PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY
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Transcript of PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY
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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
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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)
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∞ 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
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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
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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
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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
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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
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∞ 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
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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
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SYMMETRIES
LIE SYMMETRY ANALYSIS
PERTURBATIVE EXPANSION - RESONANT TERMS
SOLUTIONS OF LINEARIZATION OF EVOLUTION EQUATION
ut =F0 u[ ] ∂tSn =∂F0 u + ν Sn[ ]
∂νν =0
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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
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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
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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{ }
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PROPERTIES OF SYMMETRIES
ut =F0 u[ ]
F0 u[ ] ⇒ Sn u[ ]
ut =Sn u[ ]
SAME WAVE SOLUTIONS ?
(EXCEPT FOR UPDATEDDISPERSION RELATION)
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PROPERTIES OF SYMMETRIES
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
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∞ CONSERVATION LAWS
KDV & NLS
E.G., NLS
In = ρn dx−∞
+∞
∫
ρ0 =ϕ 2
ρ1 =iϕ ϕ *x
ρ2 =ϕ 4 −ϕ x2
M
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
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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[ ]( )
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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
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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.)
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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
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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
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OBSTCACLE TO INTEGRABILITY - BURGERS
EXPLOIT FREEDOM IN EXPANSION
wt =2wwx + wxx
+ ε3α1 w
2 wx + 3α2 wwxx
+ 3α 3 wx2 +α 4 wxxx
⎛
⎝⎜
⎞
⎠⎟
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ut =S2 u[ ] + εα 4 S3 u[ ] + ...=2uux + uxx
+ εα 4 3u2 ux + 3uuxx + 3ux2 + uxxx( )
NF
NIT
w =u+ εu1( ) + ...
OBSTCACLE TO INTEGRABILITY - BURGERS
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OBSTCACLE TO INTEGRABILITY - BURGERS
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
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PART OF PERTURBATION
CANNOT BE ACOUNTED FOR
“OBSTACLE TO ASYMPTOTIC INTEGRABILITY”
TWO WAYS OUT
BOTH EXPLOITING FREEDOM IN EXPANSION
IN GENERAL
γ ≠ 0
OBSTCACLE TO INTEGRABILITY - BURGERS
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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
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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
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HOWEVER
PHYSICALSYSTEM EXPANSION
PROCEDURE
EVOLUTION EQUATION+
PERTURBATION
EXPANSIONPROCEDURE
APPROXIMATE SOLUTION
II
I
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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
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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
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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
+ 3α 3 wx2 +α 4 wxxx
⎛
⎝⎜
⎞
⎠⎟
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STAGE I - BURGERS EQUATION
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
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STAGE I - BURGERS EQUATION
RESCALE
u =cw
t→1+ γ( )
2c2ρ0
8μt x→ −
1+ γ( )cρ0
2μx
wt =2wwx + wxx
+ ε3α1 w
2 wx + 3α2 wwxx
+ 3α 3 wx2 +α 4 wxxx
⎛
⎝⎜
⎞
⎠⎟
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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
STAGE I - BURGERS EQUATION
2α1 −α2 −2α 3 +α 4 =0
FOR
b =724
γ −124
NO OBSTACLE TO INTEGRABILITY
MOREOVER a =18γ −
78⇒ α2 =α 3
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STAGE I - BURGERS EQUATION
wt =2wwx + wxx
+ ε3α1 w
2 wx + 3α2 wwxx
+ 3α 3 wx2 +α 4 wxxx
⎛
⎝⎜
⎞
⎠⎟
=∂x
w2 + wx
+ ε α1 w3 +α2 wwx +α 4 wxx( )
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
REGAIN “CONTINUITY EQUATION”STRUCTURE
α2 = α 3
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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
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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
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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( )
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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