IBVPs for linear and integrable nonlinear evolution PDEs

Transcription

1 IBVPs for linear and integrable nonlinear evolution PDEs Dionyssis Mantzavinos Department of Applied Mathematics and Theoretical Physics, University of Cambridge. Edinburgh, May 31, 212. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

2 Main advantages of the new method Main advantages of the new method i. Unification of classical approaches. ii. Main expressions hold at the level of general boundary conditions. iii. Uniform convergence large t-asymptotics. iv. Non-separable boundary conditions. v. Integrable nonlinear PDEs. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

3 Main advantages of the new method Example: Heat equation on the half-line, with a Dirichlet b.c. Solution via sine transform: u t = u xx, < x <, t >, u(x, ) = u (x) u(x, t) = 2 π u(, t) = D(t). [ t ] sin(kx)e k2 t u(k) s + k e k2s D(s)ds dk, ( ) where u s (k) = sin(kx)u(x, t)dx, k R. Not uniformly convergent at the boundary. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

4 Main advantages of the new method Higher order linear equations? E.g. or even u t = u xxx u t = u xx + βu x, β >. There does not exist a classical x-transform! More complicated boundary conditions? E.g. oblique Robin b.c. u t (, t) + αu x (, t) + βu(, t) = γ(t). Separation of variables fails. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

5 Main advantages of the new method Alternative classical techniques: Laplace transform in t: u L (x, r) = e rt u(x, t)dt. Integral up to (not natural for evolution PDEs). Indicial equation complicated for higher order problems. Method of images: need domain with certain symmetries. Green s functions representations: not uniformly convergent. None of these methods apply to nonlinear PDEs. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

6 The new method Main advantages of the new method Based on the simultaneous spectral analysis of a relevant Lax pair. For linear equations, shortcut throught divergence form. Main components: Spectral functions: certain integral transforms of the initial condition and of known & unknown boundary values. Global relation: an identity relating an integral transform of the solution to the spectral functions. Integral representation: an expression for the solution, depending on the initial condition and the boundary values exclusively through the spectral functions. Gives uniformly convergent and numerically effective solution formulae. Deals with non-separable b.c., higher order PDEs and integrable nonlinear PDEs. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

7 Non-separable boundary conditions I. Heat equation on the half-line u t = u xx, < x <, t >, u(x, ) = u (x). Divergence form: ( [ ] e ikx+k2t u )t = e ikx+k2t (u x + ik u). x Fourier transform û(k, t) = and integrating gives the global relation: e ikx u(x, t)dx e k2tû(k, t) = û (k) g 1 (k 2 ) ik g (k 2 ), k C, where g (k 2 ) = t e k2s u(, s)ds, g 1 (k 2 ) = t e k2s u x (, s)ds. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

8 Non-separable boundary conditions Inverting: u(x, t) = 1 2π e ikx k2tû (k)dk 1 e ikx k2 t [ g 1 (k 2 ) + ikg (k 2 ) ] dk. 2π Cauchy & Jordan for k C + \ D + imply the integral representation: u(x, t) = 1 e ikx k2tû (k)dk 1 e ikx k 2 t [ g 1 (k 2 ) + ikg (k 2 ) ] dk 2π 2π D + where D + = { k C + : Re ( k 2) }. k D + π 4 Spectral functions g (k 2 ) and g 1 (k 2 ) are unknown. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

9 Oblique Robin b.c. Non-separable boundary conditions T (t) A l A s liquid r(t) x solid rod u(x, t) Infinite length temperature uniform across cross-section A s : u = u(x, t). Point of contact: u(, t) = r(t). Heat flow: dr(t) + αu x (, t) + βu(, t) = γ(t), dt where α, β depend on the specific heat, mass, conductivity, A l and A s, and γ also depends on T (t). Thus, obtain the oblique Robin boundary condition: Non-separable for general α, β, γ. u t (, t) + αu x (, t) + βu(, t) = γ(t). Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

10 Non-separable boundary conditions It follows that e k2t u(, t) + αg 1 (k 2 ) + βg (k 2 ) = δ(k 2 ), where δ(k 2 ) = t Symmetry of the spectral functions: k k. Global relation becomes: e k2s γ(s)ds + u (). ( ) e k2tû( k, t) = û ( k) g 1 (k 2 ) + ikg (k 2 ), k C +. ( ) Combine ( ) and ( ) to eliminate g and g 1 : u(x, t) = 1 2π + 1 2π + 1 2π e ikx k2tû (k)dk D + e ikx k D + e ikx k 2 t k 2 + iαk β k 2 iαk β 2 t 2ik k 2 iαk β [ e k2tû( k, ] t) û ( k) dk [ ] δ(k 2 ) e k2t u(, t) dk Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

11 Non-separable boundary conditions Cauchy & Jordan in D + imply solution: u(x, t) = 1 2π + 1 2π e ikx k2tû (k)dk D + e ikx k 2 t k 2 + iαk β [ δ(k 2 k 2 ) û ( k) ] dk iαk β + 2k j 2 2k j iα eik j x kj t [ δ(kj 2 ) α û ( k j ) ], where k 2 j iαk j β =, k j D +. k j D + k π 4 Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

12 Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24 Return to the real line Non-separable boundary conditions Simplified version: α = 1, β = γ = u(x, t) = 1 2π e ikx k2tû (k)dk + 1 2π + 2e x+t [u () û ( i)], D + e ikx k 2 t k + i k i [u () û ( k)] dk D + π 4 *i k Deforming D + to the real line: u(x, t) = 2 [ e k2 t cos kx + k sin kx ] π k 2 (cos kξ + k sin kξ) u (ξ)dξ u () dk + 1 ] + 2e [u x+t () e ξ u (ξ)dξ [D.S. Cohen (1965)].

13 Non-separable boundary conditions II. Heat equation on the interval [, l] Oblique Robin b.c. Supplement condition at x = with condition at x = l: u t (l, t) + Au x (l, t) + Bu(l, t) = Γ(t). Solution in terms of integrals along complex contours D + and D. Returning to the real line now gives sum over discrete spectrum: { u(x, t) = 2e x+t e l e l e l[ ] } û ( i) u () + u (l) e ik nx k 2 n t [ + 2ie iknl u (l) + (k n i) û (k n ) 2l(k n i) n= n 1 ] (k n + i) û ( k n ) + 2iu (), where k n = (n + 1) π, n Z. l Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

14 Non-separable boundary conditions Non-local integral constraints and l l u(x, t)dx = F(t) (l x) u(x, t)dx = R(t), where F and R are given functions of time. Returning to the real line, we obtain the solution as a sum over k m such that (2i + lk m ) e ik ml + (2i lk m ) e ik ml 4i =. Spectrum satisfies Weyl s asymptotic law: lim m k 2 m l 2 m 2 π 2 = 1. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

15 Integrable nonlinear PDEs Initial value problems III. Integrable nonlinear PDEs Initial value problems in 1+1: via Inverse Scattering Transform (1967). q (x) direct scattering [ t independent part of Lax pair ] ˆq (k) time evolution [ t dependent part of Lax pair ] q(x, t) inverse scattering [ GLM equation or R-H problem ] ˆq(k, t) Initial value problems in 2+1: extension of the IST by Ablowitz & Fokas in 1983 [d-bar problem and non-local R-H problem]. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

16 Integrable nonlinear PDEs Initial-boundary value problems Initial-boundary value problems in 1+1: the new method via R-H formalism (1997 ). Initial-boundary value problems in 2+1: the new method via d-bar formalism (29 ). NLS [B. de Monvel, Fokas, Shepelsky (23)] iq t + q xx + λ q 2 q = KdV [Fokas (22)] q t + 6qq x + q xxx = DS [Fokas (29)] KP I & II [M & Fokas (211)] iq t + q zz + q z z + 4(f + f )q = 2f z ( q 2 ) z = q t + 6qq x + q xxx ±3 1 x q yy = Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

17 Integrable nonlinear PDEs Initial-boundary value problems The new method for integrable nonlinear IBVPs Direct problem: simultaneous spectral analysis Lax pair integral representation k global relation elimination of unknowns solution µ(x, t, k, k) in terms of q(x, t) and i.c./ b.v. Inverse problem: Riemann-Hilbert/ d-bar formalism µ(x, t, k, k) in terms of spectral functions Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

18 Integrable nonlinear PDEs Initial-boundary value problems The d-bar problem d-bar derivative: f = f (k, k), k = k R + ik I, k R, k I R, f k = 1 ( f + i f ). [departure from analyticity] 2 k R k I f k f not analytic in k. Theorem (Pompeiu): Let f (k, k) be a smooth function in some piece-wise smooth domain D R 2. Then, f (k, k) is related to its value on the boundary of D and to its d-bar derivative inside D via the equation: f (k, k) = 1 2iπ D dν 1 f (ν, ν) + ν k 2iπ D dν d ν ν k where ν = ν R + iν I, ν R, ν I R, dν d ν = 2i dν R dν I. f (ν, ν), ν Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

19 Integrable nonlinear PDEs The KPII The Kadomtsev-Petviashvili II on the half-plane where q t + 6qq x + q xxx x q yy =, < x <, < y <, t >, x x 1 q yy (x, y, t) := q yy (ξ, y, t)dξ. Lax pair for µ(x, y, t, k, k) sectionally non-analytic and bounded k C: µ y µ xx 2ikµ x = qµ µ t + 4µ xxx + 12ikµ xx 12k 2 µ x = F µ, where F (x, y, t, k) = 6q ( x + ik) 3 ( q x + 1 x q y ). Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

20 Integrable nonlinear PDEs The KPII - Direct problem Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24 Direct problem k I k Simultaneous spectral analysis of Lax pair: µ + 1 µ + 2 k R µ ± j (x, y, t, k, k) = J ± j µ 1 µ 2 [ e il(ξ x)+l(l+2k)(η y) q(ξ, η, t)µ ± j (ξ, η, t, k, k) ] + L ± j where and [ ] e il(ξ x) l(l+2k)y 4il(l 2 +3kl+3k 2 )(s t) R(ξ, s, k, l)µ ± j (ξ,, s, k, k), j = 1, 2, [ ] R(x, t, k, l) = 3 i(l + 2k)q(x,, t) + x 1 q y (x,, t) J ± 1 := 1 Z [( Z Z )Z y dξ dl + dl dη 2π 2k R ( L ± 1 := 1 Z Z )Z dl + dl 2π 2k R 8 < dξ : Z 2kR R t ds R T t dl ds Z y. ] dη,

21 Integrable nonlinear PDEs The KPII - Global relation Global relation Green s theorem on Lax pair global relation for all µ: = dx dx dx T valid for l(l + 2k R ) (boundedness). dy e ilx+l(l+2k)y 4il(l 2 +3kl+3k 2 )T q(x, y, T )µ(x, y, T, k, k) dy e ilx+l(l+2k)y q (x, y)µ(x, y,, k, k) dt e ilx 4il(l 2 +3kl+3k 2 )t R(x, t, k, l)µ(x,, t, k, k), Remark: µ is adjoined to q, q, q(x,, t), q y (x,, t). Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

22 Integrable nonlinear PDEs The KPII - Inverse problem Inverse problem ν I ν µ + 1 µ + 2 ν R µ 1 µ 2 Reconstruct µ in the complex k-plane by using Pompeiu s theorem: µ(k, k) = 1 dν 1 dν d ν µ µ(ν, ν) + (ν, ν), 2iπ ν k 2iπ ν k ν Need to compute: ( µ ± 1 µ± 2 L ) ki (x, y, t, k, k), ( µ + 1,2 µ 1,2 = R 2 ) kr (x, y, t, k, k), = µ ± 1,2 (x, y, t, k, k). k Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

23 Integrable nonlinear PDEs Conclusions & Open problems Conclusions & Open problems For a well-posed problem, we know q and either q(x,, t) or q y (x,, t). For the nonlinear KPII, the elimination of the spectral function corresponding to unknown boundary value is still open: µ is adjoined to the boundary values not yet clear how to use symmetries of the global relation in this case. Linearisable boundary conditions: can directly eliminate unknown spectral functions using the global relation, as in the linear problem. The KPI and the KPII on the half-plane ( < x <, < y < ) remain open. PDEs in 2+1 (e.g. DS, KPI, KPII) on more complicated domains, e.g. quarter-plane, square, triangle etc. expected to be much harder. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

24 Thank you for your attention! The work presented was supported by an EPSRC Doctoral Training Grant. Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24

25 Further reading Further reading A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. Ser. A 453 (1997). A.S. Fokas, A.R. Its and L.Y. Sung, The Nonlinear Schrödinger Equation on the Half-Line, Nonlinearity 18, (25). J.L. Bona and A.S. Fokas, Initial-Boundary-Value Problems for Linear and Integrable Nonlinear Dispersive PDEs, Nonlinearity 21, T195-T23 (28). A.S Fokas and J. Lenells, The Unified Method: I Non-Linearizable Problems on the Half-Line, arxiv: v1 (211). A.S. Fokas, The Davey-Stewartson equation on the half-plane, Comm. Math. Phys. 289 (29). D. Mantzavinos and A.S. Fokas, The KPII on the half-plane, Physica D 24, (211). Dionyssis Mantzavinos (University of Cambridge) Edinburgh, May 31, / 24