Transcript SPONTANEOUSLY GENERATED WAVES IN PERURBED …

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
INTEGRABLE EVOLUTION EQUATIONS
•APPROXIMATIONS TO MORE COMPLEX SYSTEMS
•∞ FAMILY OF WAVE SOLUTIONS CONSTRUCTED
EXPLICITLY
LAX PAIR
INVERSE SCATTERING
BÄCKLUND TRANSFORMATION
•∞ HIERARCHY OF SYMMETRIES
•HAMILTONIAN STRUCTURE (SOME, NOT ALL)
•∞ SEQUENCE OF CONSTANTS OF MOTION
(SOME, NOT ALL)
∞ 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
ut  2u ux  uxx
WAVE SOLUTIONS:
FRONTS
vc

c
BURGERS EQUATION

 u p  um

u t, x  
1
k
um
DISPERSION RELATION:
um  0 
vk
SINGLE FRONT
um  u p ek x  v t  x0 
1  ek x  v t  x0 
v  u p  um , k  u p  um
up
CHARACTERISTIC LINE
x   vt  x0 
x
up
u(t,x)
x
um
t
BURGERS EQUATION
M WAVES  (M + 1)
SEMI-INFINITE  SINGLE FRONTS
u t, x  

 ki e


1  e

M
0  k1  k2  ...  kM
TWO “ELASTIC” SINGLE FRONTS:
0  k1 , 0  kM vi  ki
ki x  ki t  xi , 0
i 1
M
ki x  ki t  xi , 0
i 1
M1 “INELASTIC”
SINGLE FRONTS
k1  k2 k  k j  1  k j
k2  k 3 v  k j  1  k j
...
kM  1  kM
0
k4
k3
k2
t
x
k1
k1
∞ FAMILY OF WAVE SOLUTIONS - KDV EQUATION

SHALLOW WATER WAVES
PLASMA ION ACOUSTIC WAVES
a

ONE-DIMENSIONAL LATTICE OSCILLATIONS
(EQUIPARTITION OF ENERGY? IN FPU)
ut  6u ux  uxxx
WAVE SOLUTIONS:
SOLITONS
KDV EQUATION
SOLITONS ALSO CONSTRUCTED FROM
EXPONENTIAL WAVES: “ELASTIC” ONLY
x
t
2
2k
u t, x  
2
cosh k x  vt  x0 
DISPERSION RELATION:
v  4k
2
∞ FAMILY OF WAVE SOLUTIONS - NLS EQUATION
    0
NONLINEAR OPTICS
SURFACE WAVES, DEEP FLUID + GRAVITY +
VISCOSITY
NONLINEAR KLEIN-GORDON EQN. ∞ LIMIT
 t  i  xx  2i  
2
WAVE SOLUTIONS SOLITONS
NLS EQUATION
TWO-PARAMETER FAMILY
 t, x  
k
exp i  t  V x 
cosh  k x  vt 
2

v
v
2
   k  4 , V   2 
N SOLITONS: ki, vi i, Vi
SOLITONS ALSO CONSTRUCTED FROM
EXPONENTIAL WAVES: “ELASTIC” ONLY
SYMMETRIES
LIE SYMMETRY ANALYSIS
PERTURBATIVE EXPANSION - RESONANT TERMS
SOLUTIONS OF LINEARIZATION OF EVOLUTION EQUATION
ut  F0 u 
t Sn 
F0 u   Sn 

 0
SYMMETRIES
BURGERS
t Sn  2x u Sn   x Sn
KDV
t Sn  6x u Sn   x Sn
NLS
2
3

t Sn  i  x Sn  2i 2  Sn   Sn
2
*
2
*

EACH HAS AN ∞ HIERARCHY OF SOLUTIONS - SYMMETRIES
SYMMETRIES
S1  ux
S2  2u ux  uxx
BURGERS
S3  3u ux  3u uxx  3ux  uxxx
2
2
S1  u x
S2  6 u u x  u xxx
KDV
S3  30u u x  10 u u xxx  20u x u xx  u5 x
2
S4  140u u x  70 uu xxx  280u u x u xx
3
 14 u u5 x  70 u x  42u x u4 x  70u xx u xxx  u7 x
3
NOTE: S2 = UNPERTURBED EQUATION!
PROPERTIES OF SYMMETRIES
LIE BRACKETS
S ,S   S
n
m
n
u   Sm u   Sm u   Sn u 

