂∂−−−−∂∂∂∂++++φφφφ−−−− 2222 ccccoooossss

((((φφφφ++++ ψψψψ2222

))))ssssiiiinnnn

((((φφφφ −−−− ψψψψ2222

))))==== ∂∂∂∂−−−−∂∂∂∂++++φφφφ++++ ssssiiiinnnnφφφφ−−−− ssssiiiinnnnψψψψ

IIIIIIIIffffffff ∂∂∂∂∂∂∂∂−−−−−−−−∂∂∂∂∂∂∂∂++++++++φφφφφφφφ ======== ssssssssiiiiiiiinnnnnnnn φφφφφφφφ,,,,,,,, tttttttthhhhhhhheeeeeeeennnnnnnn ∂∂∂∂∂∂∂∂−−−−−−−−∂∂∂∂∂∂∂∂++++++++ψψψψψψψψ ======== ssssssssiiiiiiiinnnnnnnn ψψψψψψψψ

SG Bäcklund transformationSG Bäcklund transformation

Start with the trivial vacuum solution: Start with the trivial vacuum solution: f=0

∂∂∂∂++++ψψψψ====−−−−2222λλλλssssiiiinnnn((((ψψψψ////2222))));;;; ∂∂∂∂−−−−ψψψψ====−−−−2222λλλλ−−−−1111 ssssiiiinnnn((((ψψψψ////2222))))Back to original coordinates:Back to original coordinates: λλλλxxxx++++ ++++ λλλλ−−−−1111xxxx−−−− ==== ±±±± xxxx−−−− vvvvtttt√√√√

1111−−−− vvvv2222v =

1− λ2

1 + λ2

ψψψψ ==== 4444 aaaarrrrccccttttaaaannnn((((eeee−−−−λλλλxxxx++++ −−−− λλλλ−−−−1111xxxx−−−−))))

Kink solution: φφφφKKKK ¯KKKK ==== ±±±±4444 aaaarrrrccccttttaaaannnn((((eeee±±±±xxxx−−−−vvvvtttt√√√√1111−−−−vvvv2222 ))))

Homework: Prove it!

SG twoSG two --solitonsoliton solution:solution:

BBääcklundcklund transformation for the sinetransformation for the sine--Gordon modelGordon model

φφφφ0000ψψψψ2222

ψψψψ1111λ1

λ2

λ2

λ1φφφφ2222

Elimitating the derivatives in the SG Bäcklund transformation, we obtain (f0=0)

ttttaaaannnn

((((φφφφ22224444

))))====

((((λλλλ1111 ++++ λλλλ2222λλλλ1111 −−−− λλλλ2222

))))ttttaaaannnn

((((ψψψψ2222 −−−− ψψψψ1111

2222

))))

Recall:Recall:2 one-soliton solutions

ψψψψ1111,,,,2222 ==== 4444 aaaarrrrccccttttaaaannnn eeeeθθθθ1111,,,,2222 θθθθ1111,,,,2222 ====1111

2222

((((λλλλiiiixxxx++++ λλλλ−−−−1111iiii tttt++++ CCCCiiii

))))

φφφφ2222 ==== 4444 aaaarrrrccccttttaaaannnn

((((λλλλ1111 ++++ λλλλ2222λλλλ1111 −−−− λλλλ2222

))))eeeeθθθθ1111 −−−− eeeeθθθθ2222

1111 ++++ eeeeθθθθ1111++++θθθθ2222two-soliton solution

Consider asymptotic:Consider asymptotic: θθθθ2222 ≫≫≫≫ 1111 eeeeθθθθ1111 −−−− eeeeθθθθ2222

1111 ++++ eeeeθθθθ1111++++θθθθ2222→→→→ eeeeθθθθ1111////eeeeθθθθ2222 −−−− 1111

eeee−−−−θθθθ2222 ++++ eeeeθθθθ1111∼∼∼∼ −−−−eeee−−−−θθθθ1111

Topological charge:

The symmetric 2The symmetric 2 --kink solution kink solution (head(head --on collision, identical velocities):on collision, identical velocities):

