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• 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; Pöppe; Sato; Segal &Wilson; Tracy & Widom...

Quoting Pöppe:

“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 Pöppe: 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

∫ x0

K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸coagulation gain

− g(x ; t)∫ ∞

0K (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 ? q̃

)(x , y ; t) :=

∫Rq(x , z ; t)q̃(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).

p̂(k, y ; t) = ed(2πik) t p̂0(k , y),

q̂′(k, y ; t) = e2πiky +

∫Rb̂(k − κ) Î (κ, t) p̂0(κ, y) dκ,

Î (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 p̂ = −ih(2πik) p̂,

2. ∂t q̂ = f̂?(p̂ ? p̂†

)? q̂,

3. p̂(k , κ; t) =

∫Rĝ(k , λ; t)q̂(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

p̂(k , κ; t) = exp(−it h(2πik)

)p̂0(k , κ),

θ̂(k , κ; t) := exp(it h(−2πik)

)q̂(k , κ; t).

∂t θ̂ =(f̂ ?(p̂0 ? p̂

†0) + ih δ

)? θ̂,

θ̂(k, κ; t) = exp?(t(f̂ ?(p̂0 ? p̂

†0) + ih δ

)),

Solve: p̂(k , κ; t) =

∫Rĝ(k, λ; t)q̂(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:

p̂(k; t) = exp(d(2πik) t

)ĝ0(k),

q(t) = 1 +

(exp(t d(0))− 1

d(0)

)ĝ0(0).

d(0) = 0 ⇒ q(t) = 1 + t ĝ0(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

∫ x0

K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸coagulation gain

− g(x ; t)∫ ∞

0K (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

∫ x0

K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸coagulation gain

− g(x ; t)∫ ∞

0K (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 dz 1q2−∫ ∞

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

∫ x0

K (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

∫ z0

H(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(x

2z+xz2) ⇒ . . .;3 Exponential of symmetric polynomials and further

generalisations.

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

∫ x0

g(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 π̃ + π̃∂s π̃ = 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: q̇ = 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

q̇ = ∇pH∗, ṗ = −∇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 q̇ = p, ṗ = 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 q̇ = π.

Now assume for q = q(a, t) and p = p(a, t) with q(a, 0) = a:

q̇ = p, ṗ = 0, and p = π(q, t).

Then 0 = ṗ = ∂tπ + (∇qπ)q̇ = ∂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 +

∫ t0Ps(a)ds +

√2νBt ,

2. ∂tπ+(∇π)π + ν∆π = 0,3. Pt(a) = π

(Qt(a), t

).

Itô: π(Qt(a), t

)= π0(a) +

∫ t0

(∂sπ + (∇π)π + ν∆π

)(Qs(a), s

)ds

+√

∫ t0

(∇π)(Qs(a), s)dBs .

⇒ π(x , t) = E[π0(Q−1t (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 +

∫ t0Ps(a) ds +

√2νBt ,

2. Pt(a) = P0(a),

3. Pt(a) = πt(Qt(a), t

),

4. u(x , t) = E[π0(Q−1t (x)

)]. (Observations—hope?)

Generalised Itô:

πt(Qt(a), t

)= π0(a) +

∫ t0

((∇πs)πs − ν∆πs

)(Qs(a), s

)ds

+√

∫ t0

(∇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.

q̇ = p, ṗ = 0 ⇒ p0 = p(t) = q̇ = (∂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 p12p12 p11

), q =

(q11 q12q12 q11

)and g =

(g11 g12g12 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 + Ẇ ∗ 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 nt cos(nx)

Suppose pt = pt(x , y) satisfies

∂tpt = ∂21pt + Ẇ ∗ pt .

pt(x , y) =1

π

∑n>0,m>0

(pssnm sin(nx) sin(my) + p

csnm 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) =

cocos(x)

cos(2x)...

pt(x , y) =(sT(x) cT(x)

)(pss pscpcs 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 1t ,

1

2W 2t , . . . , 0,W

1t ,

1

2W 2t , . . .}

D := −diag{1, 22, 32, . . . , 0, 1, 22, 32, . . .}

⇒ ∂tP = DP + Ξ̇P

⇒ ∂tpnm = −n2pnm +1

nẆ nt pnm

⇒ pnm = exp(−n2t −

√πn W

nt − 12

πn2t)pnm(0)

⇒ p0m = p0m(0)

⇒ q′nm =∫ t

0pnm(τ) dτ.

Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

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Grassmann/Riccati ﬂows PDEs 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
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