Integrable evolution equations on spaces of tensor densities

Transcription

1 Integrable evolution equations on spaces of tensor densities J. Lenells, G. Misiolek and F. Tiglay* April 11, 21 Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 1/22

2 A family of equations on λ densities µdp Lax pair and bihamiltonian structure Cauchy problem µburgers Lax pair and bihamiltonian structure µburgers equation and the L 2 -geometry of Diff s Diff s /S 1 Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 2/22

3 Two integrable equations u txx 3u x u xx uu xxx =, µ(u t ) u txx + 3µ(u)u x 3u x u xx uu xxx =. (µb) (µdp) u(t, x) is a spatially periodic real-valued function of a time variable t and a space variable x S 1 [, 1), µ(u) = 1 u dx. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 3/22

4 V. Arnold s approach G Lie group configuration space g its Lie algebra with an inner product, kinetic energy, equippes G with a right invariant metric and the motions of the system can be studied through: geodesic equations defined by the right invariant metric, or Hamiltonian reduction on the Lie algebra g. (, ) a natural pairing between g and g A : g g the associated inertia operator s.t. (Au, v) = u, v. The Euler equation on g : m t = ad A 1 m m, m = Au g, (E) Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 4/22

5 Diff(S 1 ) and density modules quadratic case G := Diff(S 1 ), g = vect(s 1 ) regular part of the dual g r F 2 = { m(x)dx 2 : m C (S 1 ) } with the pairing ( mdx 2 ) 1, v x = m(x)v(x) dx the coadjoint representation of the action of vect(s 1 ) vect(s 1 ) on the regular part on the space of of its dual space quadratic differentials ad u x mdx 2 = (um x + 2u x m) dx 2 and the Euler equation (E) on g r is m t = ad A 1 m m = um x 2u x m, m = Au. (1) Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 5/22

6 On g r : m t = ad A 1 m m = um x 2u x m, m = Au. (2) On vect(s 1 ): Au t + 2u x Au + uau x = (3) with inertia operator 1 x 2 for CH, A = µ x for µch, (4) for HS. 2 x Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 6/22

7 Diff(S 1 ) and density modules general set-up replace F 2 by F λ a tensor density of weight λ (respectively λ < ) on S 1 is a section of the bundle λ T S 1 (respectively λ TS 1 ) F λ = { m(x)dx λ : m(x) C (S 1 ) }. action of Diff(S 1 ) on each density module F λ is given by F λ mdx λ m ξ ( x ξ) λ dx λ F λ, ξ Diff(S 1 ), (5) which generalizes Ad : Diff(S 1 ) Aut ( F 2 ) Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 7/22

8 the infinitesimal generator of the action in (5) L λ u x (mdx λ ) = (um x + λu x m) dx λ (6) determines the action of vect(s 1 ) on F λ the equation for the flow of the vector field defined by (6) is substituting m = Au transforms (7) into m t = um x λu x m (7) Au t + λu x Au + uau x = (8) for λ = 3 the inertia operator is 1 x 2 for DP, A = µ x 2 for µdp, for µb. 2 x (9) Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 8/22

9 Lax pair and bihamiltonian structure µdp Lax pair { ψ xxx = λmψ, ψ t = 1 λ ψ xx uψ x + u x ψ, (1) λ C is a spectral parameter, ψ(t, x) is a scalar eigenfunction and for m = µ(u) u xx µ(u t ) u txx + 3µ(u)u x 3u x u xx uu xxx =. (µdp) Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 9/22

10 Lax pair and bihamiltonian structure µdp Bihamiltonian structure δh m t = J δm = J 2 with Hamiltonian functionals H = 9 m dx and H 2 = 2 the operators J and J 2 are given by δh 2 δm, ( 3 2 µ(u)( A 1 x u ) ) u3 dx J = m 2/3 x m 1/3 3 x m 1/3 x m 2/3 and J 2 = 3 x A = 5 x and m = µ(u) u xx. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 1/22

11 Cauchy problem µdp Periodic Cauchy problem ( x Λ 2 2 µ v(x) = 2 x x y ) 1 v(z) dzdy + ( v(x) dx + x x y is the inverse of the elliptic operator ) 1 v(z) dzdydx. Λ 2 µ : H s (S 1 ) H s 2 (S 1 ), Λ 2 µv = µ(v) v xx. x v(y) dydx We use Λ 2 µ to rewrite µdp in the nonlocal form u t + uu x + 3µ(u) x Λ 2 µ u = Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 11/22

12 Cauchy problem µdp Local wellposedness and persistence u t + uu x + 3µ(u) x Λ 2 µ u = (11) u(, x) = u (x) (12) Theorem (Local wellposedness and persistence) Assume s > 3/2. Then for any u H s (T) there exists a T > and a unique solution u C ( ( T, T ), H s) C 1( ( T, T ), H s 1) of the Cauchy problem (11)-(12) which depends continuously on the initial data u. Furthermore, the solution persists as long as u(t, ) C 1 stays bounded. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 12/22

13 Cauchy problem µdp Blow-up Theorem Given any smooth periodic function u with zero mean there exists T c > such that the corresponding solution of the µdp equation stays bounded for t < T c and satisfies u x (t) as t T c. Proof. ξ = u ξ implies x ξ = (u x ξ) x ξ and setting w = x ξ/ x ξ we find that w(t, x) = 1 t + ( 1/u x (x) ). Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 13/22

14 Cauchy problem µdp Global existence Theorem Let s > 3. Assume that u H s (S 1 ) has non-zero mean and satisfies the condition Λ 2 µu (or ). Then the Cauchy problem for µdp has a unique global solution u in C(R, H s (S 1 )) C 1 (R, H s 1 (S 1 )). Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 14/22

15 Lax pair and bihamiltonian structure µburgers Lax pair { ψ xxx = λmψ, ψ t = 1 λ ψ xx uψ x + u x ψ, (13) λ C is a spectral parameter, ψ(t, x) is a scalar eigenfunction and for m = u xx u txx 3u x u xx uu xxx =. (µb) Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 15/22

16 Lax pair and bihamiltonian structure µburgers Bihamiltonian structure δh m t = J δm = J δh 2 2 δm, with Hamiltonian functionals H = 9 m dx and H 2 = u 3 dx the operators J and J 2 are given by J = m 2/3 x m 1/3 3 x m 1/3 x m 2/3 and J 2 = 5 x and m = u xx. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 16/22

17 µburgers equation and the L 2 -geometry of Diff s Diff s /S 1 Burgers equation u t + uu x = (B) Diff s (S 1 ) circle diffeomorphisms of Sobolev class H s L 2 inner product on T η Diff s (S 1 ) induces a weak Riemannian metric on Diff s (S 1 ) a geodesics η(t) in Diff s (S 1 ) satisfy the equation η η = η = ( u t + uu x ) η = (14) and hence correspond to (classical) solutions of the Burgers equation. Here η(t) is the flow of u(t, x), i.e. η(t, x) = u(t, η(t, x)) Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 17/22

18 µburgers equation and the L 2 -geometry of Diff s Diff s /S 1 L 2 -geometry of Diff s Diff s /S 1 homogeneous space Diff s = Diff s /S 1 is a smooth Hilbert manifold for s > 3/2 with T [e] Diff s = H s (S 1 ). π : Diff s Diff s is a Riemannian submersion with each tangent space decomposing as T ξ Diff s = P ξ (T ξ Diff s ) L 2 Q ξ (T ξ Diff s ). the two orthogonal projections P ξ : T ξ Diff s T π(ξ) Diff s and Q ξ : T ξ Diff s R are given explicitly by the formulas P ξ (W ) = W 1 W (x) dx and Q ξ(w ) = 1 W (x) dx. a curve η(t) in Diff s (S 1 ) is an L 2 geodesic (and hence correspond to a solution of Burgers equation) P η η η = Q η η η =. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 18/22

19 µburgers equation and the L 2 -geometry of Diff s Diff s /S 1 Theorem A smooth function u = u(t, x) is a solution of the µb equation u txx + 3u x u xx + uu xxx = if and only if the horizontal component of the acceleration of the associated flow η(t) in Diff s (S 1 ) is zero i.e. P η η η =. In fact, given any u H s (S 1 ) the flow of u has the form η(t, x) = x + t ( u (x) u () ) + η(t, ) for all sufficiently small t. integrating the µb equation twice in x gives u t + uu x = µ(u t ), (15) integrating η(t, x) = 1 η η 1 (t, x) dx = η(t, ) twice in t gives the explicit formula for the flow. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 19/22

20 µburgers equation and the L 2 -geometry of Diff s Diff s /S 1 Corollary Suppose that u(t, x) is a smooth solution of the µb equation and let u(, x) = u (x). 1 The following integrals are conserved by the flow of u 1 ( ) 1 pdx ( u µ(u) = u µ(u ) ) p dx, p = 1, 2, There exists T c > such that u x (t) as t T c. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 2/22

21 Peakons µ(u t ) u txx + λµ(u)u x = λu x u xx + uu xxx, λ Z (16) Theorem λ = 2 = µch and λ = 3 = µdp. For any c R and λ, 1, equation (16) admits the peaked period-one traveling-wave solution u(t, x) = ϕ(x ct) where ϕ(x) = c 26 (12x ) (17) for x [ 1 2, 1 2 ] and ϕ is extended periodically to the real line. one-peakon solutions of (16) are the same for any λ, they travel with a speed equal to their height. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 21/22

22 Multi-peakons m(t, x) = N p i (t)δ(x q i (t)), (18) i=1 Theorem The multi-peakon (18) satisfies the µ-equation (16) in the nonlocal form in distributional sense if and only if {q i, p i } N 1 evolve according to q i = u(q i ), ṗ i = (λ 1)p i {u x (q i )} (19) where {u x (q i )} denotes the regularized value of u x at q i defined by {u x (q i )} := N j=1 p jg (q i q j ) and the Green function is given by g(x) = x(x 1) + 12 for x [, 1) S 1. Lenells, Misiolek, Tiglay* AMS Meeting, Minnesota, April 11, 21 slide 22/22