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Poisson Structures and Integrability Peter J. Olver University of Minnesota

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Poisson Structures and Integrability Peter J. Olver University of Minnesota
Poisson Structures
and Integrability
Peter J. Olver
University of Minnesota
http://www.math.umn.edu/ ∼ olver
Hamiltonian Systems
M
—
phase space; dim M = 2 n
Local coordinates: z = (p, q) = (p1, . . . , pn , q 1, . . . , q n )
Canonical Hamiltonian system:
dz
= J ∇H
dt
Equivalently:
∂H
dpi
=−
dt
∂q i
J=
!
O
I
−I
O
dq i
∂H
= i
dt
∂p
"
Lagrange Bracket (1808):
[u,v] =
n
#
i=1
∂pi ∂q i ∂q i ∂pi
−
∂u ∂v
∂u ∂v
(Canonical) Poisson Bracket (1809):
{u,v} =
n
#
i=1
∂u ∂v
∂u ∂v
−
∂pi ∂q i ∂q i ∂pi
Given functions u1, . . . , u2n , the (2 n) × (2 n) matrices with
respective entries
[ ui , uj ]
are mutually inverse.
{ ui , uj }
i, j = 1, . . . , 2 n
Canonical Poisson Bracket
T
{ F, H } = ∇F J ∇H =
n
#
i=1
∂F ∂H ∂F ∂H
−
∂pi ∂q i ∂q i ∂pi
=⇒ Poisson (1809)
Hamiltonian flow:
dz
= { z, H } = J ∇H
dt
=⇒ Hamilton (1834)
First integral:
{ F, H } = 0
⇐⇒
dF
=0
dt
⇐⇒
F (z(t)) = const.
Poisson Brackets
{ · , · } : C∞(M, R) × C∞(M, R) −→ C∞ (M, R)
Bilinear :
{ a F + b G, H } = a { F, H } + b { G, H }
{ F, a G + b H } = a { F, G } + b { F, H }
Skew Symmetric:
Jacobi Identity:
{ F, H } = − { H, F }
{ F, { G, H } } + { H, { F, G } } + { G, { H, F } } = 0
Derivation:
{ F, G H } = { F, G } H + G { F, H }
F, G, H ∈ C∞ (M, R), a, b ∈ R.
In coordinates z = (z 1 , . . . , z m),
{ F, H } = ∇F T J(z) ∇H
where J(z)T = − J(z) is a skew symmetric matrix.
The Jacobi identity imposes a system of quadratically
nonlinear partial differential equations on its entries:
#
l
!
jk
ki
ij
il ∂J
jl ∂J
kl ∂J
J
+J
+J
∂z l
∂z l
∂z l
"
=0
Given a Poisson structure, the Hamiltonian flow corresponding
to H ∈ C∞ (M, R) is the system of ordinary differential
equations
dz
= { z, H } = J(z) ∇H
dt
Lie’s Theory of Function Groups
Used for integration of partial differential equations:
{ Fi, Fj } = Gij (F1, . . . , Fn )
=⇒ predates Lie groups!!
Ausgezeichnete Functionen = distinguished functions
= Casimirs
{ F, C } = 0
""
for all
F ∈ C∞ (M, R)
All distinguished functions are first integrals (conservation
laws) of any associated Hamiltonian system.
Darboux’ Theorem
If J(z) has constant rank, then there exist local coordinates z = (p, q, y)
such that the Poisson bracket is in canonical form
{ F, H } =
n
#
i=1
∂F ∂H ∂F ∂H
−
∂pi ∂q i ∂q i ∂pi
Canonical degenerate Hamiltonian system:
dz
= J ∇H
dt
Equivalently:
dpi
∂H
=−
dt
∂q i

