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{ }
PROPERTIES OF SYMMETRIES
ut =F0 u[ ]
F0 u[ ] ⇒ Sn u[ ]
ut =Sn u[ ]
SAME WAVE SOLUTIONS ?
(EXCEPT FOR UPDATEDDISPERSION RELATION)
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
∞ 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
+ 3α 3 wx2 +α 4 wxxx
⎛
⎝⎜
⎞
⎠⎟
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
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
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
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
+ 3α 3 wx2 +α 4 wxxx
⎛
⎝⎜
⎞
⎠⎟
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
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
⎛
⎝⎜
⎞
⎠⎟
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
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
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( )