STOCHASTIC ANALYSIS AND THE KdV EQUATION Setsuo TANIGUCHI Faculty - PowerPoint PPT Presentation

STOCHASTIC ANALYSIS AND THE KdV EQUATION Setsuo TANIGUCHI Faculty of Mathematics, Kyushu Univ. Joint work with N. Ikeda + α http://www.math.kyushu-u.ac.jp/~taniguch/ 0

PDE and Stochastic Analysis ✓ ✏ u ( x, t ); ut = L V u, u ( · , 0) = f Heat eq: ✒ ✑ � � � i,j aij∂x i ∂x j + � L V = 1 i bi∂x i + V 2 � f ( X ( t, x )) e Φ( x ; V ) � � � u ( x, t ) = E X ( t, x ) : L 0 -diff.pr. 1942 K.Itˆ o; stoch. integral, Itˆ o’s formula � d � 2 − x 2 1944 R.Cameron-W.Martin; 1 2 2 dx 1947 M.Kac; the Feynman-Kac formula ⋄ Refrectionless potential, generalized refrectionless potential, n -solitons of the KdV eq. vt = 3 2 vvx + 1 4 vxxx . 1

v ( x, t ) = 2sech2( x + t − 2) v(x,y) 2 4 1.8 1.6 1.4 1.2 1 0.8 2 0.6 0.4 0.2 0 -10 0 -5 t-axis -2 0 5 -4 x-axis 10 2

v ( x, t ) = 12 3+4 cosh(2 x +2 t )+cosh(4 x +16 t ) 3 cosh( x +7 t )+cosh(3 x +9 t ) v(x,y) 5 8 7 4 6 5 4 3 3 2 1 0 2 -15 -10 1 t-axis -5 0 0 5 -1 10 x-axis -2 15 3

• u s ( s ∈ S ): reflectionless potential ✛ ✘ u s( x ) = − 2 d 2 dx 2 log det( I + G s( x )) , where ✚ ✙ � � � � S = { ηj, mj } 1 ≤ j ≤ n � n ∈ N , ηj, mj > 0 , ηi � = ηj �� mimj e − ( η i + η j ) x � G s( x ) = . ηi + ηj 1 ≤ i,j ≤ n o op. − d 2 • Schr¨ dx 2 + u s → Scattering data s ∈ S Ξ0 = { u s | s ∈ S } bijective S , ← → • Ξ ∋ u ⇔ ∃ µ > 0 , un ∈ Ξ0 s.t. Spec ( − d 2 dx 2 + un ) ⊂ [ − µ, ∞ ) , n = 1 , 2 , . . . un → u (unif on cpts) (Marchenko) 4

• W = { w : [0 , ∞ ) → R | conti , w (0)=0 } ; 1-dim W.sp • X ( x ) : W → R : X ( x, w ) = w ( x ) , w ∈ W • Σ = { σ | finite meas on R with cpt supp } • P σ : the prob meas on W under which { X ( x ) } is the cent. Gaussian pr with cov fn � � eζ ( x + y ) − eζ | x − y | X ( x ) X ( y ) dP = σ ( dζ ) . 2 ζ R W • G = { P σ | σ ∈ Σ } bijective ← → Σ � � d X ( x )2 dP σ = e 2 ζxσ ( dζ ) ∵ ) dx W R 5

� x � � � ψ ( P σ )( x )=4 d 2 − 1 X ( y )2 dy dP σ dx 2 log exp , x ≥ 0 2 0 W The Plan of talk ✬ ✩ ✬ ✩ G : Cen. Gauss Ξ : gen. rl. pot ψ P σ, σ ∈ Σ u ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ � � � � � � � � � � � � � � ❅ ❅ (3) � � � ❅ ❅ ❅ � ❅ � � ❅ ❅ � � � ❅ ❅ ❅ � ❅ unif conv. (2) on cpts ✬ ✩ ✬ ✩ Ξ0 : rl pot G 0 : P σ , � � ❅ � ❅ ❅ us, s ∈ S � ❅ � � ❅ ❅ σ = � ❅ � ❅ ❅ � � ❅ � ❅ ❅ � � j c 2 (1) jδp j ✫ ✪ ✫ ✪ ✫ ✪ ✫ ✪ Realization of P σ , Spelling out s ∈ S , Solitons 6