O

J =I
O
dq i
∂H
= i
dt
∂p
−I
O
O

O
O

O
dy j
=0
dt
Distinguished (Casimir) functions: F (y) = const.
Variable rank: Weinstein, Conn.
Symplectic Structures
A two form Ω ∈
V2
T ∗ M is called symplectic if it is
• closed: dΩ = 0, and
• nondegenerate: Ω ∧ · · · ∧ Ω += 0.
Nondegeneracy implies that Ω defines an isomorphism
T M , T ∗ M , mapping
the Hamiltonian vector field vH to dH = vH Ω.
In coordinates Ω = dz T ∧ J(z)−1dz and vH = J(z)∇H(z) ∂z .
The associated nondegenerate Poisson bracket:
{ F, H } = - vF ∧ vH ; Ω . = ∇F T J(z) ∇H
Lie–Poisson Structure
• Originally due to Lie
• Rediscovered by Kirillov, Kostant, Souriau, . . .
g
M = g∗ , R r
z ∈ g∗
∇F (z) ∈ g
—
r-dimensional Lie algebra
— dual vector space
{ F, H } = - z ; [ ∇F (z), ∇H(z) ] .
F (z), H(z) ∈ C∞(g∗ , R)
[ ∇F (z), ∇H(z) ] — Lie bracket in g
Lie–Poisson Structure
g
M = g∗ , R r
—
r-dimensional Lie algebra
— dual vector space
Lie–Poisson bracket:
z ∈ g∗
{ F, H } = - z ; [ ∇F (z), ∇H(z) ] .
F (z), H(z) ∈ C∞(g∗ , R)
∇F (z) ∈ g
In coordinates:
[ ∇F (z), ∇H(z) ] — Lie bracket in g
Jij (z) =
r
#
k=1
ckij zk ,
z = z1µ1 + · · · zr µr ∈ g∗
ckij
— structure constants
µ1 , . . . , µr
— Maurer–Cartan forms
The Euler Equations
=⇒ Arnold
Let G = SDiff(M ) be the infinite-dimensional pseudo-group
of volume preserving diffeomorphisms of a Riemannian
manifold M .
Lie algebra: g = divergence-free vector fields.
Hamiltonian functional on g∗ : the L2 norm.
"
The corresponding Lie–Poisson flow is equivalent to the Euler
equations governing the motion of an incompressible fluid.
In R n :
ut + u · ∇u = − ∇p
u — fluid velocity
div u = 0
p — pressure
=⇒ Ebin, Marsden
Vorticity Equations
u — fluid velocity
H=
*
ω = ∇ ∧ u — vorticity
∂ω
δH
= ω · ∇u − u · ∇ω = J
∂t
δω
2
1
2 | u | dx
J = ( ω · ∇ − ∇ω ) ∇ ∧ ·
—
kinetic energy
— Poisson operator
Distinguished (Casimir) functions:
2D:
**
f (ω) dx dy
3D:
***
(u · ω) dx dy dz — helicity
Camassa–Holm Equation and Euler–Poincaré
Flows
Use the H 1 norm as the Hamiltonian for the Lie–Poisson structure on the diffeomorphism pseudo-group:
H[ u ] =
*
( u2 + α | ∇u |2 ) dx
Camassa–Holm Equation:
ut ± uxxt = 3 u ux ± (u uxx + 12 u2x)x
=⇒ nonanalytic solutions — peakons or compactons
α models: regularized Euler; geophysics, magnetohydrodynamics, computational anatomy,mathematical morphology
Poisson Bivector Field
=⇒ Lichnerowicz
Θ = ∂zT ∧ J(z) ∂z ∈
V2
TM
Poisson bracket:
"
{ F, H } = - Θ ; dF ∧ dH .
Bilinearity, skew symmetry and derivation properties are
automatic.
Theorem. The Jacobi identity holds if and only if
[ Θ, Θ ] = 0
where [ · , · ] denotes the Schouten bracket.
The Schouten Bracket
The Schouten bracket [ Θ, Ψ ] is the natural extension of the Lie
^
bracket [ v, w ] to multi-vector fields (sections of n T M ):
• Bilinear.
• Super-symmetric:
[ Θ, Ψ ] = (−1)deg Θ deg Ψ [ Ψ, Θ ] ∈
^deg Θ+deg Ψ−1
TM
• Super-Jacobi identity:
(−1)deg Θ deg Ξ [ [ Θ, Ψ ], Ξ ] + (−1)deg Ξ deg Ψ [ [ Ξ, Θ ], Ψ ]+
+ (−1)deg Ψ deg Θ [ [ Ψ, Ξ ], Θ ] = 0
• Super-derivation:
[ Θ, Ψ ∧ Ξ ] = [ Θ, Ψ ] ∧ Ξ + (−1)deg Ψ(deg Θ−1)Ψ ∧ [ Θ, Ξ ]
The Poisson Complex
R→
^0
TM
δΘ
!
^1
TM
δΘ
!
^2
TM
δΘ
!
···
Suppose Θ is a Poisson bivector:
[ Θ, Θ ] = 0
Poisson derivation:
δΘ(Ψ) = [ Θ, Ψ ]
Closure: By super-Jacobi:
2
δΘ
(Ψ) = [ Θ, [ Θ, Ψ ] ] = −[ Θ, [ Θ, Ψ ] ]−(−1)deg Ψ [ Ψ, [ Θ, Θ ] ] = 0
In particular, applying δΘ to a function gives the associated
Hamiltonian vector field:
δΘ(H) = [ Θ, H ] = vH
Poisson Cohomology
Unless Θ is nondegenerate, the Poisson complex is not
locally exact. If Θ has constant rank, the local cohomology
involves the distinguished functions and forms.
"
Poisson cohomology is poorly understood in general.
A nondegenerate Θ defines an algebra isomorphism
^
T M , k T ∗ M via ω /−→ v = ω Θ. In this case, the
Poisson complex is isomorphic to the de Rham complex:
^k
R→
R→
^0
T ∗M
d
!
"
^0
TM
^1
T ∗M
d
!
"
δΘ
!
^1
TM
^2
T ∗M
d
!
···
!
···
"
δΘ
!
^2
TM
δΘ
BiHamiltonian Systems
Definition. A system of first order differential equations
is called biHamiltonian if it can be written in Hamiltonian form
in two distinct ways:
du
= J1 ∇H1 = J2 ∇H0
dt
Thus both J1 and J2 define Poisson brackets, which we assume
are not constant multiples of each other.
=⇒ Both Hamiltonians H1 and H2 are conserved.
Infinite Toda Lattice
H=
#
i
ai = 21 e(qi−1−qi)/2
( 21 p2i + eqi−1−qi )
bi = 21 pi−1
dai
= ai(bi+1 − bi)
dt
H1 =
#
( a2i
i
J1 =
!
0
a(T+ − 1)
(1 − T− )a
0
"
+
dbi
= 2(a2i − a2i−1)
dt
1 2
2 bi )
J2 =
=⇒ Flaschka
!
H0 =
#
bi
i
1
a(T+
2
− T− )a
b(1 − T−)a
T+ , T−
a(T+ − 1)b
2(a2 T+ − T−a2 )
"
— shift operators
Compatibility
• J1 , J2, and J1 + J2 are all Poisson
• The Poisson bivectors satisfy
[Θ1 , Θ1] = [Θ2 , Θ2] = [Θ1, Θ2] = 0
• The recursion operator R = J2 J1−1 satisfies the Nijenhuis torsion
condition
R2[ v, w ] − R[ Rv, w ] − R[ v, Rw ] + [ Rv, Rw ] = 0
• The symplectic forms satisfy
−1 −1
=0
dΩ1 = dΩ2 = d(Ω−1
1 + Ω2 )
Theorem. (Magri ) Suppose
du
= J1 ∇H1 = J2 ∇H0
dt
is a biHamiltonian system, where J1 , J2 form a compatible pair of Hamiltonian
operators. Assume that J1 is nondegenerate, and define the recursion operator
R = J2 J1−1 . Then there exist an infinite sequence of conserved Hamiltonians
H0 , H1 , H2 , . . . such that
• Each associated flow is a biHamiltonian system
du
= Fn = J1 ∇Hn = J2 ∇Hn−1 = R Fn−1
dt
• The Hamiltonians are in involution with respect to either Poisson bracket:
{ Hn , Hm }1 = 0 = { Hn , Hm }2
and hence conserved by all flows.
• The flows mutually commute.
Proof : Recursion: Starting at n = 1, the nth flow comes from the vector field
vn = [ Θ2 , Hn−1 ] = [ Θ1 , Hn ]
Set vn+1 = [ Θ2 , Hn ]. Then, by super-Jacobi, compatibility, and closure
[ Θ1 , vn+1 ] = [ Θ1 , [ Θ2 , Hn ] ] = − [ Θ2 , [ Θ1 , Hn−1 ] ] − [ [ Θ1 , Θ2 ], Hn−1 ]
= − [ Θ2 , vn−1 ] = − [ Θ2 , [ Θ2 , Hn−2 ] ] = 0
Then, by exactness of the Θ1 –Poisson complex vn+1 = [ Θ1 , Hn+1 ] for some
Hn+1 .
Conservation: Using compatibility:
vn (Hm ) = [ vn , Hm ] = [ [ Θ2 , Hn−1 ], Hm ] = − [ Hn−1 , [ Θ2 , Hm ] ]
= − [ Hn−1 , [ Θ1 , Hm+1 ] ] = [ [ Θ1 , Hn−1 ], Hm+1 ] = vn−1 (Hm+1 )
and repeat . . . to reduce to vn (Hn ) = vn (Hn−1 ) = 0.
Completely Integrable Systems
Definition. A nondegenerate Hamiltonian system ut = J ∇H
on an 2 n dimensional phase space is called completely integrable
if there exist n first integrals H = F1, F2, . . . Fn that are in
involution
{ Fi , Fj } = 0
Is a Completely Integrable System
Necessarily BiHamiltonian?
Theorem. (Fernandes) A completely integrable Hamiltonian system
is biHamiltonian in the neighborhood of an invariant torus if and only if the
graph of its Hamiltonian function is a hypersurface of translation relative to
the affine structure determined by the action variables (s1 , . . . , sn ):
si = ai1 (y 1 ) + · · · + ain (y n ),
H(s1 , . . . , sn ) = φ1 (y 1 ) + · · · + φn (y n ).
Example. The perturbed Kepler problem
H=
"

1 2
p +
2 r
p2θ
r2
+
p2φ
2
r2 sin θ

−
1
ε
+ 2
r 2r
completely integrable for all ε; biHamiltonian only when ε = 0
Incompatible BiHamiltonian Systems
Ω1, Ω2 — symplectic two-forms on C4
Ω1 ∧ Ω2 += 0
Canonical forms (Debever):
Ω1 = dp1 ∧ dq1 + dp2 ∧ dq2









dp1 ∧ dq1 − dp2 ∧ dq2
Ω2 =  ep1 ( dp1 ∧ dq1 − dp2 ∧ dq2 − p2dp1 ∧ dq2 )







ep1+p2 ( dp1 ∧ dq1 − dp2 ∧ dq2 + (q1 + q2 )dp1 ∧ dp2 )
=⇒ The first pair are compatible, but the latter two are not.
Bi-Hamiltonians:
H1 = f (p1 , q1) + g(p2, q2)
H2 = f (p2 ep1/2, q2)e−p1 /2 + g(p1 )
H3 = c(q1 − q2 ) + f (p1 , p2 )
∂f
∂f
∂ 2f
+
+
= 0.
∂p1 ∂p2 ∂p1 ∂p2
Poisson brackets for Field Theories
H[ u ] =
*
H[ u ] dx
— Hamiltonian functional
δH
= E(H)
δu
— variational derivative = Euler–Lagrange expression
{ F, H } =
J
* !
δF
δH
J
δu
δu
"
dx
— Poisson bracket
— Poisson (differential) operator
=⇒
=⇒
Formally skew–adjoint: J ∗ = − J
Jacobi identity
""
Korteweg–deVries Equation
J1 = Dx
∂u
δH1
δH2
= uxxx + u ux = J1
= J2
∂t
δu *
δu
H1 [ u ] =
J2 = Dx3 + 23 u Dx + 31 ux
""
H2 [ u ] =
*
( 16 u3 − 21 u2x ) dx
1
2
u2 dx
Bi–Hamiltonian system with recursion operator
R = J2 · J1−1 = Dx2 + 23 u + 13 ux Dx−1
=⇒ Gardner, Lax, Lenard, PJO, Magri, Gel’fand–Dikii, Adler, . . .
=⇒ Lie–Poisson structure on the Lie algebra of
pseudo-differential operators (Virasoro algebra)
Poisson Functional Multi-vectors
Θ=
In general,
*
*
θ ∧ J(θ) dx
DxΩ dx = 0 — work mod image of Dx
Schouten bracket:
0 = [ Θ, Θ ] =
*
pr vJ θ [ θ ∧ J(θ) ] dx
• prolonged evolutionary “vector field”:
pr vJ θ commutes with Dx and pr vJ θ (θ) = 0.
Lemma. Any constant coefficient, skew–adjoint differential
operator is Poisson.
Example. Second KdV Hamiltonian operator:
J = Dx3 + 23 u Dx + 13 ux
Functional multi-vector:
Θ=
Schouten bracket:
[ Θ, Θ ] =
=
=
since
*
*
*
*
θ ∧ θxxx + 23 u θ ∧ θx dx
pr v
(u)
2
1
θxxx + 3 u θ+ 3 ux θ
∧ θ ∧ θx dx
(θxxx + 23 u θ + 13 ux θ) ∧ θ ∧ θx dx
θxxx ∧ θ ∧ θx dx = 0
θxxx ∧ θ ∧ θx = Dx(θxx ∧ θ ∧ θx)
Example. Second KdV Hamiltonian operator:
J = Dx3 + 23 u Dx + 13 ux
Functional multi-vector:
Θ=
Schouten bracket:
[ Θ, Θ ] =
=
=
since
*
*
*
*
θ ∧ θxxx + 23 u θ ∧ θx dx
pr v
(u)
1
2
θxxx + 3 u θ+ 3 ux θ
∧ θ ∧ θx dx
(θxxx + 23 u θ + 13 ux θ) ∧ θ ∧ θx dx
θxxx ∧ θ ∧ θx dx = 0
θxxx ∧ θ ∧ θx = Dx(θxx ∧ θ ∧ θx)
Example. Second KdV Hamiltonian operator:
J = Dx3 + 23 u Dx + 13 ux
Functional multi-vector:
Θ=
Schouten bracket:
[ Θ, Θ ] =
=
=
since
*
*
*
*
θ ∧ θxxx + 23 u θ ∧ θx dx
pr v
(u)
2
1
θxxx + 3 u θ+ 3 ux θ
∧ θ ∧ θx dx
(θxxx + 23 u θ + 13 ux θ) ∧ θ ∧ θx dx
θxxx ∧ θ ∧ θx dx = 0
θxxx ∧ θ ∧ θx = Dx(θxx ∧ θ ∧ θx)
First Order Poisson Operators
=⇒ Dubrovin, Novikov
Field variables: u(x) = (u1(x), . . . , un (x))
k
Jij = g ij (u) Dx + bij
k (u) ux
Nondegenerate Poisson operator:
•
g ij = g ji
• bij
k =
/
l
g il Γjlk
— flat (pseudo-)Riemannian metric
— Christoffel symbols (connection)
Hyperbolic systems of hydrodynamic type:
∂u
δH
=J
∂t
δu
Nonlinear Transport
=⇒ Nutku, PJO
ut = u ux
Conserved densities:
Hn (u) =
1 n
u
n
Hamiltonian structures:
J0 = Dx
J1 = 2 u Dx + ux
J2 = u2 Dx + u ux
J3 = Dx
1
1
Dx
D
ux
ux x
Hamiltonian flows:
ut = Vn = un ux = J0 δHn+1 = J1 δHn = J2δHn−1 = J3 δHn+3
• J0, J1 , J2 are mutually compatible
• J0, J3 are compatible
• J1, J3 and J2 , J3 are not compatible
• J3J0−1 = R2
R = Dx u1 — recursion operator
x
uxx
0
0
ut = 0
V n = Rn−2V
u
=
V
=
Rational flows: t
2
2
ux
0 = 1
Rational conserved densities: H
1
ux
0
V
2 j+1
0 = J δH
0
= J0 δ H
j
3
j−1
2–D Hyperbolic Systems
ut = J0 δH/δu
u=
!
u
v
"
H[ u ] =
J0 = σ1Dx =
∂u
δH
= Dx
∂t
δv
!
0
1
*
H(u, v) dx
"
1
Dx
0
∂v
δH
= Dx
∂t
δu
Examples
H(u, v) = − 12 u2v + F (v)
vγ
• Polytropic:
F (v) =
γ(γ − 1)
Gas dynamics:
• Shallow water:
Elastodynamics:
F (v) = 12 v 2,
γ = 2.
H(u, v) = 12 u2 + F (v)
• van der Waals fluid; acoustic Euler equation
Born–Infeld equation:
H(u, v) =
u v
+
v u
=⇒ Chaplygin gas γ = −1
Zero-th Order Conservation Laws
Theorem. F (u, v) is a conserved density if and only if
Huu Fvv = Hvv Fuu
Separable Hyperbolic System:
λ(u)
Huu
=
Hvv
µ(v)
λ(u) ≡ 1
— generalized gas dynamics
Higher Order Hamiltonian Structure
Separable:
λ(u)
Huu
=
Hvv
µ(v)
*
L(u) =
U (u, v) =
!
u
v
λ(u) du
M (v
L(u)
"
M (v) =
V (u, v) =
*
µ(v) dv
!
L(u) M (v
v
u
Poisson operator:
J3 = Dx Vx−1 Dx Ux−1 σ1 Dx
BiHamiltonian system:
0
u1 = J0 δH = J3 δ H
"
Recursion operator (Sheftel’):
1 = J J −1 = D V −1 D U −1
R
x x
3 0
x x
1 = R2 where R = D U −1 is
For gas dynamics, U = V and R
x x
a recursion operator
There are also two other first order Poisson operators J1 , J2,
along with hierarchies of zeroth order polynomial conservation
laws and higher order rational conservation laws, e.g.
vx
0 =
H
1
u2x − µ(v) vx2
=⇒ Verosky
Deformed Lie–Poisson Structure
u(x) ∈ C∞ (R, g∗ ) — curve in g∗ — dual to Lie algebra
Poisson bracket:
{ F, H } =
* !
δF
δH
P
δu
δu
"
dx
Hamiltonian curve flow:
δH
δH
∂u
=P
= B Dx
+ ad∗
δH/δu (u)
∂t
δu
δu
B : g −→ g∗
•
— ad∗ –invariant symmetric linear map
If g is semi-simple, B is a multiple of the Killing form
Noncanonical Perturbation Theory
dv
= J(v)∇H(v)
dt
Perturbation expansion:
v = u + ε ϕ(u) + ε2 ψ(u) + · · ·
H(v) = H0 (u) + ε H1 (u) + ε2 H2(u) + · · ·
J(v) /−→ J0 (u) + ε J1 (u) + ε2 J2 (u) + · · ·
Perturbed system:
du
= J0∇H0 + ε( J1 ∇H0 + J0 ∇H1 ) + · · ·
dt
First order Hamiltonian perturbation
du
= (J0 + εJ1 )∇(H0 + εH1 )
dt
= J0 ∇H0 + ε( J1∇H0 + J0 ∇H1 ) + ε2J1 ∇H1
"
There is no guarantee that J0 + εJ1 is a Poisson operator.
Theorem. If J1 ∇H0 = λ J0∇H1 and J0 , J1 are compatible, then the first order perturbation equation
du
= J0 ∇H0 + ε( J1 ∇H0 + J0∇H1 )
dt
and the first order Hamiltonian perturbation
du
= J0 ∇H0 + ε( J1∇H0 + J0 ∇H1 ) + ε2J1 ∇H1
dt
are biHamiltonian systems.
=⇒ KdV equation
2D Water Waves
h
y = h + η(t, x)
surface elevation
φ(t, x, y)
velocity potential
2D Water Waves
• Incompressible, irrotational fluid.
• No surface tension
φt +
1
2
φ2x
+
1
2
φ2y
+gη =
ηt = φy − ηxφx

0

y = h + η(t, x)
φxx + φyy = 0
0 < y < h + η(t, x)
φy = 0
y=0
√
c = gh
a
/
h
Small parameters — long waves in shallow water
(KdV regime)
2
a
h
α=
β = 2 = O(α)
h
/
Rescale:
x /−→ / x
y /−→ h y
η /−→ a η
ga/φ
φ −
/ →
c
Rescaled water wave system:

α 2
α 2
φy + η = 0 
φt + φx +


2
2β
1



ηt = φy − α ηx φx
β
t /−→
c=
5
/t
c
gh
y = 1 + αη
β φxx + φyy = 0
0 < y < 1+αη
φy = 0
y=0
Boussinesq expansion:
w(t, x) = φ(t, x, 0)
u(t, x) = φx (t, x, θ)
0≤θ≤1
Solve Laplace equation:
β2 2
β4 4
y wxx +
y wxxxx + · · ·
φ(t, x, y) = w(t, x) −
2
4!
Plug expansion into free surface conditions: To first order
α 2 β
w − w
=0
2 x 2 xxt
β
ηt + wx + α(ηwx)x − wxxxx = 0
6
wt + η +
Bidirectional Boussinesq system:
ut + ηx + α u ux − 21 β (θ2 − 1) uxxt = 0
""
ηt + ux + α (η u)x − 16 β(3 θ2 − 1) uxxx = 0
at θ = 1 this system is integrable
(in fact tri-Hamiltonian!!)
=⇒ Kaup–Kupershmidt
Unidirectional waves:
1
4
2
u = η − αη +
6
1
3
Korteweg-deVries (1895) equation:
1
2
− θ
2
7
β ηxx
ηt + ηx + 32 α η ηx + 61 β ηxxx = 0
""
Due to Boussinesq in 1877!
Hamiltonian Water Waves
u=
!
φS
η
"
=⇒ Zakharov
φS (x, t) = φ(x, h + η(x, t), t) — surface potential
Hamiltonian functional:
H[u] =
**
2
D
| ∇φ | dx dy +
kinetic
*
1
S 2
g η 2 dx
potential
energy
∂u
δH
=J
=
∂t
δu
!
0 1
−1 0
"!
δH/δφS
δH/δη
"
Water Waves
Symmetries and Conservation Laws
# spatial
dimensions
surface
tension
2
2
√
3
3
√
dim. symm.
group
# cons.
laws
9
8
8
7
13
12
12
13
=⇒ T.B. Benjamin - PJO
Symmetries of 2D Water Waves
(1) Horizontal translation:
(2) Time translations:
(3) Change in potential:
(4) Vertical translation:
∂
(x + α, y, t, ϕ)
∂x
∂
(x, y, t + α, ϕ)
∂t
∂
(x, y, t, ϕ + α)
∂ϕ
∂
∂
−gt
(x, y + α, t, ϕ − α g t)
∂y
∂t
(5) Horizontal Galilean boost:
t
∂
∂
+x
∂x
∂ϕ
( x + α t, y, t, ϕ + α x + 21 α2 t )
(6) Vertical Galilean boost:
t
∂
∂
+ ( y − 12 g t2 )
∂y
∂ϕ
( x, y + α t, t, ϕ + α ( y − 12 g t2 ) + 12 α2 t )
(7) Vertical acceleration:
g t2
∂
∂
∂
−t
+ ( ϕ + 2 g t y − 12 g 2 t3 )
∂y
∂t
∂ϕ
( x, y + 12 g t2 (1 − λ−2 ), λ−1 t, λ ϕ + g t y(λ − λ−1 ) + 61 g 2 t3 (λ − 3 λ−1 + 2 λ−3 ) )
(8) Gravity–free rotation:
( y + 12 g t2 )
∂
∂
∂
−x
+ gtx
∂x
∂y
∂ϕ
( x cos α + ( y + 12 g t2 ) sin α, − x sin α + ( y + 12 g t2 ) cos α − 12 g t2 ,
t, ϕ + g t(x sin α + ( y + 21 g t2 ) cos α) )
(9) Rescaling:
x
∂
∂
∂
∂
+y
+ 21 t
+ 23 ϕ
∂x
∂y
∂t
∂ϕ
( λ x, λ y,
√
λ t, λ3/2 ϕ )
Conservation Laws of 2D Water Waves
— Infinite Depth
*
(1) Horizontal momentum:
ϕ dy = Px
S
*
(2) Energy:
*
(3) Mass:
*
(4) Vertical momentum:
(5) Horizontal Center of Mass:
*
S
S
S
*
(6) Potential Energy:
S
( 21 ϕ ν ds + 21 g 2 dx ) = E
y dx = M
ϕ dx = − g M t + Py
x y dx = − Px t + Cx +∂
1
2
S
y 2 dx = − 12 g M t2 + Py t + U +∂
(7) Radial *Momentum:
ϕ(x dy − y dx) = − 76 g 2 M t3 + 72 g Py t2 + ( 27 g U − 4 E ) t + Pr +∂
S
(8) Angular Momentum:
*
S
ϕ(x dx − y dy) =
1
2
y Px t2 − g Cx t + Pθ +∂
Infinite Depth Ocean
*
(1) Horizontal momentum:
ϕ dy = Px
S
*
(2) Energy:
*
(3) Mass:
*
(4) Vertical momentum:
(5) Horizontal Center of Mass:
*
S
S
y dx = M
ϕ dx = − g M t + Py
5
x y dx = − Px t + Cx +B∞
S
*
(6) Potential Energy:
S
( 21 ϕ ν ds + 21 g 2 dx ) = E
1
2
S
6
y 2 dx = − 12 g M t2 + Py t + U +B∞
(7) Radial *Momentum:
7
ϕ(x dy − y dx) = − 76 g 2 M t3 + 72 g Py t2 + ( 27 g U − 4 E ) t + Pr +B∞
S
(8) Angular Momentum:
6
B∞
*
S
ϕ(x dx − y dy) =
= y lim
→∞
* t* ∞
0
−∞
1
2
8
y Px t2 − g Cx t + Pθ +B∞
y [ v(x, y, t) − v(x, y, 0) ] dx dτ
First Order Hamiltonian KdV Model
For surface elevation η(t, x):
Energy:
H[ η ] =
*
[ 21 η 2 + 18 α η 3 ] dx
Poisson operator:
J = Dx + 16 β Dx3 + 41 α (η Dx + Dx η)
Hamiltonian flow:
ηt + J
δH
=0
δη
Unidirectional model:
ηt + ηx + 23 α η ηx + 16 β ηxxx +
1
16
α β (η 2 )xxx +
15
32
α2 η 2 ηx = 0
First Order Hamiltonian KdV Model
For horizontal velocity at height 0 ≤ θ ≤ 1:
u(t, x) = ϕx (t, x, y)
Energy:
H[ u ] =
*
[ 12 u2 + 83 α u3 + 16 β (2 − 3 θ 2 ) u2x ] dx
Poisson operator:
J = Dx + β ( 56 − θ 2 ) Dx3 − 14 α (u Dx + Dx u)
Hamiltonian flow:
ut + J
δH
=0
δu
Unidirectional model:
ut + ux + 32 α u ux + 16 β uxxx −
+ α β ( 53
24 −
11
4
1
18
β 2 ( 95 − 32 θ 2 + θ 4 ) uxxxxx
2
θ 2 ) u uxxx + α β ( 139
24 − 7 θ ) ux uxx −
45
32
α2 u2 ux = 0
A “Magic” Depth
At depth
%
θ =
5
11
12
− 34 τ
τ — surface tension
• The bidirectional Boussinesq model has a Hamiltonian structure
• The unidirectional Hamiltonian model is a version of KdV 5,
and has a family of exact sech2 soliton solutions
• The first order expansion of the water wave Poisson structure
gives the Korteweg-deVries biHamiltonian structure exactly.
Vortex Filaments
The motion of a vortex filament C ⊂ R 3 is described by
∂C
= κ b, where κ is the curvature, and t, n, b are the Frenet
∂t
frame on the curve.
Theorem. (Hasimoto) The curvature and torsion of a
vortex filament evolve according to the completely integrable
nonlinear Schrödinger equation
∂ψ
∂ 2ψ 1
2
−i
=
+
|
ψ
|
ψ
2
∂t
∂s2
where
ψ(t, s) = κ(t, s) exp
8
i
*
9
τ (t, s) ds .
"
Integrable biHamiltonian systems appear in a surprising
range of geometric curve flows. (Lamb, Langer, Singer,
Perline, Marı́–Beffa, Sanders, Wang, Qu, Chou, Anco, . . . )
Euclidean plane curves
∂C
= J n,
∂t
∂κ
= R(J)
∂t
A = D2 + κ2 ,
G = SE(2) = SO(2) ! R 2
B = −κ
R = A − κs D−1B = D2 + κ2 + κs D−1 · κ
κt = R(κs) = κsss + 32 κ2 κs
=⇒ modified Korteweg-deVries equation
Equi-affine plane curves
∂C
= J n,
∂t
∂κ
= R(J)
∂t
G = SA(2) = SL(2) ! R 2
A = D4 + 53 κ D2 + 53 κsD + 31 κss + 94 κ2,
B = 13 D2 − 29 κ,
R = A − κs D−1B
= D4 + 35 κ D2 + 43 κsD + 31 κss + 94 κ2 + 29 κs D−1 · κ
κt = R(κs ) = κ5s + 35 κ κsss + 53 κs κss + 59 κ2κs
=⇒ Sawada–Kotera equation
Euclidean space curves
∂C
= J1 n + J2 b,
∂t
A=






∂
∂t
!
κ
τ
"
= R(J)
G = SE(3) = SO(3) ! R 3
Ds2 + (κ2 − τ 2 )
κτss − κs τs + 2κ3 τ
2τ 2 3κτs − 2κs τ
Ds +
D
+
s
κ
κ2
κ2
−2τ Ds − τs





κs τ 2 − 2κτ τs 
1 3 κs 2 κ2 − τ 2
Ds − 2 Ds +
Ds +
κ
κ
κ
κ2
!
"
!
"
!
"
κt
κ
κs
Recursion operator:
=R s
R=A−
D−1 ( κ 0 )
τt
τs
τs
=⇒ vortex filament flow
Tri–Hamiltonian Duality
=⇒ Fokas–Fuchssteiner; Camassa–Holm; PJO–Rosenau
There are, in fact, three mutually compatible Poisson structures
associated with the Korteweg–deVries equation:
J1 = Dx,
J2 = u Dx + Dx u,
J3 = Dx3
• The Hamiltonian pair J1, J2 + J3 produces the KdV biHamiltonian flow.
• The Hamiltonian pair J1 + J3 , J2 produces the Camassa–Holm
biHamiltonian flow.
"
The same construction applies to other integrable systems,
producing compacton/peakon integrable duals.
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