pdes with nonlocal nonlinearities: generation and solutiongrassmann/riccati ows pdes with nonlocal...
TRANSCRIPT
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
PDEs with nonlocal nonlinearities:Generation and solution
Margaret Beck, Anastasia Doikou, Simon J.A. Malham,Ioannis Stylianidis and Anke Wiese
Sheffield 2018: November 21st
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Outline
1 Motivation.
2 PDEs with nonlocal nonlinearities (examples).
3 Infinite dimensional Riccati flows.
4 PDEs with nonlocal nonlinearities (revisited in detail).
5 Nonlinear PDEs and SPDEs.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Motivation: Integrable systems and Fredholm determinants
Dyson; Miura; Ablowitz, Ramani & Segur; Poppe; Sato; Segal &Wilson; Tracy & Widom...
Quoting Poppe:
“For every soliton equation, there exists a linear PDE(called a base equation) such that a map can be definedmapping a solution f of the base equation to a solutionu of the soliton equation. The properties of the solitonequation may be deduced from the corresponding proper-ties of the base equation which in turn are quite simpledue to linearity. The map f → u essentially consists ofconstructing a set of linear integral operators using f andcomputing their Fredholm determinants.”
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Motivation: Marchenko equation
Ablowitz, Ramani & Segur: Marchenko equation, y > x :
p(x , y) = g(x , y) +
∫ ∞x
g(x , z)q′(z , y ; x)dz ,
1 Scattering data: p = p(x + y) and q′.
2 KdV q′ = −p:
∂tp + ∂3p = 0 ⇒ u = −2(d/dx)g(x , x) satisfies KdV.
3 Poppe: operator level.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Example PDEs with nonlocal nonlinearities
1 Fisher–Kolmogorov–Petrovskii–Piskunov (FKPP):
∂tg(x ; t) = d(∂x)g(x ; t)− g(x ; t)
∫Rb(z , ∂z) g(z ; t) dz .
2 Quadratic:
∂tg(x , y ; t) = d(∂x) g(x , y ; t)−∫Rg(x , z ; t) g(z , y ; t) dz .
3 Odd degee:
∂tg = −ih(∂1) g − g ? f ?(g ? g †).
4 Rational:
∂tg(x , y ; t) = d(√−∆x
)g(x , y ; t)− g(x , y ; t)b(y)g(y , y ; t).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Smoluchowski’s coagulation equation
∂tg(x ; t) = 12
∫ x
0K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸
coagulation gain
− g(x ; t)
∫ ∞0
K (x , y) g(y ; t) dy︸ ︷︷ ︸coagulation loss
.
Applications: polymerisation, aerosols, clouds/smog, clusteringstars/galaxies, schooling/flocking.
Special cases: (i) K = 2; (ii) K = x + y and (iii) K = xy .
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Riccati Flows: Quadratic degree
1. ∂tQ = AQ + BP,
2. ∂tP = CQ + DP,
3. P = G Q.
⇒(∂tG
)Q = ∂tP − G ∂tQ
= CQ + DP − G (AQ + BP)
= (C + DG )Q − G (A + BG )Q
⇒ ∂tG = C + DG − G (A + BG ).
(Q0 = I , P0 = G0.)
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Riccati Flows: Higher odd degree
1. ∂tQ = f (PP†)Q,
2. ∂tP = DP,
3. P = G Q.
Require:
QQ† = id,
f (x) = i∑m>0
αmxm.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Fredholm Grassmann Flows: Higher odd degree II
f † = −f ⇒ ∂t(QQ†
)= f (QQ†)− (QQ†) f
⇒ QQ† = id
⇒ PP† = GG †
(∂tG
)Q = ∂tP − G ∂tQ
= DG Q − G f (PP†)Q
= DG Q − G f (GG †)Q.
⇒ ∂tG = D G − G f (GG †).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
“Big matrix” PDEs
∂tG = DG − G BG
⇔ ∂tg(x , y ; t) = d(∂x) g(x , y ; t)−∫Rg(x , z ; t) b(z) g(z , y ; t)dz .
We set (big matrix product)(q ? q
)(x , y ; t) :=
∫Rq(x , z ; t)q(z , y ; t) dz .
