On the Whitham Equation Henrik Kalisch Department of Mathematics University of Bergen, Norway Joint work with: Handan Borluk, Denys Dutykh, Mats Ehrnström, Daulet Moldabayev, David Nicholls Research partially supported by the Research Council of Norway.
Introduction
Surface gravity waves Assumptions: incompressible inviscid two-dimensional irrotational z g η( x,t) x h 0 river bed
Surface gravity waves Assumptions: incompressible inviscid two-dimensional irrotational Hamiltonian system: H = η t = δh δφ, 1 2 gη2 + 1 2ΦG(η)Φ dx Φ t = δh δη z x g η( x,t) h 0 river bed
Possible simplifications Long wavelength: h 0 λ 1 Small amplitude: a h 0 1 λ z x a h 0 river bed
Possible simplifications Long wavelength: h 0 λ 1 a Small amplitude: h 0 1 λ z x a h 0 river bed
Possible simplifications Long wavelength: h 0 λ 1 a Small amplitude: h 0 1 = Shallow water λ z x a h 0 river bed
Shallow-water approximation: h 0 λ 1 Assumptions: p = (η z)g (hydrostatic) u = u(x, t) Shallow-water equations: η t + h 0 u x + (ηu) x = 0 u t + gη x + uu x = 0 A wave propagating to the right into undisturbed conditions satisfies the equation η t + (3 g(η + h 0 ) 2 ) gh 0 η x = 0. Developing the square root yields c 0 η t + c 0 η x + 3 2 h 0 η η x = 0.
Possible simplifications Long wavelength: h 0 λ 1 a Small amplitude: h 0 1 λ z x a h 0 river bed
Possible simplifications Long wavelength: h 0 λ 1 a Small amplitude: h 0 1 = Airy theory λ z x a h 0 river bed
Potential formulation In terms of velocity potential φ : φ = 0 in 0 < y < h 0 + η Boundary conditions: η t + φ x η x = φ z ψ t + 1 2 φ2 x + 1 2 φ2 z + gη = 0 at z = h 0 + η(x, t) ψ z = 0 at z = 0
Linearized Formulation: Dispersion relation: η = Ae ikx iωt a h 0 1 φ = Z (z)e ikx iωt ω 2 = gk tanh (h 0 k) Linear propagation speed: c(k) = ω k = g k tanh (h 0k), for waves propagating to the right. Note: lim g k 0 k tanh(h 0k) = gh 0 = c 0, lim g k k tanh(h 0k) = 0.
Linearized Formulation: KdV - approximation 2-nd order approximation: KdV full dispersion relation c(k) = c 0 1 6 c 0h 2 0k 2 c(k) = g k tanh (h 0k) 3 2 1 K3 K2 K1 0 1 2 3 k K1 Whitham KdV
Linearized Formulation: KdV - approximation 2-nd order approximation: KdV full dispersion relation c(k) = c 0 1 6 c 0h 2 0k 2 c(k) = g k tanh (h 0k)
Linearized Formulation: KdV - approximation 2-nd order approximation: KdV full dispersion relation c(k) = c 0 1 6 c 0h 2 0k 2 c(k) = g k tanh (h 0k) η t + c 0 η x + 1 6 c 0h 2 0 η xxx = 0 η t + K h0 η x = 0
Linearized Formulation: KdV - approximation 2-nd order approximation: KdV full dispersion relation g c(k) = c 0 1 6 c 0h0k 2 2 c(k) = k tanh (h 0k) η t + c 0 η x + 1 6 c 0h 2 0 η xxx = 0 η t + K h0 η x = 0 where K h0 ( ) := F 1 g ξ tanh(h 0ξ) FK h0 = g ξ tanh(h 0ξ)
Long waves of small amplitude: KdV Combining c 0 η t + c 0 η x + 3 2 h 0 η η x = 0 and η t + c 0 η x + 1 6 c 0h0 2 η xxx = 0
Long waves of small amplitude: KdV Combining c 0 η t + c 0 η x + 3 2 h 0 η η x = 0 and η t + c 0 η x + 1 6 c 0h0 2 η xxx = 0 yields KdV: c 0 η t + c 0 η x + 3 2 h 0 η η x + 1 6 c 0h0 2 η xxx = 0
The Whitham Equation KdV: c 0 η t + c 0 η x + 3 2 h 0 η η x + 1 6 c 0h0 2 η xxx = 0
The Whitham Equation KdV: delete linear terms c 0 η t + c 0 η x + 3 2 h 0 η η x + 1 6 c 0h0 2 η xxx = 0 c 0 η t + 3 2 h 0 η η x = 0
The Whitham Equation KdV: Delete linear terms: c 0 η t + c 0 η x + 3 2 h 0 η η x + 1 6 c 0h0 2 η xxx = 0 and replace by c 0 η t + 3 2 h 0 η η x = 0 c 0 η t + 3 2 h 0 η η x + K h0 η x = 0.
