PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY

PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY

Yair ZarmiPhysics Department &

Jacob Blaustein Institutes for Desert ResearchBen-Gurion University of the Negev

Midreshet Ben-Gurion, Israel

INTEGRABLE EVOLUTION EQUATIONS

•APPROXIMATIONS TO MORE COMPLEX SYSTEMS

•∞ FAMILY OF WAVE SOLUTIONS CONSTRUCTED

EXPLICITLYLAX PAIRINVERSE SCATTERINGBÄCKLUND TRANSFORMATION

•∞ HIERARCHY OF SYMMETRIES

•HAMILTONIAN STRUCTURE (SOME, NOT ALL)

•∞ SEQUENCE OF CONSTANTS OF MOTION(SOME, NOT ALL)

∞ FAMILY OF WAVE SOLUTIONS -BURGERS EQUATION

ut =2uux + uxx

WEAK SHOCK WAVES IN:FLUID DYNAMICS, PLASMA PHYSICS:

PENETRATION OF MAGNETIC FIELD INTOIONIZED PLASMA

HIGHWAY TRAFFIC: VEHICLE DENSITY

ε =v − c

c

WAVE SOLUTIONS:FRONTS

SINGLE FRONT

u t, x( ) =um + up ek x+ vt + x0( )

1 + ek x+ vt + x0( )

v=up + um , k=up −um

um

up

x =− vt + x0( )

t

x

u(t,x)

x

up :

1

k

um

− up + um( )

CHARACTERISTIC LINE

DISPERSION RELATION:

um =0 ⇒v=k

BURGERS EQUATION

M WAVES (M + 1)SEMI-INFINITE SINGLE FRONTS

0 < k1 < k2 < ... < kMTWO “ELASTIC” SINGLE FRONTS:

0 → k1 , 0 → kM

M1 “INELASTIC”SINGLE FRONTS

k1 → k2

k2 → k3

...kM −1 → kM 0 k1

k2

k3

k4

u t, x( ) =

ki eki x+ ki t + xi , 0( )

i=1

M

∑

1 + eki x+ ki t + xi , 0( )

i=1

M

∑

x

t

vi =ki

k1

k =kj +1 −kj

v=kj +1 + kj

BURGERS EQUATION

ut =6uux + uxxx

SHALLOW WATER WAVES

PLASMA ION ACOUSTIC WAVES

ONE-DIMENSIONAL LATTICE OSCILLATIONS(EQUIPARTITION OF ENERGY? IN FPU)

ε =a

λ

WAVE SOLUTIONS:SOLITONS

∞ FAMILY OF WAVE SOLUTIONS - KDV EQUATION

SOLITONS ALSO CONSTRUCTED FROMEXPONENTIAL WAVES: “ELASTIC” ONLY

u t, x( ) =2k 2

cosh2 k x+ vt + x0{ }( )

t

x

DISPERSION RELATION:

v =4k 2

KDV EQUATION

∞ FAMILY OF WAVE SOLUTIONS - NLS EQUATION

NONLINEAR OPTICS

SURFACE WAVES, DEEP FLUID + GRAVITY +VISCOSITY

NONLINEAR KLEIN-GORDON EQN. ∞ LIMIT

ε =δω ω0

ϕ t = iϕ xx + 2 i ϕ2ϕ

WAVE SOLUTIONS SOLITONS

NLS EQUATION

ϕ t, x( ) =

kexp i ω t + V x( )⎡⎣ ⎤⎦cosh k x + vt( )⎡⎣ ⎤⎦

ω = k2 −v2

4, V = −

v

2

⎛

⎝⎜⎞

⎠⎟

TWO-PARAMETER FAMILY

N SOLITONS: ki, vi ωi, Vi

SOLITONS ALSO CONSTRUCTED FROMEXPONENTIAL WAVES: “ELASTIC” ONLY

SYMMETRIES

LIE SYMMETRY ANALYSIS

PERTURBATIVE EXPANSION - RESONANT TERMS

SOLUTIONS OF LINEARIZATION OF EVOLUTION EQUATION

ut =F0 u[ ] ∂tSn =∂F0 u + ν Sn[ ]

∂νν =0

SYMMETRIES

BURGERS ∂tSn = 2∂x u Sn( ) + ∂x2Sn

KDV ∂tSn = 6∂x u Sn( ) + ∂x3Sn

NLS ∂tSn = i ∂x2Sn + 2 i 2ϕ ϕ * Sn + ϕ 2 Sn

*( )

EACH HAS AN ∞ HIERARCHY OF SOLUTIONS - SYMMETRIES

SYMMETRIES

S1 =ux

S2 =2uux + uxx

S3 =3u2 ux + 3uuxx + 3ux2 + uxxx

BURGERS

NOTE: S2 = UNPERTURBED EQUATION!

