Algebraic geometry for shallow capillary-gravity waves D ENYS D UTYKH1 Charge´ de Recherche CNRS 1 Universite ´ de Savoie ´ Laboratoire de Mathematiques (LAMA) 73376 Le Bourget-du-Lac France

Seminar of Computational Technologies Institute of Computational Technologies, SB RAS October 21, 2014

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Acknowledgements

Collaborators: Didier C LAMOND: Professor (LJAD) Universite´ de Nice Sophia Antipolis Andre´ G ALLIGO: Emiritus Professor Universite´ de Nice Sophia Antipolis

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Outline

1

Shallow capillary-gravity waves Travelling waves Phase-space analysis

2

Extended phase-space analysis

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Outline

1

Shallow capillary-gravity waves Travelling waves Phase-space analysis

2

Extended phase-space analysis

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Serre–(Green–Naghdi) equations Shallow water equations

Credits: Lord R AYLEIGH (1876) [1] (only steady version) F. S ERRE (1953) [2] C. S U & C. G ARDNER (1969) [3] A. G REEN & P. N AGHDI (1976) [4] E. P ELINOVSKY & Z HELEZNYAK (1985) [5] Modern derivations: Asymptotic methods: Variational methods:

Green & Naghdi (1976) [4]

Particle description: Miles & Salmon (1985) [6] Eulerian description: Clamond & DD (2012) [7]

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Serre’s equations with surface tension 1D case: the governing equations

Governing equations (the mass conservation + momentum): ¯ ]x = 0, ht + [ h u −1/2  h i   ¯t + u ¯u ¯x + g hx + 31 h−1 ∂x h2 γ˜ = τ hx 1 + hx2 u xx Vertical acceleration: ¯xt − u ¯t + u ¯x2 − u ¯u ¯xx ) = 2 h u ¯x2 − h [ u ¯u ¯ x ]x γ˜ = h (u Conservative form: h ¯ ¯2 + [ h u ]t + h u

1 2g

2

h +

1 2 ˜ 3h γ

− τR

i

x

= 0

Surface tension:  −1/2  −3/2 , + 1 + hx2 R = h hxx 1 + hx2 D ENYS D UTYKH (CNRS – LAMA)

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Serre’s equations: the conservation laws



Momentum conservation: i  h 2 1 2 1 3 2 1 3 ¯ − 13 (h3 u ¯ + 2 gh − 3 2h u ¯u ¯x −R = 0 ¯ x )x t + h u ¯x − 3 h u ¯u ¯xx −h2 hx u hu x

Tangential velocity at the free surface: i  ¯ x )x ¯ (h3 u ¯ x )x  h 1 2 τ hxx u (h3 u 1 2 ¯2 ¯ ¯− + − h u + gh − u − u
3/2 x = 0 x 2 2 t 3h 3h 1 + hx2 Energy conservation: q i 1 + hx2 + qt h
¯x i τ R τ h hx u 1 ¯2 1 2 ¯2 1 2+ p ¯ ¯ h hγ − u + u + gh + 1 + h =0 h u + τ u x x 2 6 3 h 1 + hx2 x h

1 ¯2 2h u

+

1 3 ¯2 6 h ux

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+

1 2g

h2 + τ

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Serre–CG equations: travelling waves Fr = c 2 /gd, Bo = τ /gd 2 , We = Bo / Fr = τ /c 2 d

¯ = −cd / h, d =< h >= Mass conservation: u

1 2ℓ

Rℓ

−ℓ h dx

Momentum conservations lead: 1 Bo γ˜ h2 Bo hhxx h2 Fr d + − = Fr + − Bo + K1 − + 3 1

2 h 2 2d 3gd 2 1 + hx2 2 1 + hx2 2 Tangential velocity: Fr Fr d 2 Bo d hxx Fr d 2 hx2 Fr K2 h Fr d 2 hxx = + − + 1 + + − 3
2 2 d 3h 2 2 2h 6h 1 + hx2 2 γ˜ /g = Fr d 3 hxx /h2 − Fr d 3 hx2 /h3

Integration constants: K2 =

K1 +

1 2

(3 + hx2 ) d 2

−1



3 h2 D Fr d 2 1 Bo d / h E. d  + + Fr − Bo = − .
1 2 h h2 1 + hx2 2

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Serre–CG equations: travelling waves Particular emphasis on solitary waves

Combination of two equations: (Fr + 2 + Fr K2 ) h Fr d Bo h2 Fr d hx2 + − − = Cnst − 1
2 2h 6h 2d 2d 1 + hx2 2 Consider solitary waves (K1 = K2 ≡ 0): F (h, h′ ) ≡

(2Fr + 1 − 2Bo ) h 2 Bo h/d Fr h′ 2 +   1 − Fr + 3 d 2 1 + h′ 2 −

h3 (Fr + 2) h2 + = 0 d2 d3

Property: h(x) ≡ h(−x)

At the crest of a regular wave: h(0) = d + a, h′ (0) = 0 Fr = 1 + a / d D ENYS D UTYKH (CNRS – LAMA)

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Serre–CG equations: travelling waves Particular emphasis on solitary waves

Combination of two equations: Fr d hx2 Fr d (Fr + 2 + Fr K2 ) h Bo h2 − + − − = Cnst 1
2h 6h 2d 2 d2 1 + hx2 2 Consider solitary waves (K1 = K2 ≡ 0): F (h, h′ ) ≡

Fr h′ 2 2 Bo h/d (2Fr + 1 − 2Bo ) h +  − Fr + 1  3 d 2 1 + h′ 2 −

(Fr + 2) h2 h3 + = 0 d2 d3

We are interested in solutions: h(∞) = d, h′ (∞) = 0 D ENYS D UTYKH (CNRS – LAMA)

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Serre–CG equation: Limiting cases I Pure gravity and pure capillary waves

Pure gravity waves: Bo → 0

h = d + a sech2 (κx/2),

(κd)2 = 3 a / (d + a),

Fr = 1 + a / d

Pure capillary waves: Fr → ∞, Bo → ∞, We = Const

h′ 2 2 (1 − We ) h h2 2 We h/d − 1 + + − 2 = 0 1 3 d d 2 2 ′ 1+h

In the limit We → 0:

Capillary wave equation reads: h′ = ± Solitary wave with angular crest:

√

3 (1 − h/d)

 √  h = d + a exp − 3 |x|

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Serre–CG equation: Limiting cases II Introduce a simplification to equations

Small slope approximation [8]:   1 2 −2 ≈ 1 − 1 + h′

1 2

2

h′ + . . .

Master equation becomes: (2Fr2 + 1) h (Fr 2 + 2) h2 h3 Fr 2 Bo h
′2 h = Fr 2 − − + − 3 d d d2 d3 Change of independent variables: 1

dξ = | 1 − 3 We h / d |− 2 dx

(⋆)

Analytical solution (≡ pure gravity case): h(ξ) = d + a sech2 (κξ/2) (κd)2 = 3 a / (d + a), Fr = 1 + a / d x(ξ) can be found from (⋆) D ENYS D UTYKH (CNRS – LAMA)

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Asymptotic analysis Following M C C OWAN (1891) [9]

Exponentially decaying solitary waves: h(x) ∼ d + a exp(−κx),

x → +∞,

κ>0

’Dispersion’ relation: Fr 2 =

3 − 3 Bo (κd)2 3 − (κd)2

  or (κd)2 (Fr − 3 Bo ) = 3 Fr 2 − 1

κ can be ony real or purely imaginary =⇒ Solitary waves or Periodic waves

Critical values: Fr = 1, Bo = 1/3, κd =

√ 3 or Bo = 31 Fr

Algebraic decay: h(x) ∼ d + a(κx)−α , x → +∞, Then necessarily =⇒ Fr ≡ 1 α > 2 =⇒ Bo = 13 α = 1 =⇒ Bo 6= 13 , D ENYS D UTYKH (CNRS – LAMA)

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Phase-space analysis: local behaviour Nonlinear autonomous ODE analysis

Two-parameter (Fr , Bo ) family of real algebraic curves in R2 : FFr , Bo (h, k) :=

Bo h Fr 2 k +2 + 1 3 (1 + k 2 ) 2 − Bo + (2Fr − 2Bo + 1)h − (Fr + 2)h2 + h3 = 0

With asymptotic behaviour at x → ∞ (k := h′ ): h(∞) = 1, k(∞) = 0 Typical workflow with a parametrized curve: Find multiple points (with horizontal tangent): ∂k FFr , Bo = 0 Find points with vertical tangent: ∂h FFr , Bo = 0 Decompose it into oriented branches (k ≷ 0, h րց 0)

Study the family of algebraic curves FFr , Bo (h, k) ∈ R2 with certified topology methods [10] D ENYS D UTYKH (CNRS – LAMA)

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Phase-space analysis: depression wave A particular example for Fr = 0.4, Bo = 0.9 > 1/3

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Phase-space analysis: depression wave A particular example for Fr = 0.4, Bo = 0.9 > 1/3

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 0.20; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 20.00; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 35.00; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 40.00; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 42.20; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 43.00; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 44.20; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 46.00; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 48.00; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 54.00; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Solitary waves collision Solitary waves of depression: a/d = −1/2, Fr = 1/2, Bo = 1/2

Free surface η(x,t) at t = 60.40; dt = 0.010 0

η(x, t)

−0.2 −0.4 −0.6 −0.8 −1 −60

Free surface SW amplitude Bottom −40

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Phase-space analysis: peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3

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Phase-space analysis: peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3

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Phase-space analysis: multi-peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3

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Phase-space analysis: multi-peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3

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Outline

1

Shallow capillary-gravity waves Travelling waves Phase-space analysis

2

Extended phase-space analysis

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Phase space for roots of polynomials Geometric interpretation for quadratic equations

Polynomial quadratic equation: x 2 + ax + b = 0 Coefficients as parameters: P : R3

(a, b, x)

7→

R x 2 + ax + b

7→

Consider an algebraic surface S ⊆ R3 : (a, b, x) ∈ S

P(a, b, x) = x 2 + ax + b = 0

⇐⇒

Consider the fibers of the map: F : (a, b) 7→ x ∈ S,

F (a, b) :=



x | (a, b, x) ∈ S



Count the cardinals: |F (a, b)| ∈ 0, 1, 2. D ENYS D UTYKH (CNRS – LAMA)

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Corresponding algebraic surface Quadratic polynomial: x 2 + ax + b = 0

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Corresponding algebraic surface Quadratic polynomial: x 2 + ax + b = 0

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Cubic equations Geometric interpretation for cubic equations

Consider a generic cubic polynomial equation: x 3 + rx 2 + sx + t = 0 Obtain the reduced form by substitution x → x − 31 r x 3 + ax + b = 0, a := s −

r2 , 3

b :=

2r 3 sr − + t. 27 3

By following Girolamo C ARDANO (1501 – 1576), the discriminant is D2 :=

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a3 b2 + 4 27

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Corresponding algebraic surface Cubic polynomial: x 3 + ax + b = 0

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Corresponding algebraic surface Cubic polynomial: x 3 + ax + b = 0

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Phase space partition for higher order polynomials The notion of the resultant of two polynomials

Consider two polynomials P, Q ∈ R[x]: deg P = n,

deg Q = m

P(x) = a0 x n + . . . ,

Q(x) = b0 x m + . . .