∂tG = D G − G f (GG †)
⇔ ∂tg = −ih(∂1) g + g ? f ?(g ? g †).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
In practice: Quadratic “Big matrix” PDE
Suppose we wish to solve the PDE
∂tg(x , y ; t) = d(∂x) g(x , y ; t)−∫Rg(x , z ; t) b(z) g(z , y ; t) dz .
Then our prescription says set up:
1. ∂tp(x , y ; t) = d(∂x) p(x , y ; t),
2. ∂tq(x , y ; t) = b(x) p(x , y ; t).
3. p(x , y ; t) = g(x , y ; t) +
∫Rg(x , z ; t) q′(z , y ; t)dz .
Here we set q(x , y ; t) = δ(x − y) + q′(x , y ; t).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
In practice: What have we gained?
∂tp(x , y ; t) = d(∂x) p(x , y ; t),
∂tq(x , y ; t) = b(x) p(x , y ; t).
p(k, y ; t) = ed(2πik) t p0(k , y),
q′(k, y ; t) = e2πiky +
∫Rb(k − κ) I (κ, t) p0(κ, y) dκ,
I (k , t) :=(ed(2πik) t − 1
)/d(2πik).
We can solve the linear equations for p and q explicitly andevaluate them for any given time t > 0. Then we can determinethe solution to the PDE at that time by solving the linearFredholm equation for g .
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
In practice: Higher odd degree
Suppose we wish to solve the PDE
∂tg = −ih(∂1) g + g ? f ?(g ? g †).
Our prescription says set up:
1. ∂tp = −ih(∂1) p,
2. ∂tq = f ?(p ? p†
)? q,
3. p = g ? q.
1. ∂t p = −ih(2πik) p,
2. ∂t q = f ?(p ? p†
)? q,
3. p(k , κ; t) =
∫Rg(k , λ; t)q(λ, κ; t)dλ.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
In practice: Higher odd degree II
p(k , κ; t) = exp(−it h(2πik)
)p0(k , κ),
θ(k , κ; t) := exp(it h(−2πik)
)q(k , κ; t).
∂t θ =(f ?(p0 ? p
†0) + ih δ
)? θ,
θ(k, κ; t) = exp?(t(f ?(p0 ? p
†0) + ih δ
)),
Solve: p(k , κ; t) =
∫Rg(k, λ; t)q(λ, κ; t) dλ.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Generalized NLS
h(x) = x4, f (x) = sin(x).
⇒ equation for g = g(x , y ; t):
i∂tg = ∂41 g + g ? sin?
(g ? g †
),
g0(x , y) := sech(x + y) sech(y).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Generalized NLS figures
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
real(det)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ima
g(d
et)
Determinant in complex plane
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Classical example
1. ∂tp(y) = d(∂y ) p(y),
2. ∂tq(y) = b(∂y ) p(y),
3. p(y) =
∫Rg(z) q(z + y) dz .
⇒∫R∂tg(z) q(z + y)dz
=∂tp(y)−∫Rg(z) ∂tq(z + y) dz
= d(∂y ) p(y)−∫Rg(z) b(∂z) p(z + y) dz
=
∫Rg(z) d(∂y ) q(z + y) dz −
∫Rg(z) b(∂z)
∫Rg(ζ) q(ζ + z + y)dζ dz
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Classical example cont’d
⇒∫R∂tg(z) q(z + y) dz
=
∫R
(d(−∂z) g(z)
)q(z + y) dz
−∫R
(b(−∂z) g(z)
) ∫Rg(ζ) q(ζ + z + y) dζ dz
=
∫R
(d(−∂z) g(z ; t)
)q(z + y ; t) dz
−∫R
(b(−∂z) g(z ; t)
) ∫Rg(ξ − z ; t) q(ξ + y ; t)dξ dz .
⇒ ∂tg(η) = d(−∂η) g(η)−∫R
(b(−∂z) g(z)
)g(η − z) dz .
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Fisher–Kolmogorov–Petrovskii–Piskunov equation (FKPP)
Consider nonlocal FKPP (see Britton or Bian, Chen & Latos):
∂tg(x ; t) = d(∂x)g(x ; t)− g(x ; t)
∫Rb(z , ∂z) g(z ; t) dz .