KDVS1 =ux

S2 =6uux + uxxx

S3 =30u2 ux +10uuxxx + 20uxuxx + u5x

S4 =140u 3 ux + 70uuxxx + 280uuxuxx

+14uu5x + 70ux3 + 42uxu4x + 70uxxuxxx + u7x

PROPERTIES OF SYMMETRIES

LIE BRACKETS

Sn ,Sm[ ] ≡∂ Sn u+ Sm u[ ]⎡⎣ ⎤⎦−Sm u+ Sn u[ ]⎡⎣ ⎤⎦( )=0

=0

SAME SYMMETRY HIERARCHY

ut =F0 u[ ]

⇓

Sn u[ ]{ }

ut =Sm u[ ]

⇓

Sn u[ ]{ }

Sn{ } ≡ Sn{ }


ut =F0 u[ ]

F0 u[ ] ⇒ Sn u[ ]

ut =Sn u[ ]

SAME WAVE SOLUTIONS ?

(EXCEPT FOR UPDATEDDISPERSION RELATION)


ut =S2 u[ ] + εα S3 u[ ] + ε 2 βS4 u[ ] + ...

BURGERS v =k→ v=k+ εα k2 + ε 2 β k3 + ...KDV

v =4k2 → v=4k2 + εα 4k2( )2+ ε 2 β 4k2( )

3+ ...

ut =S2 u[ ] → ut =Sn u[ ]

SAME!!!! WAVE SOLUTIONS, MODIFIED kv RELATION

BURGERS S2 → Sn v=k→ v=kn−1

KDV S2 → Sn v=4k2 → v= 4k2( )n−1

NF

∞ CONSERVATION LAWS

KDV & NLS

E.G., NLS

In = ρn dx−∞

+∞

∫

ρ0 =ϕ 2

ρ1 =iϕ ϕ *x

ρ2 =ϕ 4 −ϕ x2

M

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

EVOLUTION EQUATIONS AREAPPROXIMATIONS TO MORE COMPLEX SYSTEMS

NIT w =u+ εu1( ) + ε 2 u 2( ) + ...NF ut =S2 u[ ] + εU1 + ε 2U2 + ...

IN GENERAL, ALL NICE PROPERTIES BREAK DOWNEXCEPT FOR u - A SINGLE WAVE

UNPERTURBED EQN. RESONANT TERMSAVOID UNBOUNDED TERMS IN u(n)

wt =F w[ ] =

F0 w[ ] + ε F1 w[ ] + ε 2 F2 w[ ] + ...

F0 w[ ] =S2 w[ ]( )

BREAKDOWN OF PROPERTIES

•∞ FAMILY OF CLOSED-FORM WAVE SOLUTIONS

•∞ HIERARCHY OF SYMMETRIES

•∞ SEQUENCE OF CONSERVATION LAWS

FOR PERTURBED EQUATION

CANNOT CONSTRUCT

EVEN IN A PERTURBATIVE SENSE(ORDER-BY-ORDER IN PERTURBATION EXPANSION)

“OBSTACLES” TO ASYMPTOTIC INTEGRABILITY

OBSTACLES TO ASYMPTOTIC INTEGRABILITY - BURGERS

2α1 −α2 −2α 3 +α 4 =0

wt =2wwx + wxx

+ ε3α1 w

2 wx + 3α2 wwxx

+ 3α 3 wx2 +α 4 wxxx

⎛

⎝⎜

⎞

⎠⎟

(FOKAS & LUO, KRAENKEL, MANNA ET. AL.)

OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV

wt =6wwx + wxxx

+ ε30α1 w

2 wx +10α2 wwxxx

+ 20α 3 wx wxx +α 4 w5x

⎛

⎝⎜⎞

⎠⎟

+ ε 2

140β1 w3 wx + 70β2 w2 wxxx + 280β3 wwx wxx

+14β4 ww5x + 70β5 wx3 + 42β6 wx w4x +

70β7wxx wxxx + β8 w7x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

1009 3α1α2 + 4α2

2 −18α1α 3 + 60α2α 3 −24α 32 +18α1α 4 −67α2α 4 + 24α 4

2( )

+ 1403 3β1 −4β2 −18β3 +17β4 +12β5 −18β6 +12β7 −4β8( ) =0

KODAMA, KODAMA & HIROAKA

ψ t = iψ xx + 2 i ψ2ψ

+ ε α 1ψ xxx + α 2 ψ2ψ x + α 3ψ 2 ψ x

*( )

+ ε 2 iβ1ψ xxxx + β2 ψ

2ψ xx + β 3ψ * ψ x

2( )

+ β 4ψ2 ψ xx

* + β5 ψ ψ x

2+ β6 ψ

4ψ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

18α12 −3α1α2 +α2α 3 −2α 3

2

+ 24β1 −2β2 −4β3 −8β4 + 2β5 + 4β6 =0

OBSTACLES TO ASYMPTOTIC INTEGRABILITY - NLS

KODAMA & MANAKOV

OBSTCACLE TO INTEGRABILITY - BURGERS

EXPLOIT FREEDOM IN EXPANSION

wt =2wwx + wxx

+ ε3α1 w

2 wx + 3α2 wwxx


⎛

⎝⎜

⎞

⎠⎟

ut =S2 u[ ] + εα 4 S3 u[ ] + ...=2uux + uxx

+ εα 4 3u2 ux + 3uuxx + 3ux2 + uxxx( )

NF

NIT

w =u+ εu1( ) + ...



u 1( ) =au2 + bqux + cux

q=∂x−1u( )

u 1( )t =2 uu1( )( )

x+ u1( )

xx

+ 3 α1 −α 4( )u2 ux

+ 3 α2 −α 4( )uuxx

+ 3 α 3 −α 4( )ux2

TRADITIONALLY:

DIFFERENTIALPOLYNOMIAL

γ =2α 1 − α 2 − 2α 3 + α 4 = 0

PART OF PERTURBATION

CANNOT BE ACOUNTED FOR

“OBSTACLE TO ASYMPTOTIC INTEGRABILITY”

TWO WAYS OUT

BOTH EXPLOITING FREEDOM IN EXPANSION

IN GENERAL

γ ≠ 0


WAYS TO OVERCOME OBSTCACLES

I. ACCOUNT FOR OBSTACLE BY ZERO-ORDER TERM

ut =S2 u[ ] + εα 4 S3 u[ ] ⇒

ut =S2 u[ ] + εα 4 S3 u[ ] + γ R u[ ]( )

GAIN: HIGHER-ORDER CORRECTION BOUNDED POLYNOMIAL

LOSS: NF NOT INTEGRABLE,ZERO-ORDER UNPERTURBED SOLUTION

KODAMA, KODAMA & HIROAKA - KDVKODAMA & MANAKOV - NLS

OBSTACLE

WAYS TO OVERCOME OBSTCACLES

II. ACCOUNT FOR OBSTACLE BY FIRST-ORDER TERM

ut =S2 u[ ] + εα 4 S3 u[ ]LOSS: HIGHER-ORDER CORRECTION IS NOT POLYNOMIAL

HAVE TO DEMONSTRATE THAT BOUNDED

GAIN: NF IS INTEGRABLE,ZERO-ORDER UNPERTURBED SOLUTION

ALLOW NON-POLYNOMIAL PART IN u(1)

u 1( ) =au2 + bqux + cux + ξ t,x( )

VEKSLER + Y.Z.: BURGERS, KDVY..Z.: NLS

HOWEVER

PHYSICALSYSTEM EXPANSION

PROCEDURE

EVOLUTION EQUATION+

PERTURBATION

EXPANSIONPROCEDURE

APPROXIMATE SOLUTION

II

I

FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION

USUAL DERIVATION ONE-DIMENSIONAL IDEAL GAS

1. ∂τρ + ∂ξ ρv( ) =0

2. ∂τ ρv( ) + ∂ξ ρv2 + P −μ∂ξv( ) =0

P =c2 ρ0

γρρ0

⎛

⎝⎜⎞

⎠⎟

γ

γ =cp

cv

⎛

⎝⎜⎞

⎠⎟

c = SPEED of SOUND

ρ0 = REST DENSITY

τ → t = ε 2 τ ξ → x = ε ξ

ρ = ρ0 + ε ρ1 v = ε u

I - BURGERS EQUATION

1. SOLVE FOR ρ1 IN TERMS OF u FROM EQ. 1 :

POWER SERIES IN ε2. EQUATION FOR u: POWER SERIES IN ε

FROM EQ.2

RESCALE

u =cw

t→1+ γ( )