Resultant of two polynomials is R(P, Q) := a0m b0n

n Y m Y i=1 j=1

(ri − sj )

where ri are zeros of P(x) and sj are zeros of Q(x). Resultant is the determinant of the Sylvester matrix! Discriminant of a polynomial equation P(x) = 0 is D(P) := (−1) D ENYS D UTYKH (CNRS – LAMA)

n(n−1) 2

R(P, P ′ )

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Phase space analysis: global behaviour Detection of multiple points

Points with horizontal tangent satisfy: FFr , Bo (k, h) = 0 ∂k FFr , Bo (k, h) = 0

The 2nd equation can be solved analytically: k =0 Fr (k 2 + 1)3 = 9Bo 2 h2

To avoid cubic roots, change of variables: k 2 = y 2 − 1, y ≥ 1 Wave height can be expressed as h =

Fr 3Bo

Y3

Polynomial equation in y: f := Fr 2 y 9 − (3Fr − 2)Fr Bo y 6 +

9Bo 2 (1 + 2Fr − 2Bo )y 3 + 27Bo 3 y 2 − 36Bo 3

Adapted tool to describe the real roots in (Fr , Bo ) space: discriminant locus! D = (Fr − 3Bo )2 × P10 D ENYS D UTYKH (CNRS – LAMA)

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Phase space analysis: global behaviour Contains ≈ 11 cells with 0 to 3 real roots such as y > 1

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Phase space analysis: global behaviour Contains ≈ 11 cells with 0 to 3 real roots such as y > 1

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Phase-space analysis: weakly singular solitary wave A particular example for Fr = 0.8, Bo = 0.3538557 > 1/3

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Phase-space analysis: weakly singular solitary wave A particular example for Fr = 0.8, Bo = 0.3538557 > 1/3

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Phase-space analysis: wave with algebraic decay A particular example for Fr = 1.0, Bo = 1/3

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Conclusions & Perspectives

Conclusions: Capillary-gravity solitary waves were analyzed in shallow water regime Fully nonlinear & weakly dispersive model Phase-space analysis using the methods of the algebraic geometry

Only two types of regular solitary waves (+a, −a) Perspectives: Analysis of periodic CG-waves Go to 3D ! Compute fully nonlinear lump-solitary waves

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Thank you for your attention!

http://www.denys-dutykh.com/ D ENYS D UTYKH (CNRS – LAMA)

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References I J. W. S. Lord Rayleigh. On Waves. Phil. Mag., 1:257–279, 1876. F. Serre. ´ ´ Contribution a` l’etude des ecoulements permanents et variables dans les canaux. La Houille blanche, 8:830–872, 1953. C. H. Su and C. S. Gardner. KdV equation and generalizations. Part III. Derivation of Korteweg-de Vries equation and Burgers equation. J. Math. Phys., 10:536–539, 1969. A. E. Green and P. M. Naghdi. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech., 78:237–246, 1976. D ENYS D UTYKH (CNRS – LAMA)

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References II M. I. Zheleznyak and E. N. Pelinovsky. Physical and mathematical models of the tsunami climbing a beach. In E N Pelinovsky, editor, Tsunami Climbing a Beach, pages 8–34. Applied Physics Institute Press, Gorky, 1985. J. W. Miles and R. Salmon. Weakly dispersive nonlinear gravity waves. J. Fluid Mech., 157:519–531, 1985. D. Clamond and D. Dutykh. Practical use of variational principles for modeling water waves. Phys. D, 241(1):25–36, 2012. F. Dias and P. Milewski. On the fully-nonlinear shallow-water generalized Serre equations. Phys. Lett. A, 374(8):1049–1053, 2010. D ENYS D UTYKH (CNRS – LAMA)

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References III

J. McCowan. On the solitary wave. Phil. Mag. S., 32(194):45–58, 1891. L. Gonzalez-Vega and I. Necula. Efficient topology determination of implicitly defined algebraic plane curves. Computer Aided Geometric Design, 19(9):719–743, December 2002.

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