Algebraic geometry for shallow capillary-gravity waves
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1 Algebraic geometry for shallow capillary-gravity waves DENYS DUTYKH 1 Chargé de Recherche CNRS 1 Université de Savoie Laboratoire de Mathématiques (LAMA) Le Bourget-du-Lac France Seminar of Computational Technologies Institute of Computational Technologies, SB RAS October 21, 2014 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
2 Acknowledgements Collaborators: Didier CLAMOND: Professor (LJAD) Université de Nice Sophia Antipolis André GALLIGO: Emiritus Professor Université de Nice Sophia Antipolis DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
3 Outline 1 Shallow capillary-gravity waves Travelling waves Phase-space analysis 2 Extended phase-space analysis DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
4 Outline 1 Shallow capillary-gravity waves Travelling waves Phase-space analysis 2 Extended phase-space analysis DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
![Serre (Green Naghdi) equations Shallow water equations Credits: Lord RAYLEIGH (1876) [1] (only steady version) F.](/docs-images/94/119448937/images/5-1.jpg "SERRE (1953) [2] C. SU & C. GARDNER (1969) [3] A. GREEN & P. NAGHDI (1976) [4] E.")
![PELINOVSKY & ZHELEZNYAK (1985) [5] Modern derivations: Asymptotic methods: Green & Naghdi (1976) [4] Variational methods:](/docs-images/94/119448937/images/5-2.jpg "Particle description: Miles & Salmon (1985) [6] Eulerian description: Clamond & DD (2012) [7] DENYS DUTYKH (CNRS LAMA) Shallow")
5 Serre (Green Naghdi) equations Shallow water equations Credits: Lord RAYLEIGH (1876) [1] (only steady version) F. SERRE (1953) [2] C. SU & C. GARDNER (1969) [3] A. GREEN & P. NAGHDI (1976) [4] E. PELINOVSKY & ZHELEZNYAK (1985) [5] Modern derivations: Asymptotic methods: Green & Naghdi (1976) [4] Variational methods: Particle description: Miles & Salmon (1985) [6] Eulerian description: Clamond & DD (2012) [7] DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
6 Serre s equations with surface tension 1D case: the governing equations Governing equations (the mass conservation + momentum): h t + [ h ū] x = 0, [ ū t + ū ū x + g h x h 1 x h 2 γ ] = τ [ 1/2] h x (1+hx) 2 xx Vertical acceleration: γ = h(ū 2 x ū xt ūū xx ) = 2 h ū 2 x h[ū t + ū ū x ] x Conservative form: [ ] [ h ū] t + h ū g h h2 γ τ R = 0 x Surface tension: R = h h xx (1+h 2 x ) 3/2 + ( 1+h 2 x) 1/2, DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
7 Serre s equations: the conservation laws Momentum conservation: [ ] ] hū 1 3 (h3 ū x ) x [hū t gh h3 ūx h3 ūū xx h 2 h x ūū x R = 0 x Tangential velocity at the free surface: [ū (h 3 ū x ) [ x 3 h ]t + 1 2ū2 + gh 1 2 h2 ūx 2 ū(h3 ū x ) x 3 h Energy conservation: [ ] 1 2 h ū h3 ūx g h2 + τ 1+hx 2 [ ( 1 2ū h2 ū 2 x +gh+ 1 3 hγ τr h t + τ h xx ( 1+h 2 x ) 3/2 ]x = 0 ) h ū+τ ū 1+h 2 x + τ h h x ū x 1+h 2 x ]x = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
8 Serre CG equations: travelling waves Fr = c 2 /gd, Bo = τ/gd 2, We = Bo/Fr = τ/c 2 d Fr d h Mass conservation: ū = cd / h, d =< h >= 1 2l Momentum conservations lead: + h2 2 d 2 + γh2 3gd 2 Bo hh xx ( 1+h 2 x ) 3 2 Tangential velocity: Fr d 2 2 h 2 + h d + Fr d 2 h xx 3 h Integration constants: Bo ( 1+h 2 x ) 1 2 Fr d 2 h 2 x 6 h 2 Bo d h xx ( 1+h 2 x ) 3 2 γ/g = Fr d 3 h xx /h 2 Fr d 3 h 2 x/h 3 K 2 = (3+h 2 x) d 2 3 h 2 1 K Fr Bo = Fr d 2 h Bo d / h ( 1+h 2 x ) 1 2 l l h dx = Fr Bo + K 1 = Fr Fr K 2 2 / d. h DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
9 Serre CG equations: travelling waves Particular emphasis on solitary waves Combination of two equations: Fr d 2 h h2 2 d 2 Fr d h x 2 6 h Bo ( 1+h 2 x ) (Fr + 2+Fr K 2) h 2 d = Cnst Consider solitary waves (K 1 = K 2 0): F(h, h ) Fr h Bo h/d ( 1+h 2 ) 1 2 Fr + (2Fr + 1 2Bo)h d Property: h(x) h( x) (Fr + 2) h2 d 2 At the crest of a regular wave: h(0) = d + a, h (0) = 0 Fr = 1 + a/d + h3 d 3 = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
10 Serre CG equations: travelling waves Particular emphasis on solitary waves Combination of two equations: Fr d 2 h h2 2 d 2 Fr d h x 2 6 h Bo ( 1+h 2 x ) (Fr + 2+Fr K 2) h 2 d = Cnst Consider solitary waves (K 1 = K 2 0): F(h, h ) Fr h Bo h/d ( 1+h 2 ) 1 2 Fr + (2Fr + 1 2Bo)h d (Fr + 2) h2 d 2 + h3 d 3 = 0 We are interested in solutions: h( ) = d, h ( ) = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
11 Serre CG equation: Limiting cases I Pure gravity and pure capillary waves Pure gravity waves: Bo 0 h = d + a sech 2 (κx/2), (κd) 2 = 3 a/(d + a), Fr = 1 + a/d Pure capillary waves: Fr, Bo, We = Const h We h/d ( 1+h 2 ) (1 We)h d h2 d 2 = 0 In the limit We 0: Capillary wave equation reads: h = ± 3(1 h/d) Solitary wave with angular crest: ( h = d + a exp ) 3 x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
12 Serre CG equation: Limiting cases II Introduce a simplification to equations Small slope approximation [8]: ( ) 1+h h Master equation becomes: (Fr 2 3 Bo h d ) h 2 = Fr 2 (2Fr2 + 1) h d Change of independent variables: + (Fr 2 + 2) h 2 d 2 h3 d 3 dξ = 1 3 We h/d 1 2 dx ( ) Analytical solution ( pure gravity case): h(ξ) = d + a sech 2 (κξ/2) (κd) 2 = 3 a/(d + a), Fr = 1 + a/d x(ξ) can be found from ( ) DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
13 Asymptotic analysis Following MCCOWAN (1891) [9] Exponentially decaying solitary waves: h(x) d + a exp( κx), x +, κ > 0 Dispersion relation: Fr 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 = 1 3 Fr Algebraic decay: h(x) d + a(κx) α, x +, α > 1 Then necessarily = Fr 1 α = 1 = Bo 1 3, α > 2 = Bo = 1 3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
14 Phase-space analysis: local behaviour Nonlinear autonomous ODE analysis Two-parameter (Fr,Bo) family of real algebraic curves in R 2 : F Fr,Bo (h, k) := Fr 3 k 2 Bo h (1+k 2 ) 1 2 Bo +(2Fr 2Bo + 1)h (Fr + 2)h 2 + h 3 = 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 F Fr,Bo = 0 Find points with vertical tangent: h F Fr,Bo = 0 Decompose it into oriented branches (k 0, h րց 0) Study the family of algebraic curves F Fr,Bo (h, k) R 2 with certified topology methods [10] DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
15 Phase-space analysis: depression wave A particular example for Fr = 0.4, Bo = 0.9 > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
16 Phase-space analysis: depression wave A particular example for Fr = 0.4, Bo = 0.9 > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
17 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
18 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
19 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
20 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
21 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
22 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
23 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
24 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
25 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
26 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
27 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 = η(x, t) Free surface SW amplitude Bottom x DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
28 Phase-space analysis: peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
29 Phase-space analysis: peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
30 Phase-space analysis: multi-peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
31 Phase-space analysis: multi-peakon of depression A particular example for Fr = 0.4, Bo = 0.9 > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
32 Outline 1 Shallow capillary-gravity waves Travelling waves Phase-space analysis 2 Extended phase-space analysis DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
33 Phase space for roots of polynomials Geometric interpretation for quadratic equations Polynomial quadratic equation: x 2 + ax + b = 0 Coefficients as parameters: P : R 3 R (a, b, x) x 2 + ax + b Consider an algebraic surface S R 3 : (a, b, x) S P(a, b, x) = x 2 + ax + b = 0 Consider the fibers of the map: F : (a, b) x S, F(a, b) := { x (a, b, x) S } Count the cardinals: F(a, b) 0, 1, 2. DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
34 Corresponding algebraic surface Quadratic polynomial: x 2 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
35 Corresponding algebraic surface Quadratic polynomial: x 2 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
36 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 1 3 r a := s r 2 3 x 3 + ax + b = 0,, b := 2r 3 27 sr 3 + t. By following Girolamo CARDANO ( ), the discriminant is D 2 := b2 4 + a3 27 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
37 Corresponding algebraic surface Cubic polynomial: x 3 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
38 Corresponding algebraic surface Cubic polynomial: x 3 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
39 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) = a 0 x n +..., Q(x) = b 0 x m +... Resultant of two polynomials is R(P, Q) := a m 0 bn 0 n i=1 j=1 m (r i s j ) where r i are zeros of P(x) and s j 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) n(n 1) 2 R(P, P ) DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
40 Phase space analysis: global behaviour Detection of multiple points Points with horizontal tangent satisfy: F Fr,Bo (k, h) = 0 k F Fr,Bo (k, h) = 0 The 2nd equation can be solved analytically: k = 0 Fr(k 2 + 1) 3 = 9Bo 2 h 2 To avoid cubic roots, change of variables: k 2 = y 2 1, y 1 Wave height can be expressed as h = Fr 3Bo Y 3 Polynomial equation in y: f := Fr 2 y 9 (3Fr 2)Fr Bo y 6 + 9Bo 2 (1+2Fr 2Bo)y Bo 3 y 2 36Bo 3 Adapted tool to describe the real roots in (Fr, Bo) space: discriminant locus! D = (Fr 3Bo) 2 P 10 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
41 Phase space analysis: global behaviour Contains 11 cells with 0 to 3 real roots such as y > 1 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
42 Phase space analysis: global behaviour Contains 11 cells with 0 to 3 real roots such as y > 1 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
43 Phase-space analysis: weakly singular solitary wave A particular example for Fr = 0.8, Bo = > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
44 Phase-space analysis: weakly singular solitary wave A particular example for Fr = 0.8, Bo = > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
45 Phase-space analysis: wave with algebraic decay A particular example for Fr = 1.0, Bo = 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
46 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 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
47 Thank you for your attention! DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
48 References I J. W. S. Lord Rayleigh. On Waves. Phil. Mag., 1: , F. Serre. Contribution à l étude des écoulements permanents et variables dans les canaux. La Houille blanche, 8: , 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: , A. E. Green and P. M. Naghdi. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech., 78: , DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
49 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 Applied Physics Institute Press, Gorky, J. W. Miles and R. Salmon. Weakly dispersive nonlinear gravity waves. J. Fluid Mech., 157: , D. Clamond and D. Dutykh. Practical use of variational principles for modeling water waves. Phys. D, 241(1):25 36, F. Dias and P. Milewski. On the fully-nonlinear shallow-water generalized Serre equations. Phys. Lett. A, 374(8): , DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
50 References III J. McCowan. On the solitary wave. Phil. Mag. S., 32(194):45 58, L. Gonzalez-Vega and I. Necula. Efficient topology determination of implicitly defined algebraic plane curves. Computer Aided Geometric Design, 19(9): , December DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Novosibirsk, October 21, / 32
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