0
0
ut  F0 u 
ut  Sm u 


S u 
n

ˆ
S

S
 n n
SAME SYMMETRY HIERARCHY


Sˆn u 
PROPERTIES OF SYMMETRIES
ut  F0 u 
F0 u   Sn u 
ut  Sn u 
SAME WAVE SOLUTIONS ?
(EXCEPT FOR UPDATED
DISPERSION RELATION)
PROPERTIES OF SYMMETRIES
ut  S2 u  ut  Sn u 
SAME!!!! WAVE SOLUTIONS, MODIFIED kv RELATION
BURGERS
S2  Sn
S2  Sn
KDV
v  k  v  kn  1
 
v  4k  v  4k
2
2 n 1
ut  S2 u   S3 u    S4 u  ... NF
2
BURGERS
v  k  v  k   k    k  ...
2
2
3
KDV
     4 k   ...
v  4 k  v  4 k   4 k
2
2
2 2
2
2 3
∞ CONSERVATION LAWS
KDV & NLS
E.G., NLS

In 


n
dx
2

0  

*


i


1
x

   4  
x
 2




2


EVOLUTION EQUATIONS ARE
APPROXIMATIONS TO MORE COMPLEX SYSTEMS
wt  F w  
F0 w    F1 w    F2 w   ...
2
F w   S w 
0
NIT
NF
2
1
w  u  u   u
2
2
 ...
ut  S2 u   U1   U2  ...
UNPERTURBED EQN.
2
RESONANT TERMS
AVOID UNBOUNDED TERMS IN u(n)
IN GENERAL, ALL NICE PROPERTIES BREAK DOWN
EXCEPT FOR
u - A SINGLE WAVE
BREAKDOWN OF PROPERTIES
FOR PERTURBED EQUATION
CANNOT CONSTRUCT
•∞ FAMILY OF CLOSED-FORM WAVE SOLUTIONS
•∞ HIERARCHY OF SYMMETRIES
•∞ SEQUENCE OF CONSERVATION LAWS
EVEN IN A PERTURBATIVE SENSE
(ORDER-BY-ORDER IN PERTURBATION EXPANSION)
“OBSTACLES” TO ASYMPTOTIC INTEGRABILITY
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - BURGERS
wt  2 w wx  wxx
 31 w wx  3 2 w wxx 
 

2
  3 3 wx   4 wxxx 
2
2 1   2  2  3   4  0
(FOKAS & LUO, KRAENKEL, MANNA ET. AL.)
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV
wt  6 w wx  wxxx
 30 1 w 2 wx  10  2 w wxxx 



20

w
w


w


3
x
xx
4
5x
 140 1 w 3 wx  70  2 w 2 wxxx  280  3 w wx wxx 

2
3
    14  4 w w5 x  70  5 wx  42  6 wx w4 x  
 70  7 wxx wxxx   8 w7 x

100
9
3 
1
2
 4  2 2  18 1  3  60  2  3  24  3 2  18 1  4  67  2  4  24  4 2
 140
3 3 1  4  2  18  3  17  4  12  5  18  6  12  7  4  8   0
KODAMA, KODAMA & HIROAKA

OBSTACLES TO ASYMPTOTIC INTEGRABILITY - NLS
 t  i  xx  2i  
2

  1  xxx   2   x   3   x
2
2
*

 
 1  xxxx  2  2  xx   3  *  x 2 
 2 i

2
    2  *        4  
4
xx
5
x
6
18   31  2   2  3  2  3
 24 1  2 2  4 3  8 4  2 5  4 6  0
2
1
2
KODAMA & MANAKOV
OBSTCACLE TO INTEGRABILITY - BURGERS
wt  2 w wx  wxx
 31 w wx  3 2 w wxx 
 

2
  3 3 wx   4 wxxx 
2
EXPLOIT FREEDOM IN EXPANSION
OBSTCACLE TO INTEGRABILITY - BURGERS
NIT
1
w  u   u  ...
NF
ut  S2 u     4 S3 u   ...
 2u u x  u xx

   4 3u u x  3u u xx  3u x  u xxx
2
2

OBSTCACLE TO INTEGRABILITY - BURGERS
u
1
t
  u
 2 uu
1
1
xx
x
 31   4 u u x
2
 3 2   4 u u xx
 3 3   4 u x
TRADITIONALLY:
DIFFERENTIAL
POLYNOMIAL
1
2
u  au  b qux  cux
2
q   u 
1
x
  2 1   2  2  3   4  0
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
WAYS TO OVERCOME OBSTCACLES
I. ACCOUNT FOR OBSTACLE BY ZERO-ORDER TERM
ut  S2 u 

