PERTURBED NONLINEAR EVOLUTION EQUATIONS AND ASYMPTOTIC INTEGRABILITY

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PERTURBED NONLINEAR EVOLUTION EQUATIONS AND
ASYMPTOTIC INTEGRABILITY
Yair Zarmi Physics Department Jacob Blaustein
Institutes for Desert Research Ben-Gurion
University of the Negev Midreshet Ben-Gurion,
Israel
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INTEGRABLE EVOLUTION EQUATIONS

• APPROXIMATIONS TO MORE COMPLEX SYSTEMS
• 8 FAMILY OF WAVE SOLUTIONS CONSTRUCTED
• EXPLICITLY
• LAX PAIR
• INVERSE SCATTERING
• BÄCKLUND TRANSFORMATION
• 8 HIERARCHY OF SYMMETRIES
• HAMILTONIAN STRUCTURE (SOME, NOT ALL)
• 8 SEQUENCE OF CONSTANTS OF MOTION
• (SOME, NOT ALL)

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8 FAMILY OF WAVE SOLUTIONS -BURGERS EQUATION
WEAK SHOCK WAVES IN FLUID DYNAMICS, PLASMA
PHYSICS PENETRATION OF MAGNETIC FIELD
INTO IONIZED PLASMA HIGHWAY TRAFFIC
VEHICLE DENSITY
WAVE SOLUTIONS FRONTS
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BURGERS EQUATION
SINGLE FRONT
up
CHARACTERISTIC LINE
x
um
up
DISPERSION RELATION
u(t,x)
x
um
t
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BURGERS EQUATION
M WAVES ? (M 1) SEMI-INFINITE ? SINGLE FRONTS
TWO ELASTIC SINGLE FRONTS
M?1 INELASTIC SINGLE FRONTS
k4
k3
k2
t
k1
0
k1
x
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8 FAMILY OF WAVE SOLUTIONS - KDV EQUATION
SHALLOW WATER WAVES PLASMA ION ACOUSTIC
WAVES ONE-DIMENSIONAL LATTICE OSCILLATIONS (EQUI
PARTITION OF ENERGY? IN FPU)
WAVE SOLUTIONS SOLITONS
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KDV EQUATION
SOLITONS ALSO CONSTRUCTED FROMEXPONENTIAL WAVES
ELASTIC ONLY
x
t
DISPERSION RELATION
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8 FAMILY OF WAVE SOLUTIONS - NLS EQUATION
NONLINEAR OPTICS SURFACE WAVES, DEEP FLUID
GRAVITY VISCOSITY NONLINEAR KLEIN-GORDON
EQN. ??8 LIMIT
WAVE SOLUTIONS SOLITONS
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NLS EQUATION
TWO-PARAMETER FAMILY
N SOLITONS ki, vi ??i, 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
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SYMMETRIES
BURGERS
KDV
NLS
EACH HAS AN 8 HIERARCHY OF SOLUTIONS - SYMMETRIES
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SYMMETRIES
BURGERS
KDV
NOTE S2 UNPERTURBED EQUATION!
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PROPERTIES OF SYMMETRIES
LIE BRACKETS
SAME SYMMETRY HIERARCHY
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PROPERTIES OF SYMMETRIES
SAME WAVE SOLUTIONS ? (EXCEPT FOR
UPDATED DISPERSION RELATION)
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PROPERTIES OF SYMMETRIES
SAME!!!! WAVE SOLUTIONS, MODIFIED k?v RELATION
BURGERS
KDV
NF
BURGERS
KDV
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8 CONSERVATION LAWS
KDV NLS
E.G., NLS
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EVOLUTION EQUATIONS AREAPPROXIMATIONS TO MORE
COMPLEX SYSTEMS
NIT
NF
UNPERTURBED EQN.
RESONANT TERMS AVOID UNBOUNDED TERMS IN u(n)
IN GENERAL, ALL NICE PROPERTIES BREAK DOWN EXCEPT
FOR u - A SINGLE WAVE
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BREAKDOWN OF PROPERTIES
FOR PERTURBED EQUATION CANNOT CONSTRUCT