The Whitham equation Whitham equation: c 0 η t + 3 2 h 0 η η x + K h0 η x = 0 small amplitude: a h 0 1 long-wave assumption?? formal asymptotic expansion??
Formal derivation
Total energy of the water-wave system: η η 1 H = z dzdx + 2 φ 2 dzdx R 0 R 1 Trace of the potential at free surface is Φ(x, t) = φ(x, η(x, t), t) H = 1 [ η 2 + ΦG(η)Φ ] dx 2 R G(η) is the Dirichlet-Neumann operator, and can be written as G(η) = G j (η) j=0 G 0 (η) = D tanh(d) G 1 (η) = DηD D tanh(d)ηd tanh(d) V.E. Zakharov, Stability of periodic waves of finite amplitude... J. Appl. Mech. Tech. (1968) W. Craig, C. Sulem, Numerical simulation of gravity waves J. Comp. Phys (1993)
Formulate Hamiltonian in terms of u = Φ x : G(η) = DK(η)D. K(η) = K j (η) K j (η) = D 1 G j (η)d 1. j=0 The Hamiltonian is expressed as H = 1 [ η 2 + uk(η)u ] dx. 2 Note: R K 0 = tanh D tanh D D If h 0 = 1, then K h0 = D K 0 = K 2 h 0
Hamiltonian equations for water-wave problem: η t = x δh δu, u t = x δh δη Structure map: ( ) 0 x J η,u = x 0 Expand Hamiltonian following ideas of Craig and Groves, Wave Motion, 1994 Wavefield with characteristic wavelength l and amplitude a Nondimensional amplitude: α = a h 0 Nondimensional wavelength: λ = l h 0
Derivation of Whitham-type system Scaling: x = 1 λx, η = α η Hamiltonian is expanded in terms of α and µ = 1 λ : H = 1 2 η 2 dx R [ ] + 1 2 ũ 1 1 3 µ2 D2 + 2 15 µ4 D4 + ũ dx + α 2 ηũ 2 dx R R [ α 2 ũ µ D ] [ 1 3 µ3 D3 + η µ D ] 1 3 µ3 D3 + ũ dx R keep terms of O(µ n ) for all n keep terms of O(α) DO NOT keep terms of O(αµ)
Whitham number W = a h 0 e (l/h 0) O(1) α e λ O(1) α e λ α e 1/µ Hamiltonian is H = 1 2 η 2 dx + 1 R 2 e 1/µ 2 R R [ ] ũ 1 1 3 µ2 D2 + ũ dx + e 1/µ 2 [ ũ µ D 1 3 µ3 D3 +... ] [ η µ D 1 3 µ3 D3 + R ηũ 2 dx ] ũ dx Disregarding terms of O(µ 2 e 1/µ ), but not of O(e 1/µ ) yields [ ] H = 1 2 η 2 dx + 1 2 ũ 1 1 3 µ2 D2 +... ũ dx + e 1/µ ηũ 2 dx R R 2 R
An arbitrary number of terms of O(µ n ) can be kept the approximation: H N = 1 [ η 2 + uk0 N (η)u + uηu ] dxdz 2 R The Whitham system is δh N η t = x δu = KN 0 u x (ηu) x, δh N u t = x δη = η x uu x But we have K N 0 K 0 O(µ N ) so we may change system to η t = K 0 u x (ηu) x, u t = η x uu x