1. ∂tp(x) = d(∂x) p(x),
2. ∂tq(x) = b(x , ∂x) p(x),
3. p(x) = g(x)
∫Rq(z) dz .
(= g(x)q
).
(∂tg(x ; t)
)q(t) = ∂tp(x ; t)− g(x ; t) ∂tq(t)
= d(∂x)p(x ; t)− g(x ; t)
∫Rb(z , ∂z) p(z ; t) dz
= d(∂x)g(x ; t) q(t)− g(x ; t)
∫Rb(z , ∂z) g(z ; t)dz q(t).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
FKPP II
Given data g0, set q(0) = 1 & p(x ; 0) = g0(x) ⇒
g(x ; t) =p(x ; t)
q(t)
Consider the case b = 1:
p(k; t) = exp(d(2πik) t
)g0(k),
q(t) = 1 +
(exp(t d(0))− 1
d(0)
)g0(0).
d(0) = 0 ⇒ q(t) = 1 + t g0(0).
⇒ explicit solution for any diffusive or dispersive d = d(∂x).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Smoluchowski’s coagulation equation
∂tg(x ; t) = 12
∫ x
0K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸
coagulation gain
− g(x ; t)
∫ ∞0
K (x , y) g(y ; t) dy︸ ︷︷ ︸coagulation loss
.
g(x , t) = density of clusters of mass x ;
Tag cluster mass x : rate it merges with cluster mass yproportional to the density of clusters;
Constant of proportionality is K = K (x , y) or frequency;
Rate coalesce (y , x−y)→ x is 12K (y , x−y) g(y ; t)g(x−y ; t);
Loss rate is K (x , y) g(x ; t)g(y ; t).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Smoluchowski’s coagulation equation II
∂tg(x ; t) = 12
∫ x
0K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸
coagulation gain
− g(x ; t)
∫ ∞0
K (x , y) g(y ; t) dy︸ ︷︷ ︸coagulation loss
.
Consider the case K = 1. With q(t) :=∫∞
0 q(z) dz :
1. ∂tp(x) = 0,
2. ∂tq(x) = −12p(x) q2,
3. p(x) =
∫ ∞0
g(z) q(z + x)dz1
q2.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Smoluchowski’s coagulation equation III
⇒ ∂tq = −12g q.
⇒∫ ∞
0∂tg(z) q(z + x)dz
1
q2
= −∫ ∞
0g(z) ∂tq(z + x) dz
1
q2+ 2
∫ ∞0
g(z) q(z + x) dz∂tq
q3
= 12
∫ ∞0
g(z)
∫ ∞0
g(ζ) q(ζ + z + x)dζ dz1
q2−∫ ∞
0g(z) q(z + x)dz
g
q2
= 12
∫ ∞0
g(z)
∫ ∞z
g(ξ − z) q(ξ + x)dξ dz1
q2−∫ ∞
0g(z) q(z + x)dz
g
q2.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Smoluchowski’s gain-only coagulation equation
∂tg(x ; t) = 12
∫ x
0K (y , x − y) g(y ; t)g(x − y ; t) dy
1. ∂tp(x) = 0,
2. ∂tq(x) = −12p(x),
3. p(x) =
∫ ∞0
H(x , z) g(z) q(z + x)dz ,
Inversion:
∫ ∞0
H(x , z) q(z + x)q∗(x , y) dx = δ(y − z).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Smoluchowski’s gain-only coagulation equation II
Similar calculation ⇒∫ ∞0
H(x , z) ∂tg(z) q(z + x) dz
= 12
∫ ∞0
∫ z
0H(x , ξ)H(x + ξ, z − ξ)︸ ︷︷ ︸
=K(z,ξ)H(x ,z)
g(ξ)g(z − ξ) dξ q(z + x) dz .
Example cases:
1 H(x , z) = eαxz ⇒ K (ξ, z − ξ) = eαξ(z−ξ);
2 H(x , z) = eα(x2z+xz2) ⇒ . . .;
3 Exponential of symmetric polynomials and furthergeneralisations.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Smoluchowski’s coagulation equation: Add/Multive kernels
Kernel cases K (x , y) = x + y and K (x , y) = xy intimately related.