• 8 FAMILY OF CLOSED-FORM WAVE SOLUTIONS
• 8 HIERARCHY OF SYMMETRIES
• 8 SEQUENCE OF CONSERVATION LAWS

EVEN IN A PERTURBATIVE SENSE (ORDER-BY-ORDER IN
PERTURBATION EXPANSION)
OBSTACLES TO ASYMPTOTIC INTEGRABILITY
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OBSTACLES TO ASYMPTOTIC INTEGRABILITY - BURGERS
(FOKAS LUO, KRAENKEL, MANNA ET. AL.)
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OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV
KODAMA, KODAMA HIROAKA
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OBSTACLES TO ASYMPTOTIC INTEGRABILITY - NLS
KODAMA MANAKOV
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OBSTCACLE TO INTEGRABILITY - BURGERS
EXPLOIT FREEDOM IN EXPANSION
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OBSTCACLE TO INTEGRABILITY - BURGERS
NIT
NF
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OBSTCACLE TO INTEGRABILITY - BURGERS
TRADITIONALLY DIFFERENTIAL POLYNOMIAL
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OBSTCACLE TO INTEGRABILITY - BURGERS
IN GENERAL ? ? 0
PART OF PERTURBATION CANNOT BE ACOUNTED
FOR OBSTACLE TO ASYMPTOTIC INTEGRABILITY
TWO WAYS OUT BOTH EXPLOITING FREEDOM IN
EXPANSION
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WAYS TO OVERCOME OBSTCACLES
I. ACCOUNT FOR OBSTACLE BY ZERO-ORDER TERM
OBSTACLE
GAIN HIGHER-ORDER CORRECTION BOUNDED POLYNOMIAL
LOSS NF NOT INTEGRABLE, ZERO-ORDER ?UNPERTURBED
SOLUTION
KODAMA, KODAMA HIROAKA - KDV KODAMA MANAKOV -
NLS
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WAYS TO OVERCOME OBSTCACLES
II. ACCOUNT FOR OBSTACLE BY FIRST-ORDER TERM
ALLOW NON-POLYNOMIAL PART IN u(1)
GAIN NF IS INTEGRABLE, ZERO-ORDER ?UNPERTURBED
SOLUTION
LOSS HIGHER-ORDER CORRECTION IS NOT POLYNOMIAL
? HAVE TO DEMONSTRATE THAT BOUNDED
VEKSLER Y.Z. BURGERS, KDV Y..Z. NLS
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HOWEVER
I
PHYSICAL SYSTEM
EXPANSION PROCEDURE
EVOLUTION EQUATION PERTURBATION
EXPANSION PROCEDURE
II
APPROXIMATE SOLUTION
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FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION
USUAL DERIVATION
ONE-DIMENSIONAL IDEAL GAS
c SPEED of SOUND
?0 REST DENSITY
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I - BURGERS EQUATION

• SOLVE FOR ?1 IN TERMS OF u FROM EQ. 1 POWER
  SERIES IN ?
• EQUATION FOR u POWER SERIES IN ?
• FROM EQ.2

RESCALE
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STAGE I - BURGERS EQUATION
OBSTACLE TO ASYMPTOTIC INTEGRABILITY
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STAGE I - BURGERS EQUATION
HOWEVER, EXPLOIT FREEDOM IN EXPANSION

• SOLVE FOR ?1 IN TERMS OF u FROM EQ. 1 POWER
  SERIES IN ?
• EQUATION FOR u POWER SERIES IN ?
• FROM EQ.2

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STAGE I - BURGERS EQUATION
RESCALE
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STAGE I - BURGERS EQUATION
FOR
NO OBSTACLE TO INTEGRABILITY
MOREOVER
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STAGE I - BURGERS EQUATION
REGAIN CONTINUITY EQUATION STRUCTURE
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STAGE I - KDV EQUATION
ION ACOUSTIC PLASMA WAVE EQUATIONS
SECOND-ORDER OBSTACLE TO INTEGRABILITY
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STAGE I - KDV EQUATION
EXPLOIT FREEDOM IN EXPANSION
CAN ELIMINATE SECOND-ORDER OBSTACLE IN PERTURBED
KDV EQUATION
MOREOVER, CAN REGAIN CONTINUITY EQUATION
STRUCTURE THROUGH SECOND ORDER
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OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV
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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