  4 S3 u  
ut  S2 u     4 S3 u    R u 
OBSTACLE
GAIN: HIGHER-ORDER CORRECTION BOUNDED POLYNOMIAL
LOSS: NF NOT INTEGRABLE,
ZERO-ORDER UNPERTURBED SOLUTION
KODAMA, KODAMA & HIROAKA - KDV
KODAMA & MANAKOV - NLS
WAYS TO OVERCOME OBSTCACLES
II. ACCOUNT FOR OBSTACLE BY FIRST-ORDER TERM
ALLOW NON-POLYNOMIAL PART IN u(1)
u  au  bqux  cux   t, x 
1
2
GAIN: NF IS INTEGRABLE,
ZERO-ORDER UNPERTURBED SOLUTION
ut  S2 u    4 S3 u 
LOSS: HIGHER-ORDER CORRECTION IS NOT POLYNOMIAL 
HAVE TO DEMONSTRATE THAT BOUNDED
VEKSLER + Y.Z.: BURGERS, KDV
Y..Z.: NLS
HOWEVER
I
PHYSICAL
SYSTEM
II
EXPANSION
PROCEDURE
EXPANSION
PROCEDURE
APPROXIMATE SOLUTION
EVOLUTION EQUATION
+
PERTURBATION
FREEDOM IN EXPANSION STAGE I - BURGERS EQUATION
USUAL DERIVATION
1.
ONE-DIMENSIONAL IDEAL GAS
     v   0


c = SPEED of SOUND
2.   v     v  P    v  0 0 = REST DENSITY
c 0   
P
  0 
2
2

cp 

   c 
v
 t  
  x  
  0   1
v  u
2
I - BURGERS EQUATION
1. SOLVE FOR 1 IN TERMS OF u FROM EQ. 1 :
POWER SERIES IN 
2. EQUATION FOR u: POWER SERIES IN 
FROM EQ.2
RESCALE
u  cw
1    c 0

t
t
2
8
2
1   c0

x
x
2
STAGE I - BURGERS EQUATION
wt  2 w wx  wxx
 31 w wx  3 2 w wxx 
 

2
  3 3 wx   4 wxxx 
2
1  0
1
2  
3
1 
3  
4 12
1 
4  
8 8
1
7
2 1   2  2  3   4  

0
24 24
OBSTACLE TO ASYMPTOTIC INTEGRABILITY
STAGE I - BURGERS EQUATION
HOWEVER,
EXPLOIT FREEDOM IN EXPANSION
  0   1   2
v   u   u2
2
2
1. SOLVE FOR 1 IN TERMS OF u FROM EQ. 1 :
POWER SERIES IN 
2. EQUATION FOR u: POWER SERIES IN 
FROM EQ.2
u2  au  bux
2
STAGE I - BURGERS EQUATION
RESCALE
u  cw
1    c 0

t
t
2
1   c0

x
x
2
8
2
wt  2 w wx  wxx
 31 w wx  3 2 w wxx 
 

2
  3 3 wx   4 wxxx 
2
STAGE I - BURGERS EQUATION
2
1  a
2 1   2  2  3   4  0
3
2
1
FOR
2  b 
3
3
1 2 2
1
7
1
3    b  
b
 
4 3 3
12
24
24
1
 4    1  b
NO OBSTACLE TO INTEGRABILITY
8
MOREOVER
1
7
a     2  3
8
8
STAGE I - BURGERS EQUATION
wt  2 w wx  wxx
2  3
 31 w wx  3 2 w wxx 
 

2
  3 3 wx   4 wxxx 
2
w  wx
 x 
3
  1 w   2 w wx   4 wxx
2

REGAIN “CONTINUITY EQUATION”
STRUCTURE




STAGE I - KDV EQUATION
ION ACOUSTIC PLASMA WAVE EQUATIONS
 n   n v   0
v

 v        0
 2

2

   e  n
2
 t   
  x  
3
n  1   n1
2
   1
2
v  1   u
2
SECOND-ORDER OBSTACLE TO INTEGRABILITY
STAGE I - KDV EQUATION
EXPLOIT FREEDOM IN EXPANSION:
n  1   n1   n2   n3
2
4
6
   1    2    3
2
4
6
v  1   u   u2   u3
2
4
6
CAN ELIMINATE SECOND-ORDER OBSTACLE IN
PERTURBED KDV EQUATION
MOREOVER, CAN REGAIN
“CONTINUITY EQUATION” STRUCTURE
THROUGH SECOND ORDER
OBSTACLES TO ASYMPTOTIC INTEGRABILITY - KDV
wt  6 w wx  wxxx
 30 1 w 2 wx  10  2 w wxxx 



20

w
w


w


3
x
xx
4
5x
 140 1 w 3 wx  70  2 w 2 wxxx  280  3 w wx wxx 

2
3
    14  4 w w5 x  70  5 wx  42  6 wx w4 x  
 70  7 wxx wxxx  8 w7 x

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