Water wave analysis with nonlinear Fourier transforms Peter Prins p.j.prins@tudelft.nl 7 September 2018 This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716669). 1
Nonlinear water waves “ I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation , a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed . I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation. “ John Scott Russell (1808-1882) 2
https://youtu.be/w-oDnvbV8mY 3
https://youtu.be/D14QuUL8x60?t=51s 4
Wave equations • Water surface elevation: u = u ( x, t ) • Linear wave equation: u tt − u xx = 0 • Korteweg-De Vries equation (KdV): − u t + 6 uu x + u xxx = 0 • Many other non-linear wave equations exist. 5
Why a Fourier transform? Hard calculations u(t) y(t) ☹ Inverse Fourier Fourier Transform Transform Easy calculations " 6
Why a Fourier transform? d t + ↵ 2 d 2 u y ( t ) = ↵ 0 u ( t ) + ↵ 1 d u d t 2 =: ( ↵ 0 + ↵ 1 � t + ↵ 2 � tt ) u ( t ) | {z } | K M v i = � i v i K exp( j ! 0 t ) = H ( j ! 0 )exp( j ! 0 t ) Z ∞ X w = c i v i u ( t ) = U ( j ! ) exp( j ! t ) d ! −∞ i Z ∞ X y ( t ) = Ku ( t ) = U ( j ! ) K exp( j ! t ) d ! y = M w = c i M v i −∞ i Z ∞ X = U ( j ! ) H ( j ! ) exp( j ! t ) d ! = c i � i v i | {z } −∞ Y ( j ω ) i 7
➡ ⬆ Solitons overtaking: No superposition Source: http://lie.math.brocku.ca/~sanco /solitons/kdv_solitons.php Adapted from: https://carretero.sdsu.edu/ teaching/M-639/lectures/ nonlinear_waves.html 8
Lax pair � u t + 6 uu x + u xxx = 0 ⇕ − L t + LA − AL = 0 − − A := 4 � xxx + 6 u � x + 3 u x L := � xx + u • A and L are linear operators. • The eigenvalues of L are constant in time iff u(x,t) satisfies the KdV equation. 9
Schrödinger eigenvalue problem • The Non-linear Fourier transform of an input u(x,t0) w.r.t. the KdV equation, is the eigenvalue problem of L. " • It can be shown that " – All are real. λ – All <0 are an eigenvalue (cf. ordinary λ " Fourier transform) – Some isolated >0 may be eigenvalues (cf. λ matrix eigenvalues) 10
Schrödinger eigenvalue problem � � δ xx + u ( x ) w ( x ) = λ w ( x ) � � w xx ( x ) = λ − u ( x ) w ( x ) " # " # " # d w ( x ) 0 1 w ( x ) = · w x ( x ) λ − u ( x ) 0 w x ( x ) d x d d x w ( x ) = A ( x ) w ( x ) � ¯ � d x w ( x ) = ¯ d A w ( x ) ⇒ w ( x ) = exp · w (0) A x 11
Staircase approximation 1 0.8 u(x,t 0 ) 0.6 0.4 0.2 0 -6 -4 -2 0 2 4 6 x u i := u ( x i ) ¯ " # 0 1 ¯ A i = λ − ¯ u i 0 � � � ¯ � � ¯ � � ¯ � � � x D + ε / 2 , λ = exp A D ( λ ) · ε · · · exp A 2 ( λ ) · ε · exp A 1 ( λ ) · ε x 1 − ε / 2 , λ w · w | {z } | {z } | {z } G D ( λ ) G 2 ( λ ) G 1 ( λ ) 12
Computational complexity • D gridpoints in x => (D-1) matrix " multiplications. λ • D gridpoints in => Repeat the calculation D times. • Hence complexity O(D^2). • We have improved it to O(D log^2(D)), see next slide. 13
Fast Nonlinear Fourier Transform = p 1 ↘ p 1 p 2 ↗ p 2 ↘ " # p 11 i ( z ) p 12 i ( z ) p 1 p 2 p 3 p 4 G i ( λ ) ≈ p 3 ↗ p 21 i ( z ) p 22 i ( z ) ↘ p 3 p 4 ↘ ↗ p · i ( z ) = c 0 + c 1 z + c 2 z 2 + . . . p 4 p z = z ( λ ) p 5 ↘ " p 5 p 6 ↗ ↗ p 6 ↘ Main benefit: No need to repeat " p 5 p 6 p 7 p 8 λ for every . We can fill out the p 7 ↗ resulting 4 polynomials for ↘ p 7 p 8 ↗ λ every value of we need. p 8 Source: Wahls, Sander, and H. Vincent Poor. "Fast numerical nonlinear Fourier transforms." IEEE Transactions on Information Theory 61.12 (2015): 14 6957-6974.
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