∂tg(x ; t) = 12x
∫ x
0g(y ; t)g(x−y ; t) dy−g(x ; t)
∫ ∞0
(x+y) g(y ; t) dy .
Desingularised LT: π(s, t) =
∫ ∞0
(1− e−sx) g(x , t) dx .
Menon & Pego (2003) ⇒
K = 1 : ∂tπ + 12π
2 = 0;
K = x + y : ∂tπ + π∂sπ = −π;
K = xy : ∂t π + π∂s π = 0.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Optimal nonlinear control: Riccati PDEs
Byrnes (1998) and Byrnes & Jhemi (1992) ⇒
Nonlinear state evolution: q = b(q) + σ(q)u;
Nonlinear cost function:∫ T
0 L(q, u) dt + Q(q(T )
);
Bolza problem ⇒ optimal u∗ = u∗(q, p) with
q = ∇pH∗, p = −∇qH∗ and pT = −∇Q(q(T )
);
Goal: Find map π s.t. p = π(q, t) ⇒ u∗ = u∗(q, π(q, t)
);
Generates ∂tπ = ∇qH∗(q, π) + (∇qπ)(∇πH∗(q, π)
); (offline)
L(q, u) = |u|2 ⇒ p = u and q = p, p = 0 so ∂tπ = (∇π)π.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Inviscid Burgers flow
Characteristics: Assume π sat. ∂tπ + (∇π)π = 0; set q = π.
Now assume for q = q(a, t) and p = p(a, t) with q(a, 0) = a:
q = p, p = 0, and p = π(q, t).
Then 0 = p = ∂tπ + (∇qπ)q = ∂tπ + (∇π)π. To find π = π(x , t):
π(q(a, t), t) = π0(a) and q(a, t) = a + tπ0(a).
Hence if
q(a, t) = x ⇔ x = a + tπ0(a) ⇔ a = (id + tπ0)−1 ◦ x ,
then π(x , t) = π0 ◦ (id + tπ0)−1 ◦ x .
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Riccati flow is a Burgers subflow
Riccati flow is Burgers subflow corres. to linear data π0(a) = π0a.
In this case:
x = a + tπ0a ⇔ a = (id + tπ0)−1x ,
π(x , t) = π0a = π0(id + tπ0)−1x .
Note π(x , t) is linear in x so set πR(t) := π0(id + tπ0)−1:
∂tπ(x , t) = −π0(id + tπ0)−1π0(id + tπ0)−1x
= −(πR(t)
)2x ,
(∇xπ)π = πR(t)πR(t)x .
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Burgers flow/SPDEs
1. Qt(a) = a +
∫ t
0Ps(a)ds +
√2νBt ,
2. ∂tπ+(∇π)π + ν∆π = 0,
3. Pt(a) = π(Qt(a), t
).
Ito: π(Qt(a), t
)= π0(a) +
∫ t
0
(∂sπ + (∇π)π + ν∆π
)(Qs(a), s
)ds
+√
2ν
∫ t
0(∇π)(Qs(a), s)dBs .
⇒ π(x , t) = E[π0
(Q−1
t (x))].
Constantin & Iyer (2008,. . . )Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Burgers flow/SPDEs II
1. Qt(a) = a +
∫ t
0Ps(a) ds +
√2νBt ,
2. Pt(a) = P0(a),
3. Pt(a) = πt(Qt(a), t
),
4. u(x , t) = E[π0
(Q−1
t (x))]. (Observations—hope?)
Generalised Ito:
πt(Qt(a), t
)= π0(a) +
∫ t
0
((∇πs)πs − ν∆πs
)(Qs(a), s
)ds
+√
2ν
∫ t
0(∇πs)(Qs(a), s) dBs +
∫ t
0πs(Qs(a), ds
).
⇒ dπt+((∇πt)πt − ν∆πt
)dt +
√2ν(∇πt) dBt = 0.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Looking forward
Coordinate patches:(qp
)=
(q
πt ◦ q
)=
(idπt
)◦ q.