reflectionless potential and n -soliton � � � σ = � n � j =1 c 2 Σ0= jδp j � n ∈ N , pj ∈ R , pj � = pi, cj > 0 • σ ∈ Σ0 . { b ( x ) } x ≥ 0 ; an n-dim B.m.on (Ω , F , P ) � x ξσ ( x ) = exD σ e − yD σ db ( y ) ( Dσ = diag [ pj ]) 0 Xσ ( x ) = � c , ξσ ( x ) � = � n i =1 ciξi σ ( x ) ( c=( ci ) ) P σ = P ◦ X − 1 σ ( Xσ : Ω → W ) • σ ∈ Σ0 ; ∃ m < n, 1 ≤ j (1) < · · · < j ( m ) ≤ n s.t. | pj | ≤ | pj +1 | , pj ( ℓ ) > 0 , pj ( ℓ )+1 = − pj ( ℓ ) # {| p 1 | , . . . , | pn |} = n − m . 7

r r r r p 2 ψ : Σ0 ∋ σ �→ { ηj, mj } ∈ S ✉ ✉ ✉ ✉ n p 2 1 − 1 { η 1 < · · · < ηn } = { pj (1) ,. . ., pj ( m ) , √ r 1 ,. . ., √ rn − m } ( 0 < r 1 < · · · < rn − m : � n j =1 c 2 j/ ( p 2 j − r ) = − 1 )  c 2   � �  ηk + ηi pk + ηi j ( ℓ )+1   2 ηi ,   c 2  ηk − ηi pk − ηi   j ( ℓ ) k � = i k � = j ( ℓ ) ,j ( ℓ )+1  mi = if i = j ( ℓ ) ,    n  � �  ηk + ηi pk + ηi    − 2 ηi , otherwise.   ηk − ηi pk − ηi  k � = i k =1 8

Thm 1. Let P σ ∈ G 0 = { P σ | σ ∈ Σ0 } . Then � x � � � − 1 X ( y )2 dy dP σ 4 log exp 2 0 W � � = − 2 log det I + Gψ ( σ )( x ) n � � � + 2 log det I + Gψ ( σ )(0) − 2 x ( pi + ηi ) . i =1 Moreover, ψ : G 0 → Ξ0 and ψ ( P σ ) = uψ ( σ ) . Finally, ψ : G 0 → Ξ0 is bijective. � • ψ ( P σ )( x ) = 4 d 2 exp( · · · ) dP σ dx 2 log W • ψ ( P σ ) = uψ ( σ ) on [0 , ∞ ) ; “ ψ ( G 0) ⊂ Ξ0 ” The real analyticity does the rest of job 9

Cor. (i) If µ ( A ) = σ ( − A ) , then uψ ( σ )( x ) = ψ ( P µ )( − x ) for x ∈ ( −∞ , 0] . (ii) For y ≤ 0 , let b ( y ) = b ( − y ) , and � 0 ξσ ( y ) = − eyD σ e − zD σ db ( z ) y Xσ ( y ) = � c , ξσ ( y ) � . Then u = ψ ( P σ ) is represented as � � 0 ∨ x � � u ( x ) = 4 d 2 − 1 Xσ ( y )2 dy dx 2 log exp dP 2 Ω 0 ∧ x for every x ∈ R . 10

• The τ -function of the KdV hierarchy is τ ( x,� t ) = det( I + A ( x,� t )) t = ( tj ) ∈ RN with # { tj � = 0 } < ∞ , where x ∈ R , � �� mimj � e −{ ζ i ( x,� t )+ ζ j ( x,� t ) } A ( x,� t ) = , ηi + ηj 1 ≤ i,j ≤ n ∞ � tαη 2 α +1 { ηj, mj } ∈ S , ζi = ζi ( x,� . t ) = xηi + i α =1 t = ( t, 0 , . . . ) , then v ( x, t ) = 2 ∂ 2 • If � x log τ ( x,� t ) solves the KdV eq; vt = 3 2 vvx + 1 4 vxxx . 11

• For σ ∈ Σ0 , let ψ ( σ ) = { ηj, mj } ∈ Ξ0 . Define � ( ∂xφ ) φ − 1 � (0 ,� t ) , ζ = diag [ ζj ] , β� t = − � � cosh( ζ ) − sinh( ζ ) R − 1 U − 1 DσU U − 1 , φ ( x,� t ) = U σ + c ⊗ c = UR 2 U − 1 ( R = diag [ ηj ] ), U ∈ O ( n ) ; D 2 � � x � − 1 Xσ ( y )2 dy Iσ ( x,� t ) = exp 2 Ω 0 � +1 2 � ( β� t − Dσ ) ξσ ( x ) , ξσ ( x ) � dP. � � = − 1 Iσ ( x,� 2 log τ ( x,� Thm 2 (i) log t ) t ) � n +1 t ) − x 2 log τ (0 ,� i =1( pi + ηi ) 2 � � t )= − 4 ∂ 2 (ii) If � t = ( t, 0 , . . . ) , then vσ ( x,� x log Iσ is an n -soliton of the KdV eq. (Super pos) 12

Change of variables formulae on W ; Prop. Let φ ( y ) ∈ R n × n be a sol of φ ′′ − Eσφ = 0 , where Eσ = D 2 σ + c ⊗ c . Let x > 0 and assume (A.1) det φ ( y ) � = 0 , (A.2) β ( y ) = − ( φ ′ φ − 1)( y ) is symm (0 ≤ y ≤ x ) . Then � � x � − 1 Xσ ( y )2 dy exp 2 Ω 0 � +1 2 � ( β (0) − Dσ ) ξσ ( x ) , ξσ ( x ) � dP � � 1 / 2 � � − 1 / 2 . ex tr D σ det φ ( x ) = det φ (0) 13

γ : [0 , x ] → R n × n • OU pr � H.Osc; � x � � − 1 Xσ ( y )2 dy exp 2 Ω 0 � +1 2 � ( γ ( x ) − Dσ ) ξσ ( x ) , ξσ ( x ) � dP � � x � = e − tr D σ / 2 − 1 exp � Eσb ( y ) , b ( y ) � dy 2 Ω 0 � +1 2 � γ ( x ) b ( x ) , b ( x ) � dP • (C-M) Itˆ o ⊕ Girsanov; � x � � − 1 � ( γ 2 + γ ′ ) b ( y ) , b ( y ) � dy exp 2 Ω 0 � x � � 1 � +1 0 tr γ 2 � γ ( x ) b ( x ) , b ( x ) � dP = exp 2 γ 2 + γ ′ = Eσ , • γ ( x ) = β (0) − Dσ Cole-Hopf; γ ( y )= − ( φ ′ φ − 1)( x − y ) ⇒ φ ′′ − Eσφ = 0 14

Pf of Thm 1: φ ′′− Eσφ =0 , φ (0)= I, φ ′ (0)= − Dσ ; φ ( y ) = cosh( yE 1 / 2 ) − E − 1 / 2 sinh( yE 1 / 2 ) Dσ σ σ σ (Case1) | pi | < | pi +1 | , i = 1 , . . . , n − 1 . � � φ ( y )= − 1 2 UV R − 1 B eyRB − 1 XC I + Gψ ( σ )( y ) � σ − rjI ) − 1 c |− 1 � , | ( D 2 V = diag � � n � , R = diag [ ηj ] , a ( i ) = sgn β =1( pβ − ηi ) � � � 1 / 2 , α � = i ( η 2 α − η 2 i ) � n b ( i ) = a ( i ) − 2 ηi β =1 ( p 2 β − η 2 i ) � � − 1 . B = diag [ b ( j )] , Xij = pj + ηi j = pj − ε � m (Case2) pε i =1 δj,ℓ ( i )+1 , ε → 0 . 15

Pf of Thm 2: φ ( y ) = φ ( y,� t ) ; (A.1),(A.2) are fulfilled (Case1) | pi | < | pi +1 | , i = 1 , . . . , n − 1 . t ) } eζ ( y,� t ) = − 1 2 UR − 1 V B { I + A ( y,� t ) B − 1 XC φ ( y,� j = pj − ε � m (Case2) pε i =1 δj,ℓ ( i )+1 , ε → 0 . 16

Bijectivity & Cor • Let u = u s ∈ Ξ0 (s ∈ S ) , and e +( x ; ζ ) be the right Jost sol of L = − ( d/dx )2 + u s ; L e +( ∗ ; ζ ) = ζ 2 e +( ∗ ; ζ ) , e +( x ; ζ ) ∼ eiζx ( x → ∞ ) Then ∃ λj ∈ C ∞ ( R ; R ) , 1 ≤ j ≤ n, s.t. ζ −√− 1 λ j ( x ) √− 1 ζx � e +( x ; ζ )= e ζ + √− 1 η j . j Define κ : Ξ0 → Σ0 by κ (s) = � j ( − λ ′ j (0)) δλ j (0) Then ψ ( κ (s)) = s , κ ( ψ ( σ )) = σ. • Let � u ( x ) = u s( − x ) . Then � s ( κ ( � s) = µ ) u = u � 17

generalized reflectionless potentials • u ∈ Ξ ⇔ ∃ µ > 0 , un ∈ Ξ0 s.t. � � − d 2 Spec ⊂ [ − µ, ∞ ) , n = 1 , 2 , . . . dx 2 + un un → u (unif on cpts) � � x � � − 1 X ( y )2 dy dP σ • Φ σ ( x ) = exp 2 0 W ψ ( P σ ) = 4 d 2 dx 2 log Φ σ G ⊃ G 0 ∋ P σ �→ ψ ( P σ ) ∈ Ξ0 ⊂ Ξ : bijective Question: “ P σ n → P σ ” � “ un → u ”, ψ ( G ) � Ξ 18