2c2ρ0

8μt x→ −

1+ γ( )cρ0

2μx

STAGE I - BURGERS EQUATION

α1 = 0

α 2 = −1

3

α 3 =1

4−

γ

12

α 4 =1

8+

γ

8

2α1 −α2 −2α 3 +α 4 =−124

+7γ24

≠0

OBSTACLE TO ASYMPTOTIC INTEGRABILITY

wt =2wwx + wxx

+ ε3α1 w

2 wx + 3α2 wwxx


⎛

⎝⎜

⎞

⎠⎟


HOWEVER,EXPLOIT FREEDOM IN EXPANSION

ρ =ρ0 + ε ρ1 + ε 2 ρ2 v = ε u + ε 2 u2

u2 =au2 + bux

1. SOLVE FOR ρ1 IN TERMS OF u FROM EQ. 1 :

POWER SERIES IN ε2. EQUATION FOR u: POWER SERIES IN ε

FROM EQ.2


RESCALE

u =cw

t→1+ γ( )

2c2ρ0

8μt x→ −

1+ γ( )cρ0

2μx

wt =2wwx + wxx

+ ε3α1 w

2 wx + 3α2 wwxx


⎛

⎝⎜

⎞

⎠⎟

α1 =2

3a

α 2 =2

3b −

1

3

α 3 =1

4+

2

3+

2

3b −

1

12γ

α 4 =1

8γ + 1( ) + b


2α1 −α2 −2α 3 +α 4 =0

FOR

b =724

γ −124

NO OBSTACLE TO INTEGRABILITY

MOREOVER a =18γ −

78⇒ α2 =α 3


wt =2wwx + wxx

+ ε3α1 w

2 wx + 3α2 wwxx


⎛

⎝⎜

⎞

⎠⎟

=∂x

w2 + wx

+ ε α1 w3 +α2 wwx +α 4 wxx( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

REGAIN “CONTINUITY EQUATION”STRUCTURE

α2 = α 3

STAGE I - KDV EQUATION

ION ACOUSTIC PLASMA WAVE EQUATIONS

∂τn + ∂ξ n v( ) = 0

∂τ v + ∂ξ

v2

2+ ϕ

⎛

⎝⎜⎞

⎠⎟= 0

∂ξ2ϕ = eϕ − n

SECOND-ORDER OBSTACLE TO INTEGRABILITY

τ → t = ε 3 τ ξ → x = ε ξ

n =1+ ε 2 n1

ϕ =ε 2ϕ 1

v=±1+ ε 2 u

STAGE I - KDV EQUATION

EXPLOIT FREEDOM IN EXPANSION:

n =1+ ε 2 n1 + ε 4 n2 + ε 6 n3

ϕ =ε 2ϕ 1 + ε 4 ϕ 2 + ε 6ϕ 3

v=±1+ ε 2 u+ ε 4 u2 + ε 6 u3

CAN ELIMINATE SECOND-ORDER OBSTACLE INPERTURBED KDV EQUATION

MOREOVER, CAN REGAIN“CONTINUITY EQUATION” STRUCTURE

THROUGH SECOND ORDER

OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV

wt =6wwx + wxxx

+ μ30α1 w

2 wx +10α2 wwxxx

+ 20α 3 wx wxx +α 4 w5x

⎛

⎝⎜⎞

⎠⎟

+ μ2

140β1 w3 wx + 70β2 w2 wxxx + 280β3 wwx wxx

+14β4 ww5x + 70β5 wx3 + 42β6 wx w4x +

70β7wxx wxxx + β8 w7x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

μ =ε 2( )

SUMMARY

STRUCTURE OF PERTURBED EVOLUTION EQUATIONS

DEPENDS ON

FREEDOM IN EXPANSION

IN DERIVING THE EQUATIONS

IF RESULTING PERTURBED EVOLUTION EQUATION

CONTAINS AN OBSTACLE TO ASYMPTOTIC INTERABILITY

DIFFERENT WAYS TO HANDLE OBSTACLE:

FREEDOM IN EXPANSION

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PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY