Dispersive Quantization of Linear and Nonlinear Waves Peter J. - PowerPoint PPT Presentation

Dispersive Quantization of Linear and Nonlinear Waves Peter J. Olver University of Minnesota http://www.math.umn.edu/ ∼ olver ICMAT — July, 2017

Happy 70 th , Darryl!!!

Peter J. Olver Introduction to Partial Di ff erential Equations Undergraduate Texts, Springer, 2014 —, Dispersive quantization, Amer. Math. Monthly 117 (2010) 599–610. Gong Chen & —, Dispersion of discontinuous periodic waves, Proc. Roy. Soc. London A 469 (2012), 20120407. Gong Chen & —, Numerical simulation of nonlinear dispersive quantization, Discrete Cont. Dyn. Syst. A 34 (2013), 991–1008.

Dispersion A linear partial di ff erential equation is called Definition. dispersive if the di ff erent Fourier modes travel unaltered but at di ff erent speeds. Substituting u ( t, x ) = e i ( k x − ω t ) produces the dispersion relation ω = ω ( k ) , ω , k ∈ R relating frequency ω and wave number k . c p = ω ( k ) Phase velocity: k c g = d ω Group velocity: (stationary phase) dk

A Simple Linear Dispersive Wave Equation: ∂ t = ∂ 3 u ∂ u ∂ x 3 = linearized Korteweg–deVries equation ⇒ ω = k 3 Dispersion relation: c p = ω k = k 2 Phase velocity: c g = d ω dk = 3 k 2 Group velocity: Thus, wave packets (and energy) move faster (to the right) than the individual waves.

Linear Dispersion on the Line ∂ t = ∂ 3 u ∂ u u (0 , x ) = f ( x ) ∂ x 3 Fourier transform solution: � ∞ 1 f ( k ) e i ( k x − k 3 t ) dk � u ( t, x ) = √ 2 π −∞ Fundamental solution u (0 , x ) = δ ( x ) � � � ∞ u ( t, x ) = 1 1 x −∞ e i ( k x − k 3 t ) dk = 3 t Ai √ √ − 2 π 3 3 3 t

t = . 03 t = . 1 t = 1 / 3 t = 1 t = 5 t = 20

Linear Dispersion on the Line ∂ t = ∂ 3 u ∂ u u (0 , x ) = f ( x ) ∂ x 3 Superposition solution formula: � ξ − x 1 � ∞ � u ( t, x ) = −∞ f ( ξ ) Ai d ξ √ √ 3 3 3 t 3 t � 0 , x < 0 , Step function initial data: u (0 , x ) = σ ( x ) = 1 , x > 0 . u ( t, x ) = 1 � � x 3 − H √ − 3 3 t � 2 � 1 � 2 � 2 z 2 Γ � 9 z 3 � � 9 z 3 � 3 ; 2 3 , 4 3 ; 1 3 ; 4 3 , 5 3 ; 1 z Γ 1 F 2 1 F 2 3 3 H ( z ) = � 2 � 4 − 3 5 / 3 Γ � � 3 7 / 3 Γ � 4 � � 5 � Γ Γ 3 3 3 3 = — via Meijer G functions Mathematica ⇒

t = . 005 t = . 01 t = . 05 t = . 1 t = . 5 t = 1 .

Periodic Linear Dispersion ∂ t = ∂ 3 u ∂ u ∂ x 3 ∂ 2 u ∂ x 2 ( t, − π ) = ∂ 2 u ∂ u ∂ x ( t, − π ) = ∂ u u ( t, − π ) = u ( t, π ) ∂ x ( t, π ) ∂ x 2 ( t, π ) Step function initial data: � 0 , x < 0 , u (0 , x ) = σ ( x ) = 1 , x > 0 . Fourier series solution formula: sin( (2 j + 1) x − (2 j + 1) 3 t ) u ⋆ ( t, x ) ∼ 1 2 + 2 ∞ � . 2 j + 1 π j =0

Periodic linearized KdV — irrational times t = 0 . t = . 1 t = . 2 t = . 3 t = . 4 t = . 5

Periodic linearized KdV — rational times t = 1 t = 1 t = 1 30 π 15 π 10 π t = 2 t = 1 t = 1 15 π 6 π 5 π

t = 1 t = 1 t = π 2 π 3 π t = 1 t = 1 t = 1 4 π 5 π 6 π t = 1 t = 1 t = 1 7 π 8 π 9 π

At rational time t = 2 π p/q , the solution u ⋆ ( t, x ) is Theorem. constant on every subinterval 2 π j/q < x < 2 π ( j + 1) /q . At irrational time u ⋆ ( t, x ) is a non-di ff erentiable continuous fractal function.

Lemma. � ∞ c k e i k x f ( x ) ∼ k = −∞ is piecewise constant on intervals 2 π j/q < x < 2 π ( j + 1) /q if and only if c k = � c l , k ≡ l ̸≡ 0 mod q, c k = 0 , 0 ̸ = k ≡ 0 mod q. � � where 2 π k c k c k = k ̸≡ 0 mod q. � i q ( e − 2 i π k/q − 1) = DFT ⇒

The Fourier coe ffi cients of the solution u ⋆ ( t, x ) at rational time t = 2 π p/q are c k = b k e − 2 π i k 3 p/q ( ∗ ) where, for the step function initial data,  − i / ( π k ) , k odd,    b k = 1 / 2 , k = 0 ,   0 , 0 ̸ = k even.  Crucial observation: if k ≡ l mod q then k 3 ≡ l 3 mod q which implies e − 2 π i k 3 p/q = e − 2 π i l 3 p/q and hence the Fourier coe ffi cients ( ∗ ) satisfy the condition in the Lemma. Q.E.D.

The Fundamental Solution: F (0 , x ) = δ ( x ) At rational time t = 2 π p/q , the fundamental Theorem. solution F ( t, x ) is a linear combination of finitely many periodically extended delta functions, based at 2 π j/q for integers − 1 2 q < j ≤ 1 2 q . At rational time, any solution profile u (2 π p/q, x ) Corollary. to the periodic initial-boundary value problem is a linear combination of ≤ q translates of the initial data, namely f ( x + 2 π j/q ), and hence its value depends on only finitely many values of the initial data.

The same quantization/fractalization phenomenon ⋆ ⋆ appears in any linearly dispersive equation with “integral polynomial” dispersion relation: n � ω ( k ) = c m k m m =0 where c m = α n m n m ∈ Z

Linear Free-Space Schr¨ odinger Equation ∂ t = − ∂ 2 u i ∂ u ∂ x 2 ω = k 2 Dispersion relation: c p = ω Phase velocity: k = k c g = d ω Group velocity: dk = 2 k

The Talbot E ff ect ∂ t = − ∂ 2 u i ∂ u ∂ x 2 ∂ u ∂ x ( t, − π ) = ∂ u u ( t, − π ) = u ( t, π ) ∂ x ( t, π ) Michael Berry et. al. • Oskolkov • Kapitanski, Rodnianski • “Does a quantum particle know the time?” Michael Taylor • Bernd Thaller, Visual Quantum Mechanics •

William Henry Fox Talbot (1800–1877)

⋆ Talbot’s 1835 image of a latticed window in Lacock Abbey = oldest photographic negative in existence. ⇒

A Talbot Experiment Fresnel di ff raction by periodic gratings (1836): “It was very curious to observe that though the grating was greatly out of the focus of the lens . . . the appearance of the bands was perfectly distinct and well defined . . . the experiments are communicated in the hope that they may prove interesting to the cultivators of optical science.” — Fox Talbot = Lord Rayleigh calculates the Talbot distance (1881) ⇒ = Lord Rayleigh calculates the Talbot distance (1881) ⇒

The Quantized/Fractal Talbot E ff ect • Optical experiments — Berry & Klein • Di ff raction of matter waves (helium atoms) — Nowak et. al.

Quantum Revival • Electrons in potassium ions — Yeazell & Stroud • Vibrations of bromine molecules — Vrakking, Villeneuve, Stolow

Periodic Linear Schr¨ odinger Equation ∂ t = − ∂ 2 u i ∂ u ∂ x 2 ∂ u ∂ x ( t, − π ) = ∂ u u ( t, − π ) = u ( t, π ) ∂ x ( t, π ) Integrated fundamental solution: e i ( k x − k 2 t ) ∞ u ( t, x ) = 1 � . 2 π k 0 ̸ = k = −∞ For x/t ∈ Q , this is known as a Gauss sum (or, more generally, k n , a Weyl sum), of great importance in number theory = Hardy, Littlewood, Weil, I. Vinogradov, etc. ⇒ ⋆ ⋆ The Riemann Hypothesis!

Dispersive Carpet Schr¨ odinger Carpet

Periodic Linear Dispersion ∂ u ∂ t = L ( D x ) u, u ( t, x + 2 π ) = u ( t, x ) Dispersion relation: u ( t, x ) = e i ( k x − ω t ) = ω ( k ) = − i L ( − i k ) assumed real ⇒ Riemann problem: step function initial data � 0 , x < 0 , u (0 , x ) = σ ( x ) = 1 , x > 0 . Solution: u ( t, x ) ∼ 1 2 + 2 sin[ (2 j + 1) x − ω ( k ) t ] ∞ � . 2 j + 1 π j =0 ⋆ ⋆ ω ( − k ) = − ω ( k ) odd Polynomial dispersion, rational t = Weyl exponential sums ⇒

2D Water Waves h y = h + η ( t, x ) surface elevation φ ( t, x, y ) velocity potential

2D Water Waves • Incompressible, irrotational fluid. • No surface tension φ t + 1 2 φ 2 x + 1 2 φ 2 y + g η = 0 ⎫ ⎬ y = h + η ( t, x ) η t = φ y − η x φ x ⎭ φ xx + φ yy = 0 0 < y < h + η ( t, x ) φ y = 0 y = 0 c = √ g h • Wave speed (maximum group velocity): 6 c h 2 k 3 + · · · � g k tanh( h k ) = c k − 1 • Dispersion relation:

c = √ g h a ℓ h Small parameters — long waves in shallow water (KdV regime) β = h 2 α = a ℓ 2 = O ( α ) h

Rescale: → ℓ t x �− → ℓ x y �− → h y t �− c → g a ℓ φ � η �− → a η φ �− c = g h c Rescaled water wave system: φ t + α x + α ⎫ 2 φ 2 2 β φ 2 y + η = 0 ⎪ ⎪ ⎬ y = 1 + α η η t = 1 β φ y − α η x φ x ⎪ ⎪ ⎭ β φ xx + φ yy = 0 0 < y < 1 + α η φ y = 0 y = 0

Boussinesq expansion Set ψ ( t, x ) = φ ( t, x, 0) u ( t, x ) = φ x ( t, x, θ ) 0 ≤ θ ≤ 1 Solve Laplace equation: 2 β 2 y 2 ψ xx + 1 4! β 4 y 4 ψ xxxx + · · · φ ( t, x, y ) = ψ ( t, x ) − 1 Plug expansion into free surface conditions: To first order 2 α ψ 2 ψ t + η + 1 x − 1 2 β ψ xxt = 0 η t + ψ x + α ( ηψ x ) x − 1 6 β ψ xxxx = 0

Bidirectional Boussinesq systems: 2 β ( θ 2 − 1) u xxt = 0 u t + η x + α u u x − 1 6 β (3 θ 2 − 1) u xxx = 0 η t + u x + α ( η u ) x − 1 ⋆ ⋆ at θ = 1 this system is integrable = Kaup, Kupershmidt ⇒ Boussinesq equation u tt = u xx + 1 2 α ( u 2 ) xx − 1 6 β u xxxx Regularized Boussinesq equation u tt = u xx + 1 2 α ( u 2 ) xx − 1 6 β u xxtt = DNA dynamics (Scott) ⇒

Unidirectional waves: � 1 4 α η 2 + 2 θ 2 � u = η − 1 3 − 1 β η xx + · · · Korteweg-deVries (1895) equation: η t + η x + 3 2 α η η x + 1 6 β η xxx = 0 = Due to Boussinesq in 1877! ⇒ Benjamin–Bona–Mahony (BBM) equation: η t + η x + 3 2 α η η x − 1 6 β η xxt = 0