QQQQ ====1111

2222ππππ

∞∞∞∞∫∫∫∫

−−−−∞∞∞∞

ddddxxxx∂∂∂∂φφφφ2222∂∂∂∂xxxx

==== 2222φφφφ2222 ==== 4444 aaaarrrrccccttttaaaannnn

[[[[vvvv ssssiiiinnnnhhhh xxxx√√√√

1111−−−−vvvv2222

ccccoooosssshhhh vvvvtttt√√√√1111−−−−vvvv2222

]]]]λλλλ2222 ==== −−−−

1111

λλλλ1111;;;; vvvv ====

1111−−−− λλλλ222211111111 ++++ λλλλ22221111

,,,, λλλλ1111 >>>> 0000

sinesine--Gordon model: 2Gordon model: 2--soliton interactionssoliton interactions

φφφφ2222 ==== 4444 aaaarrrrccccttttaaaannnn

[[[[vvvv ssssiiiinnnnhhhh xxxx√√√√

1111−−−−vvvv2222

ccccoooosssshhhh vvvvtttt√√√√1111−−−−vvvv2222

]]]]KKKK --collisoncollisonv=0.8

x

t

Asymptotic: The phase shift:

θθθθ →→→→ ±±±±∞∞∞∞δδδδ ==== 2222

√√√√vvvv2222 −−−− 1111 llllnnnn vvvv

KKKK --collisoncollisonv=0.8

λλλλ2222 ====1111

λλλλ1111;;;; vvvv ====

1111−−−− λλλλ222211111111 ++++ λλλλ22221111

,,,, λλλλ1111 >>>> 0000

x

t

φφφφφφφφ22222222 ======== 44444444 aaaaaaaarrrrrrrrccccccccttttttttaaaaaaaannnnnnnn

[[[[[[[[ssssssssiiiiiiiinnnnnnnnhhhhhhhh vvvvvvvvtttttttt√√√√√√√√

11111111−−−−−−−−vvvvvvvv22222222

vvvvvvvv ccccccccoooooooosssssssshhhhhhhh xxxxxxxx√√√√√√√√11111111−−−−−−−−vvvvvvvv22222222

]]]]]]]]

Breather:Breather:

vvvv ====iiiiωωωω√√√√1111−−−− ωωωω2222

φφφφ2222 ==== 4444 aaaarrrrccccttttaaaannnn

[[[[√√√√1111 −−−− ωωωω2222

ωωωω

ssssiiiinnnn ωωωωtttt

ccccoooosssshhhh√√√√1111 −−−− ωωωω2222xxxx

]]]]

φφφφ2222

φφφφ2222

t

x

sinesine--Gordon model: Lax pair formulationGordon model: Lax pair formulationRecall: Lax pair is given by two linear equations

ψψψψxxxx ==== LLLLψψψψ;;;; ψψψψtttt ==== AAAAψψψψ ψψψψ ====

((((ψψψψ11111111 ψψψψ11112222ψψψψ22221111 ψψψψ22222222

))))

ψψψψxxxxtttt ==== LLLLttttψψψψ ++++ LLLLψψψψtttt;;;;ψψψψttttxxxx ==== AAAAxxxxψψψψ ++++ AAAAψψψψxxxx....

LLLLttttψψψψ ++++ LLLLAAAAψψψψ ==== AAAAxxxxψψψψ ++++ AAAALLLLψψψψ;;;; LLLLtttt −−−−AAAAxxxx ==== [[[[AAAA,,,, LLLL]]]]

Zero curvature conditionsinesine --Gordon:Gordon:

LLLL ==== iiiiλλλλ

((((1111 00000000 −−−−1111

))))++++

iiii

2222

((((0000 uuuuxxxx

uuuuxxxx 0000

))))==== iiiiλλλλ ···· σσσσ3333 ++++

iiii

2222uuuuxxxx ···· σσσσ1111;;;; λλλλ ∈∈∈∈ CCCC

AAAA ====ccccoooossss uuuu

4444iiiiλλλλ

((((1111 00000000 −−−−1111

))))++++

1111

4444iiiiλλλλ

((((0000 −−−−iiiiiiii 0000

))))====

ccccoooossss uuuu

4444iiiiλλλλ···· σσσσ3333 ++++

1111

4444iiiiλλλλ···· σσσσ2222

LLLLtttt ====iiiiuuuuttttxxxx

2222···· σσσσ1111;;;; AAAAxxxx ==== −−−− 1111

4444iiiiλλλλuuuuxxxx ssssiiiinnnn uuuu ···· σσσσ3333 ++++

1111

4444iiiiλλλλuuuuxxxx ···· σσσσ2222

[[[[AAAA,,,, LLLL]]]] ====iiii

4444λλλλ···· σσσσ2222 −−−−

iiii

4444λλλλ···· σσσσ3333 ++++

iiii

2222ssssiiiinnnn uuuu ···· σσσσ1111

in 0th order in λ

iiiiuuuuttttxxxx

2222···· σσσσ1111 ====

iiii

2222ssssiiiinnnn uuuu ···· σσσσ1111

sine-Gordon equation is recovered!

sine-Gordon ↔massive Thirring model

SSSSSSSS ========

∫∫∫∫∫∫∫∫dddddddd22222222xxxxxxxx

[[[[[[[[11111111

22222222∂∂∂∂∂∂∂∂µµµµµµµµφφφφφφφφ∂∂∂∂∂∂∂∂

µµµµµµµµφφφφφφφφ −−−−−−−− αααααααα

ββββββββ22222222((((((((11111111 −−−−−−−− ccccccccoooooooossssssss ββββββββφφφφφφφφ))))))))

]]]]]]]]

sine-Gordon model

ThirringThirring modelmodel

Meson Meson statesstates →→ fermionfermion--antianti fermionfermion boundbound statesstates

SolitonSoliton →→ fundamental fundamental fermionfermion

γγγγγγγγ00000000 ======== σσσσσσσσ11111111 ,,,,,,,, γγγγγγγγ11111111 ======== −−−−−−−−iiiiiiiiσσσσσσσσ22222222 ,,,,,,,, γγγγγγγγ55555555 ======== γγγγγγγγ00000000γγγγγγγγ11111111 ======== σσσσσσσσ33333333

SSSSSSSS ========

∫∫∫∫∫∫∫∫dddddddd22222222xxxxxxxx

[[[[[[[[iiiiiiii ¯ψψψψψψψψγγγγγγγγµµµµµµµµ∂∂∂∂∂∂∂∂

µµµµµµµµψψψψψψψψ ++++++++ mmmmmmmm ¯ψψψψψψψψψψψψψψψψ −−−−−−−− gggggggg

22222222(((((((( ¯ψψψψψψψψγγγγγγγγµµµµµµµµψψψψψψψψ))))))))(((((((( ¯ψψψψψψψψγγγγγγγγ

µµµµµµµµψψψψψψψψ))))))))]]]]]]]]

BosonizationBosonization:: ββββββββ22222222

44444444ππππππππ ======== 1111111111111111++++++++gggggggg////////ππππππππ

InvarianciesInvariancies::

mmmmmmmm ¯ψψψψψψψψ11111111 ∓∓∓∓∓∓∓∓ γγγγγγγγ55555555

22222222ψψψψψψψψ ======== −−−−−−−− αααααααα

ββββββββ22222222eeeeeeee±±±±±±±±iiiiiiiiφφφφφφφφ

φφφφφφφφ →→→→→→→→ φφφφφφφφ′′′′′′′′ ======== φφφφφφφφ ++++++++ 22222222ππππππππnnnnnnnnββββββββ ;;;;;;;; ψψψψψψψψ →→→→→→→→ ψψψψψψψψ′′′′′′′′ ======== eeeeeeeeiiiiiiiiααααααααVVVVVVVV ψψψψψψψψ;;;;;;;; ψψψψψψψψ →→→→→→→→ ψψψψψψψψ′′′′′′′′ ======== eeeeeeeeiiiiiiiiγγγγγγγγ55555555ααααααααAAAAAAAAψψψψψψψψ

The topological current of the sineThe topological current of the sine--Gordon model Gordon model

coincides with the coincides with the NoetherNoether current of the massive current of the massive ThirringThirring modelmodel

JJJJµµµµ ==== 11112222ππππεεεεµµµµνννν∂∂∂∂

ννννφφφφ

jjjjjjjjµµµµµµµµ ======== iiiiiiii ¯ψψψψψψψψγγγγγγγγµµµµµµµµ∂∂∂∂∂∂∂∂µµµµµµµµψψψψψψψψ

((S.ColemanS.Coleman, 1975), 1975)

EquationEquation SolutionSolution

YESYES

NO!NO!

How do we know if it is How do we know if it is integrableintegrable or it is a nonor it is a non --integrableintegrable ? ?

Historically, combination of Historically, combination of ““ experimental mathematicsexperimental mathematics ”” ((φφφφφφφφ44) and) andknown analytic solutions (Sknown analytic solutions (S --G), then inverse scattering transform, G), then inverse scattering transform, group theoretic structure (Kacgroup theoretic structure (Kac --Moody Algebras), PainlevMoody Algebras), Painlev éé test.test.

λλφφφφφφφφ4 4 ::

SS--G:G:

Does any part of Does any part of ““ hierarchyhierarchy ”” of of solitonssolitons in in integrableintegrabletheories (Stheories (S --G breather) exist in nonG breather) exist in non --intergrableintergrabletheories? theories?

Solitons vs. Solitary Waves

¨φφφφφφφφ −−−−−−−− φφφφφφφφ′′′′′′′′′′′′′′′′ ++++++++ ssssssssiiiiiiiinnnnnnnn φφφφφφφφ ======== 00000000 φφφφφφφφKKKKKKKK ¯KKKKKKKK ======== ±±±±±±±±44444444 aaaaaaaarrrrrrrrccccccccttttttttaaaaaaaannnnnnnn ((((((((eeeeeeee−−−−−−−−xxxxxxxx++++++++xxxxxxxx00000000 ))))))))

¨φφφφφφφφ −−−−−−−− φφφφφφφφ′′′′′′′′′′′′′′′′ −−−−−−−− φφφφφφφφ ++++++++ φφφφφφφφ33333333 ======== 00000000 φφφφφφφφKKKKKKKK ¯KKKKKKKK ======== ±±±±±±±±aaaaaaaa ttttttttaaaaaaaannnnnnnnhhhhhhhh((((((((

mmmmmmmm((((((((xxxxxxxx−−−−−−−−xxxxxxxx00000000))))))))√√√√√√√√22222222

))))))))

Topology primer: maps and windings

Kinks in 2d:Kinks in 2d:

Space:++∞∞

--∞∞Vacuum:

+1+1

-- 11Maps:

Circles: SCircles: S11 →→SS11

Space: Vacuum:

Maps:

φφφφφφφφαααααααα ======== ((((((((ssssssssiiiiiiiinnnnnnnn ϕϕϕϕϕϕϕϕ;;;;;;;; ccccccccoooooooossssssss ϕϕϕϕϕϕϕϕ))))))))

Topological charge: QQQQQQQQ ======== 1111111122222222

∞∞∞∞∞∞∞∞∫∫∫∫∫∫∫∫

−−−−−−−−∞∞∞∞∞∞∞∞ddddddddxxxxxxxx ∂∂∂∂∂∂∂∂φφφφφφφφ

∂∂∂∂∂∂∂∂xxxxxxxx ======== φφφφφφφφ((((((((∞∞∞∞∞∞∞∞)))))))) −−−−−−−− φφφφφφφφ((((((((−−−−−−−−∞∞∞∞∞∞∞∞))))))))

Circles: SCircles: S11 →→SS11

Vacuum:

Topological charge: QQQQQQQQ ======== 1111111122222222ππππππππ

22222222ππππππππ∫∫∫∫∫∫∫∫

00000000

ddddddddϕϕϕϕϕϕϕϕ εεεεεεεεααααααααββββββββddddddddφφφφφφφφαααααααα

ddddddddϕϕϕϕϕϕϕϕφφφφφφφφββββββββ

Q=0:Q=0: φφφφφφφφαααααααα ======== ((((((((00000000,,,,,,,, 11111111))))))))

φφφφφφφφαααααααα ======== ((((((((ssssssssiiiiiiiinnnnnnnn ϕϕϕϕϕϕϕϕ;;;;;;;; ccccccccoooooooossssssss ϕϕϕϕϕϕϕϕ))))))))Q=1:Q=1:

Q=2:Q=2: φφφφφφφφαααααααα ======== ((((((((ssssssssiiiiiiiinnnnnnnn 22222222ϕϕϕϕϕϕϕϕ;;;;;;;; ccccccccoooooooossssssss 22222222ϕϕϕϕϕϕϕϕ))))))))

Consider a model with scalar field in dConsider a model with scalar field in d --dimdim

Scaling agruments: Derrick’s theorem

EEEEEEEE [[[[[[[[φφφφφφφφ]]]]]]]] ========∫∫∫∫∫∫∫∫ddddddddddddddddxxxxxxxx [[[[[[[[∂∂∂∂∂∂∂∂µµµµµµµµφφφφφφφφ∂∂∂∂∂∂∂∂

µµµµµµµµφφφφφφφφ ++++++++ UUUUUUUU ((((((((φφφφφφφφ))))))))]]]]]]]] ======== EEEEEEEE22222222 ++++++++ EEEEEEEE00000000

Scale transformation:Scale transformation: DDDDxxxx→→→→ DDDDyyyy ==== λλλλDDDDxxxx;;;; ∂∂∂∂µµµµφφφφ((((DDDDxxxx)))) ====∂∂∂∂φφφφ((((####xxxx))))∂∂∂∂xxxxµµµµ

→→→→ λλλλ ∂∂∂∂φφφφ((((λλλλ####xxxx))))∂∂∂∂((((λλλλxxxxµµµµ))))

==== λλλλ ∂∂∂∂φφφφ((((####yyyy))))∂∂∂∂yyyyµµµµ

ddddddddddddddddxxxxxxxx →→→→→→→→ dddddddddddddddd((((((((λλλλλλλλxxxxxxxx))))))))λλλλλλλλ−−−−−−−−dddddddd ======== λλλλλλλλ−−−−−−−−ddddddddddddddddddddddddyyyyyyyyEEEEEEEE [[[[[[[[φφφφφφφφ]]]]]]]] →→→→→→→→ λλλλλλλλ22222222−−−−−−−−ddddddddEEEEEEEE22222222 ++++++++ λλλλλλλλ−−−−−−−−ddddddddEEEEEEEE00000000

Each term is positive. If there is a stationary point of E(Each term is positive. If there is a stationary point of E(λλ)?)?ddddddddEEEEEEEE [[[[[[[[λλλλλλλλφφφφφφφφ]]]]]]]]

ddddddddλλλλλλλλ ======== ((((((((22222222 −−−−−−−− dddddddd))))))))λλλλλλλλ11111111−−−−−−−−ddddddddEEEEEEEE22222222 −−−−−−−− ddddddddλλλλλλλλ−−−−−−−−dddddddd−−−−−−−−11111111EEEEEEEE00000000

d=1d=1 d=2d=2 d=3d=3

EEEEEEEE [[[[[[[[φφφφφφφφ]]]]]]]] ========∫∫∫∫∫∫∫∫ddddddddddddddddxxxxxxxx [[[[[[[[∂∂∂∂∂∂∂∂µµµµµµµµφφφφφφφφ∂∂∂∂∂∂∂∂

µµµµµµµµφφφφφφφφ ++++++++ UUUUUUUU ((((((((φφφφφφφφ))))))))]]]]]]]] ======== EEEEEEEE22222222 ++++++++ EEEEEEEE00000000For a simple model For a simple model

nontrivial solutions (Enontrivial solutions (E 22 ≠≠ 0, 0, EE00 ≠≠ 0 0 ) are possible only in d=1 ) are possible only in d=1

There are 3 possibilities to evade DerrickThere are 3 possibilities to evade Derrick ’’s theorem:s theorem:

•• d=2:d=2: take take EE00 = 0, then the model is scale= 0, then the model is scale --invariantinvariant•• Extend the model including higher derivatives in Extend the model including higher derivatives in ϕϕ ((SkyrmeSkyrme model in model in d=3, baby d=3, baby SkyrmeSkyrme model in d=2, model in d=2, FaddeevFaddeev --SkyrmeSkyrme model in d=3)model in d=3)

•• Extend the model including gauge fields (monopoles in d=3, Extend the model including gauge fields (monopoles in d=3, instantonsinstantons in Euclidean space d=4) in Euclidean space d=4)

DDDDDDDDxxxxxxxx →→→→→→→→ λλλλλλλλDDDDDDDDxxxxxxxx ======== DDDDDDDDyyyyyyyy ;;;;;;;; AAAAAAAAµµµµµµµµ((((((((DDDDDDDDxxxxxxxx)))))))) →→→→→→→→ λλλλλλλλAAAAAAAAµµµµµµµµ((((((((DDDDDDDDyyyyyyyy))))))));;;;;;;; DDDDDDDDµµµµµµµµφφφφφφφφ((((((((DDDDDDDDxxxxxxxx)))))))) →→→→→→→→ λλλλλλλλDDDDDDDDµµµµµµµµφφφφφφφφ((((((((DDDDDDDDyyyyyyyy))))))));;;;;;;; FFFFFFFFµµµµµµµµνννννννν ((((((((DDDDDDDDxxxxxxxx)))))))) →→→→→→→→ λλλλλλλλ22222222FFFFFFFFµµµµµµµµνννννννν ((((((((DDDDDDDDyyyyyyyy))))))))

EEEEEEEE [[[[[[[[φφφφφφφφ]]]]]]]] ========∫∫∫∫∫∫∫∫ddddddddddddddddxxxxxxxx

[[[[[[[[||||||||FFFFFFFFµµµµµµµµνννννννν ||||||||22222222 ++++++++ ||||||||DDDDDDDDµµµµµµµµφφφφφφφφ||||||||22222222 ++++++++ UUUUUUUU ((((((((φφφφφφφφ))))))))

]]]]]]]]======== EEEEEEEE44444444 ++++++++ EEEEEEEE22222222 ++++++++ EEEEEEEE00000000

EEEEEEEE [[[[[[[[φφφφφφφφ]]]]]]]] →→→→→→→→ λλλλλλλλ44444444−−−−−−−−ddddddddEEEEEEEE44444444 ++++++++ λλλλλλλλ22222222−−−−−−−−ddddddddEEEEEEEE22222222 ++++++++ λλλλλλλλ−−−−−−−−ddddddddEEEEEEEE00000000

•• d=1:d=1: there are there are solitonsoliton solutions in thesolutions in the models with gauge models with gauge and scalar fields or in pure scalar models with a potential and scalar fields or in pure scalar models with a potential U(U(ϕϕ) ) ((KinksKinks ).).

•• d=2:d=2: there are there are solitonsoliton solutions in thesolutions in the models with gauge models with gauge and scalar fields and scalar fields ((vorticesvortices )) or in pure scalar models or in pure scalar models withoutwithoutpotential U(potential U( ϕϕ) ) ((LumpsLumps ))..•• d=3: d=3: there are there are solitonsoliton solutions in thesolutions in the models with gauge models with gauge and scalar fields and scalar fields ((monopolesmonopoles ))•• d=4: d=4: there are there are solitonsoliton solutions in thesolutions in the models with gauge models with gauge field onlyfield only ((instantonsinstantons ))•• d>4: d>4: there are no there are no solitonsoliton solutions, higher derivatives are solutions, higher derivatives are necessary.necessary.

If we restrict ourselves to the models with quadratic If we restrict ourselves to the models with quadratic

terms in derivatives, there are possibilities: terms in derivatives, there are possibilities:

Alternative:Alternative: one can consider time-dependent

stationary configurations!