Fredholm Grassmannian flows and their applications to nonlinear PDEs - PowerPoint PPT Presentation

Grassmann/Riccati flows Fredholm Grassmannian flows and their applications to nonlinear PDEs and SPDEs Margaret Beck, Anastasia Doikou, Simon J.A. Malham, Ioannis Stylianidis and Anke Wiese NTNU Trondheim 10th May 2019 Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Outline 1 Motivation. 2 Canonical linear system. 3 Integrable nonlinear PDEs. 4 Smoluchowski-type equations. Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Motivation: Integrable systems and Fredholm determinants Dyson; Miura; Ablowitz, Ramani & Segur; P¨ oppe; Sato; Segal & Wilson; Tracy & Widom. . . . Quoting P¨ oppe: “For every soliton equation, there exists a linear PDE (called a base equation) such that a map can be defined mapping a solution p of the base equation to a solution g ∗ of the soliton equation. The properties of the soliton equation may be deduced from the corresponding properties of the base equation which in turn are quite simple due to linearity. The map p → g ∗ essentially consists of constructing a set of linear integral operators using p and computing their Fredholm determinants.” Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Motivation: Marchenko equation Ablowitz, Ramani & Segur: Marchenko equation, y � x : � ∞ p ( x , y ) = g ( x , y ) + g ( x , z ) q ( z , y ; x ) d z , x 1 Scattering data: p = p ( x + y ) and q . 2 KdV q = − p : ∂ t p + ∂ 3 Suppose x p = 0 , γ ( x ) := − 2( d / d x ) g ( x , x ) then ∂ t γ + ∂ 3 satisfies x γ = 6 γ∂ x γ. 3 P¨ oppe: elevated argument to operator level. Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Example PDEs with local/nonlocal nonlinearities 1 Nonlinear Schr¨ odinger equation, local nonlinearity: � � � 2 . � γ ( x ; t ) i ∂ t γ ( x ; t ) = ∂ 2 x γ ( x ; t ) − γ ( x ; t ) 2 PDE with quadratic nonlocal nonlinearity: � ∂ t g ( x , y ; t ) = d ( ∂ x ) g ( x , y ; t ) − g ( x , z ; t ) g ( z , y ; t ) d z . R Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Canonical linear system ( Q := id + ˆ ∂ t Q = AQ + BP , Q ) ∂ t P = CQ + DP , ( Q 0 = id , P 0 = G 0 ) P = G Q . � � ⇒ ∂ t G Q = ∂ t P − G ∂ t Q = CQ + DP − G ( AQ + BP ) = ( C + DG ) Q − G ( A + BG ) Q ⇒ ∂ t G = C + DG − G ( A + BG ) . Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows “Big matrix” PDEs ∂ t G = DG − G BG � ⇔ ∂ t g ( x , y ; t ) = d ( ∂ x ) g ( x , y ; t ) − g ( x , z ; t ) b ( z ) g ( z , y ; t ) d z . R Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows In practice: Quadratic “Big matrix” PDE Suppose we wish to solve the PDE � ∂ t g ( x , y ; t ) = d ( ∂ x ) g ( x , y ; t ) − g ( x , z ; t ) b ( z ) g ( z , y ; t ) d z . R Then our prescription says set up: ∂ t p ( x , y ; t ) = d ( ∂ x ) p ( x , y ; t ) , ∂ t q ( x , y ; t ) = b ( x ) p ( x , y ; t ) . � p ( x , y ; t ) = g ( x , z ; t ) q ( z , y ; t ) d z . R Here we set q ( x , y ; t ) = δ ( x − y ) + ˆ q ( x , y ; t ). Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows In practice: What have we gained? ∂ t p ( x , y ; t ) = d ( ∂ x ) p ( x , y ; t ) , ∂ t q ( x , y ; t ) = b ( x ) p ( x , y ; t ) . p ( k , y ; t ) = e d (2 π i k ) t p 0 ( k , y ) , � q ( k , y ; t ) = e 2 π i ky + b ( k − κ ) I ( κ, t ) p 0 ( κ, y ) d κ, R � � e d (2 π i k ) t − 1 I ( k , t ) := / d (2 π i k ) . ⇒ Solve the linear equations for p and q explicitly. ⇒ Evaluate them for any given time t > 0. ⇒ Solve linear Fredholm equation for g . ⇒ Solution g to the PDE at that time. Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Integrable systems: Nonlinear Schr¨ odinger equation i ∂ t P = ∂ 2 x P , ˆ Q = P † P , P = G (id + ˆ Q ) . � 0 ( P ψ )( y ; x ) := p ( y + z + x ) ψ ( z ) d z , −∞ � 0 p ∗ ( y + ξ + x ; t ) p ( ξ + z + x ; t ) d ξ, q ( y , z ; x , t ) = ˆ −∞ � 0 p ( y + z + x ; t ) = g ( y , z ; x , t ) + g ( y , ξ ; x , t )ˆ q ( ξ, z ; x , t ) d ξ. −∞ Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Product rule Assume R additive with kernel r = r ( y + z + x ). 1 Define: � G � ( x ; t ) := g (0 , 0; x , t ); 2 Product rules: Primitive: 1 � �� � [ F ∂ x ( RR ′ ) F ′ ]( y , z ) ≡ [ FR ]( y , 0) [ R ′ F ′ ](0 , z ) Complete: 2 � F ∂ x ( RR ′ ) F ′ � ≡ � FR �� R ′ F ′ � . Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Product rule: proof Proof. � � � r ( ξ 1 + ξ 2 + x ) r ′ ( ξ 2 + ξ 3 + x ) f ′ ( ξ 3 , z ) d ξ 3 d ξ 2 d ξ 1 f ( y , ξ 1 ) ∂ x R 3 � − � � r ( ξ 1 + ξ 2 + x ) r ′ ( ξ 2 + ξ 3 + x ) f ′ ( ξ 3 , z ) d ξ 3 d ξ 2 d ξ 1 = f ( y , ξ 1 ) ∂ ξ 2 R 3 � − f ( y , ξ 1 ) r ( ξ 1 + x ) r ′ ( ξ 3 + x ) f ′ ( ξ 3 , z ) d ξ 3 d ξ 1 = R 2 � � − r ′ ( ξ 3 + x ) f ′ ( ξ 3 , z ) d ξ 3 = f ( y , ξ 1 ) r ( ξ 1 + x ) d ξ 1 · R − R − � �� � [ R ′ F ′ ](0 , z ) = [ FR ]( y , 0) . Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Nonlinear Schr¨ odinger equation Proof. Q = PP † ⇒ : Set U := (id + ˆ Q ) − 1 , so G = PU , and ˆ � � i ∂ t G − ∂ 2 PU ( P † x P ) x U + P x U ( P † P ) x U + PU x ( P † P ) x U x G = 2 . � �� � i ∂ t g ( y , z ) − ∂ 2 [( PUP † ) x ]( y , 0) x g ( y , z ) = 2 [ PU ](0 , z ) � �� � [ V ( PP † ) x V ]( y , 0) = 2 [ PU ](0 , z ) �� ��� �� � [ P † V ](0 , 0) = 2 VP ]( y , 0) [ G ](0 , z ) �� ��� �� � [ G † ](0 , 0) = 2 G ]( y , 0) [ G ](0 , z ) = 2 g ( y , 0) g ∗ (0 , 0) g (0 , z ) . Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Examples Integrable hierarchy (algebra?); Nonlocal reaction-diffusion (classes of); Higher degree nonlocal nonlinearities; Nonlinear elliptic systems (classes of); SPDEs (classes of); Smoluchowski coagulation (?). Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows SPDEs with nonlocal nonlinearities On T = [0 , 2 π ] 2 with q = q ( x , y ; t ), p = p ( x , y ; t ), g = g ( x , y ; t ): 1 p + γ ˙ ∂ t p = ∂ 2 W ∗ p , ∂ t q = ǫ p , p = g ⋆ q . ∂ t g = ∂ 2 1 g + γ ˙ ⇒ W ∗ g − ǫ g ⋆ g . Here W = W ( x ; t ) is a Wiener field. Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows SPDEs with nonlocal nonlinearities: figure Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Smoluchowski’s coagulation equation � x ∂ t g ( x ; t ) = 1 K ( y , x − y ) g ( y ; t ) g ( x − y ; t ) d y 2 0 � �� � coagulation gain � ∞ − g ( x ; t ) K ( x , y ) g ( y ; t ) d y . 0 � �� � coagulation loss g ( x , t ) = density of clusters of mass x ; Applications: polymerisation, aerosols, clouds/smog, clustering stars/galaxies, schooling/flocking. Solvable cases: (i) K = 2; (ii) K = x + y and (iii) K = xy . Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Smoluchowski’s coagulation equation: K = 1 Consider the case K = 1 (many other cases by rescaling). � x � ∞ ∂ t g ( x ; t ) = 1 g ( y ; t ) g ( x − y ; t ) d y − g ( x ; t ) g ( y ; t ) d y . 2 0 0 � �� � m 0 ( t ) m 0 = − 1 2 m 2 Direct integration ⇒ ˙ 0 ⇒ m 0 known. ∂ t p = − m 0 p , ∂ t q = − 1 2 p , p = gq . 2 g 2 − m 0 g . ∂ t g = 1 ⇒ Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Smoluchowski: more generally � x � x ∂ t g ( x ; t ) = g ( x − y ; t ) b ( ∂ y ) g ( y ; t ) d y − a ( x − y ; t ) g ( y ; t ) d y 0 0 � x � y + d ( ∂ x ) g ( x ; t ) − b 0 ( y − z ; t ) g ( z ; t ) d z g ( x − y ; t ) d y . 0 0 � ∞ (1 − e − sx ) g ( x , t ) d x . Desingularised Laplace Transf.: π ( s , t ) = 0 Menon & Pego (2003) ⇒ 2 π 2 = 0; ∂ t π + 1 K = 1 : K = x + y : ∂ t π + π∂ s π = − π ; K = xy : ∂ t ˜ π + ˜ π∂ s ˜ π = 0 . Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications

Grassmann/Riccati flows Optimal nonlinear control: Riccati PDEs Inspired by Byrnes (1998) and Byrnes & Jhemi (1992) ⇒ q = aq + bp , ˙ p = cq + dp , ˙ p = π ( q , t ) . ⇒ ∂ t π = cq + d π − ( ∇ π )( aq + b π ) Ex. q = p , ˙ p = 0 , ˙ ⇒ ∂ t π = ( ∇ π ) π. Beck, Doikou, Malham, Stylianidis, Wiese Fredholm Grassmannian flows and applications