Whitham number in terms of solitary wave scaling 0.5 0.45 0.4 Data Boussinesq regime Whitham regime 0.35 0.2 Time: 40000 0.1 0 0.2 Time: 10000 0.1 0 Amplitude 0.3 0.25 0.2 0.15 0.1 Stokes number: S = 4.46 Whitham regime parameters: κ = 1.35; ν = 0.51. Whitham number: W = 3.78. 0.2 Time: 0.05 0.1 0 4000 3000 2000 1000 0 1000 2000 3000 4000 (a) 0.05 0 2 4 6 8 10 12 14 16 18 20 Wavelength (b) (a): Formation of solitary waves of the Whitham equation from an initial surface wave (b): Fitting Whitham number and Stokes number to amplitude/wavelength data from Whitham solitary waves
Whitham equation Analyze relation between η and u in linearized Whitham system: η t = Kh 2 0 u x u t = η x Right-going part r and a left-going part s are defined by r = 1 2 (η + K h 0 u), s = 1 2 (η K h 0 u). Structure map changes to ( ) ( ) T F F J r,s = J η,u = (η, u) (η, u) ( 1 2 xk h0 0 ) 0 1 2 xk h0 See transformation theory in Craig, Guyenne, Kalisch, Hamiltonian Long Wave Expansions... Comm. Pure Appl. Math. (2005)
Scaling: x = µx, r = α r, s = α s Hamiltonian: H = 1 ( r + s) 2 dx 2 R + 1 2 R + α ( 2 ( r + s) K 1 ( r s)) dx 2 R α [ K 1 ( r s) µ 2 D 1 3 µ3 D3 + R [ ] K 1 ( r s) 1 1 3 µ2 D2 + K 1 ( r s) dx Exact equation for r: [ r t = 1 ( ] δ α 2 H) 2α2 x K δ r ] ( r + s) [ µ D 1 3 µ3 D3 + ] K 1 ( r s) dx
Assuming that s is O(µ 2 e κ/µν ), the equation for r is r t = 1 x[ 1 1 2 6 µ2 D2 + ]{ 2 r + α ( [1 + 1 2 6 µ2 D2 + ] r + α [ 1 + 1 6 µ2 D2 + ] ( r [ 1 + 1 6 µ2 D2 + ] r ) α ( [µ D 1 2 3 µ3 D3 + ][ 1 + 1 6 µ2 D2 + ] r ) 2 α [ µ D 1 3 µ3 D3 + ][ 1 + 1 6 µ2 D2 + ] ( r [ µ D 1 3 µ3 D3 + ][1 ] r) } + 1 6 µ2 D2 + + O(αµ 2 ) ) 2 Disregarding terms of O(µ 2 e 1/µ ), but not of O(e 1/µ ) yields r t = K h0 r x 3 2 rr x Disregarding terms of O(µ 4 e 1/µ ), but not of O(µ 2 e 1/µ ) yields r t = K h0 r x 3 2 rr x 13 12 r xr xx 5 12 rr xxx
Traveling waves
Traveling water waves
Traveling water waves
Traveling waves Whitham equation: Ansatz: η t + 3 2 η η x + K h0 η x = 0 (1) η(x, t) = φ(x ct)
Traveling waves Whitham equation: η t + 3 2 η η x + K h0 η x = 0 (1) Ansatz: η(x, t) = φ(x ct) Using this form, the equation (1) transforms into c φ + 3 2 φ φ + K h0 φ = 0, which may be integrated to c φ + 3 4 φ2 + K h0 φ = B.
Traveling waves Whitham equation: Ansatz: c 0 η t + 3 2 h 0 η η x + K h0 η x = 0 (1) η(x, t) = φ(x ct) Using this form, the equation (1) transforms into c 0 c φ + 3 2 h 0 φ φ + K h0 φ = 0, which may be integrated to c 0 c φ + 3 4 h 0 φ 2 + K h0 φ = B. We shall consider here only the case when B = 0
Results for traveling waves Local bifurcation (Ehrnström, Kalisch, Diff. Int. Eq. 2009) Thm: For a given L > 0 and a given depth h 0 > 0, there exists a local bifurcation curve of steady, 2L-periodic, even and continuous solutions of the Whitham equation. The wave speed at the bifurcation point is determined by c = g L π tanh (h 0π/L) = g k tanh (h 0k). Existence of solitary waves (Ehrnström, Groves, Wahlén, Nonlinearity 2012) Global bifurcation (Ehrnström, Kalisch, Math. Mod. Nat. Phenomena 2013)
Numerical Approximation
Numerical approximation: steady equation The collocation points are x n = π 2n 1 2N for n = 1,...N. 1 0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 The discrete cosine representation of φ N is where and N 1 φ N (x) = w(l)φ N (l) cos(lx), w(l) = l=0 { 1/N, l = 0, 2/N, l 1, N 1 Φ N (l) = w(l) φ N (x n ) cos(lx n ), for l = 0,..., N 1. n=0
Numerical approximation: steady equation S N = span R {cos (lx) } 0 l N 1 The discretization is defined by seeking φ N in S N satisfying µ φ N + φ 2 N + K N φ N = 0, where K N is discrete form of K. The term K N φ N is practically evaluated by [ K N ] φ N (x m ) = N [ K N ] (m, n)φ N (x n ), n=1 where the matrix [ K N] (m, n) is [ K N ] N 1 (m, n) = w 2 1 (k) l tanh l cos(lx n ) cos(lx m ). l=0
Numerical approximation: steady equation The system of N nonlinear equations µ φ N + φ 2 N + K N φ N = 0 can be efficiently solved using a Newton method. φ N 0.25 0.2 (a) 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0 1 2 3 4 5 6 x Φ N 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2 (b) 0 5 10 15 k Left: An approximate traveling-wave solution φ N with wavespeed µ = 0.789, wavelength 2π, and waveheight 0.337 using 16 collocation points. Right: Amplitudes of the discrete cosine modes.
Numerical approximation: steady equation 0.25 1.6 0.2 0.15 (a) 1.4 1.2 (b) φ N 0.1 0.05 0 0.05 0.1 Φ N 1 0.8 0.6 0.4 0.15 0.2 0.25 0 1 2 3 4 5 6 x 0.2 0 0.2 0 5 10 15 k Wavespeed µ = 0.789, waveheight 0.337 0.25 1.6 0.2 0.15 (a) 1.4 1.2 (b) φ N 0.1 0.05 0 0.05 0.1 Φ N 1 0.8 0.6 0.4 0.15 0.2 0.25 0 1 2 3 4 5 6 x 0.2 0 0.2 0 5 10 15 k Wavespeed µ = 0.772, waveheight 0.415
Numerical approximation: evolution equation S N = span C { e ikx k Z, N/2 k N/2 1 } The collocation points are x j = 2πj N for j = 0, 1,...N 1. I N is the interpolation operator from Cper ([0, 2π]) onto S N. Semi-discrete equation is { t η N + x I N (ηn 2 ) + [ ] 1 Kh0 a /a N xη N = 0, x [0, 2π], η N (, 0) = φ N.
Numerical approximation: evolution equation { t η N + x I N (ηn 2 ) + [ ] 1 Kh0 a /a N xη N = 0, η N (, 0) = φ N. x [0, 2π], 0.08 0.0651 (a) (b) 0.04 0.0654 φ N, η N 0 φ N, η N 0.0657 0.04 0.08 0 1 2 3 4 5 6 x 0.066 3 3.1 3.2 3.3 x Solid line: approximate traveling wave φ N for the Whitham equation. Dashed line: η N after time integration for 10000 periods. In this computation, N = 512 and dt = 0.0005. The L 2 error is 0.0021.
A branch of periodic traveling waves Part of a branch of solutions with h 0 = 1 and L = π: 0.25 0.25 0.15 (a) 0.15 (b) 0.05 0.05 φ N φ N 0.05 0.05 0.15 0.15 0.25 0 1 2 3 4 5 6 x 0.25 0 1 2 3 4 5 6 x Note that the highest wave is not shown here.
Evolution of highest wave, and next point on branch A branch of periodic traveling waves 0.5 0.5 (a) 0.48 (b) 0.4 0.46 waveheight 0.3 waveheight 0.44 0.2 0.42 0.1 0.4 0 0.76 0.78 0.8 0.82 0.84 0.86 0.88 µ 0.38 0.764 0.766 0.768 0.77 0.772 0.774 0.776 µ Left: Waveheight vs. wavespeed Right: close-up of turning point 0.4 1.6 0.3 (a) 1.4 1.2 (b) 0.2 1 0.1 0.8 η η 0.6 0 0.4 0.1 0.2 0.2 0 0.2 0.3 0 1 2 3 4 5 6 7 x 0.4 0 1 2 3 4 5 6 7 x
Stability 0.8 0.14 0.7 0.12 Wavehight 0.6 0.5 0.4 0.3 0.2 η 2 L 2 0.10 0.08 0.06 0.04 0.1 0.02 0.0 0.76 0.78 0.80 0.82 0.84 0.86 0.88 Wavespeed 0.00 0.76 0.78 0.80 0.82 0.84 0.86 0.88 Wavespeed Waveheight Energy
Comparison with Euler h 0 = 0.1 and L = 1: 0.1 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 x Euler (D. Nicholls, JCP 1998) KdV
Comparison with Euler h 0 = 0.1 and L = 1: 0.1 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 x Euler (D. Nicholls, JCP 1998) Whitham
Convergence to solitary wave 0.3 (a) 0.3 (b) 0.25 0.25 0.2 0.2 φ N 0.15 φ N 0.15 0.1 0.1 0.05 0.05 0 0 0.05 0 5 10 15 20 25 30 35 40 45 50 x 0.05 0 5 10 15 20 25 30 35 40 45 50 x Approximate solitary waves for the Whitham equation with h 0 = 1 after a Galilean shift φ φ + γ and c c + 2γ a) L = 4π, c = 3.5640, and γ = 0.4320 b) L = 8π, c = 3.5642, and γ = 0.4321
Comparison: Euler vs. KdV, BBM, Whitham
Comparison: S = 1, α = 0.2, λ = 5 0.1 0 Time: 20 0.1 0 Time: 10 0.1 0 Time: 0.1 0.05 40 20 0 20 40 Euler - - KdV - - BBM Whitham
Comparison: positive wave, S = 0.2 A, positive: S = 0.2, α = 0.1, λ = 2 B, positive: S = 0.2, α = 0.2, λ = 1 0.45 0.4 KdV BBM Whitham 0.8 0.7 KdV BBM Whitham Normalized L 2 error 0.35 0.3 0.25 0.2 0.15 Normalized L 2 error 0.6 0.5 0.4 0.3 0.1 0.2 0.05 0.1 0 1 2 [ 3 4 5 Time t/ ] h0/g 0 0.5 1 [ 1.5 2 2.5 3 Time t/ ] h0/g L 2 errors in approximation of solutions to full Euler equations by different model equations
Comparison: positive wave, S = 1 C, positive: S = 1, α = 0.1, λ = 10 D, positive: S = 1, α = 0.2, λ = 5 0.06 KdV BBM Whitham 0.16 0.14 KdV BBM Whitham Normalized L 2 error 0.05 0.04 0.03 0.02 Normalized L 2 error 0.12 0.1 0.08 0.06 0.01 0 5 10 [ 15 20 25 30 Time t/ ] h0/g 0.04 0.02 0 5 [ 10 15 20 Time t/ ] h0/g L 2 errors in approximation of solutions to full Euler equations by different model equations
Comparison: positive wave, S = 5 Normalized L 2 error 5.5 5 4.5 4 3.5 6 x 10 3 E, positive: S = 5, α = 0.1, λ = 50 KdV BBM Whitham 0 5 10 [ 15 20 25 30 Time t/ ] h0/g Normalized L 2 error 0.045 0.04 0.035 0.03 0.025 0.02 0.015 F, positive: S = 5, α = 0.2, λ = 5 KdV BBM Whitham 0 5 [ 10 15 20 Time t/ ] h0/g L 2 errors in approximation of solutions to full Euler equations by different model equations
Euler - - KdV - - BBM Whitham Comparison: S = 1, α = 0.2, λ = 5 0.1 0 Time: 20 0.1 0 Time: 10 0 0.1 Time: 0.1 0.2 40 20 0 20 40
Comparison: negative wave, S = 0.2 A, negative: S = 0.2, α = 0.1, λ = 2 B, negative: S = 0.2, α = 0.2, λ = 1 0.6 0.5 KdV BBM Whitham 0.6 KdV BBM Whitham Normalized L 2 error 0.4 0.3 0.2 Normalized L 2 error 0.5 0.4 0.3 0.2 0.1 0.1 0 2 4 [ 6 8 10 Time t/ ] h0/g 0 1 2 3 [ 4 5 6 7 Time t/ ] h0/g L 2 errors in approximation of solutions to full Euler equations by different model equations
Comparison: negative wave, S = 1 0.3 0.25 C, negative: S = 1, α = 0.1, λ = 10 KdV BBM Whitham 0.55 0.5 0.45 D, negative: S = 1, α = 0.2, λ = 5 KdV BBM Whitham Normalized L 2 error 0.2 0.15 0.1 Normalized L 2 error 0.4 0.35 0.3 0.25 0.2 0.15 0.05 0.1 0.05 0 5 10 [ 15 20 25 30 Time t/ ] h0/g 0 5 [ 10 15 20 Time t/ ] h0/g L 2 errors in approximation of solutions to full Euler equations by different model equations
Comparison: negative wave, S = 5 0.018 0.016 E, negative: S = 5, α = 0.1, λ = 50 KdV BBM Whitham 0.12 0.11 F, negative: S = 5, α = 0.2, λ = 5 KdV BBM Whitham 0.1 Normalized L 2 error 0.014 0.012 0.01 0.008 Normalized L 2 error 0.09 0.08 0.07 0.06 0.05 0.006 0.04 0.03 0.004 0.02 0 5 10 [ 15 20 25 30 Time t/ ] h0/g 0 5 [ 10 15 20 Time t/ ] h0/g L 2 errors in approximation of solutions to full Euler equations by different model equations
References
H. Borluk, H. Kalisch and D.P. Nicholls A numerical study of the Whitham equation as a model for steady surface water waves to appear in Journal of Computational and Applied Mathematics M. Ehrnström, M. Groves and E. Wahlén Solitary waves of the Whitham equation Nonlinearity 25 (2012) M. Ehrnström, and H. Kalisch Traveling waves for the Whitham equation Differential Integral Equations 22 (2009) M. Ehrnström, and H. Kalisch Global bifurcation for the Whitham equation Math. Modeling Natural Phenomena 8 (2013) V.M. Hur and M. Johnson Modulational instability in the Whitham equation of water waves Stud. Appl. Math. 134 (2015) C. Klein and J.-C. Saut A numerical approach to blow-up issues for dispersive perturbations of Burgers equation Physica D 295 (2015) D. Moldabayev, H. Kalisch and D. Dutykh The Whitham Equation as a model for surface water waves Physica D 309 (2015) N. Sanford, K. Kodama, J. D. Carter and H. Kalisch Stability of traveling wave solutions to the Whitham equation Physics Letters A 378 (2014) G.B. Whitham Variational methods and applications to water waves, Proc. R. Soc. Lond., Ser. A 299 (1967)