Eg. instead choose q = π′t(p), i.e. π′ = π−1.
q = p, p = 0 ⇒ p0 = p(t) = q = (∂tπ′t) ◦ p(t) = (∂tπ
′t) ◦ p0.
Setting y = p0 ⇒ π′t ◦ y = π′0 ◦ y + ty .
Girsanov and Cole–Hopf transformations;
Q, P and P = π(Q, t) all operators;
SPDEs, Madelung, dispersion...
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Thank you
Thank you for listening!
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Nonlocal reaction-diffusion system
With d11 = ∂21 + 1, d12 = −1/2, b12 = 0 and b11 = N(x , σ):
∂tu = d11u + d12v − u ? (b11u)− u ? (b12v)− v ? (b12u)− v ? (b11v),
∂tv = d11v + d12u − u ? (b11v)− u ? (b12v)− v ? (b12v)− v ? (b11u),
u0(x , y) := sech(x+y) sech(y) and v0(x , y) := sech(x+y) sech(x).
p =
(p11 p12
p12 p11
), q =
(q11 q12
q12 q11
)and g =
(g11 g12
g12 g11
).
Similar forms for d and b. Riccati equation: ∂tG = dG − G (bG )
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Nonlocal reaction-diffusion system II
-0.1
10
0
0.1
5 10
0.2
u
0.3
Direct method: T=0.5
5
y
0
0.4
x
0.5
0-5
-5-10 -10
-0.2
10
-0.1
0
5 10
v
0.1
Direct method: T=0.5
5
y
0.2
0
x
0.3
0-5
-5-10 -10 0 0.1 0.2 0.3 0.4 0.5
t
0
20
40
60
80
100
120
140
de
t, n
orm
Determinant and Hilbert--Schmidt norm
Determinant
HS-norm
-0.1
10
0
0.1
5 10
0.2
g11
0.3
Riccati method: T=0.5
5
y
0
0.4
x
0.5
0-5
-5-10 -10
-0.2
10
-0.1
0
5 10
g12
0.1
Riccati method: T=0.5
5
y
0.2
0
x
0.3
0-5
-5-10 -10
0
10
1
5 10
2
ab
s-r
ea
l
10 -5
Euclidean Difference: T=0.5
5
3
y
0
x
4
0-5
-5-10 -10
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Stochastic PDEs with nonlocal nonlinearities
∂tQ = AQ + BP,
∂tP = CQ + DP,
P = G Q.
∂tG = C + DG − G (A + BG ).
∂tg = ∂21g + W ∗ g − g ? g .
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Stochastic PDEs with nonlocal nonlinearities II
On T = [0, 2π]:
Wt(x) :=1√π
∑n>1
1
nW n
t cos(nx)
Suppose pt = pt(x , y) satisfies
∂tpt = ∂21pt + W ∗ pt .
pt(x , y) =1
π
∑n>0,m>0
(pssnm sin(nx) sin(my) + pcsnm cos(nx) sin(my)
+ pscnm sin(nx) cos(my) + pccnm cos(nx) cos(my)).
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Stochastic PDEs with nonlocal nonlinearities II
s(x) =
sin(x)sin(2x)
...
and c(x) =
co
cos(x)cos(2x)
...
pt(x , y) =(sT(x) cT(x)
)(pss psc
pcs pcc
)(s(y)c(y)
)
=(sT(x) cT(x)
)P
(s(y)c(y)
)
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
Grassmann/Riccati flowsPDEs with nonlocal nonlinearities
Stochastic PDEs with nonlocal nonlinearities III
Ξ :=1√πdiag{W 1
t ,1
2W 2
t , . . . , 0,W1t ,
1
2W 2
t , . . .}
D := −diag{1, 22, 32, . . . , 0, 1, 22, 32, . . .}
⇒ ∂tP = DP + ΞP
⇒ ∂tpnm = −n2pnm +1
nW n
t pnm
⇒ pnm = exp(−n2t −
√πn W n
t − 12πn2 t)pnm(0)
⇒ p0m = p0m(0)
⇒ q′nm =
∫ t
0pnm(τ) dτ.
Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities