Families of steady fully nonlinear shallow capillary-gravity waves
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1 Families of steady fully nonlinear shallow capillary-gravity waves DENYS DUTYKH 1 Chargé de Recherche CNRS 1 Université Savoie Mont Blanc Laboratoire de Mathématiques (LAMA) Le Bourget-du-Lac France NUMERIWAVES Seminar Basque Center for Applied Mathematics (BCAM) Otsaila 25, 2015 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
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 Bilbao, February 25, / 41
3 Outline 1 Epigraph 2 Physical problem Governing equations Phase-space analysis 3 Extended phase-space analysis Collision of roots Classification of waves DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
4 Epigraph Sir Isaac Newton ( ) Gottfried Wilhelm Leibniz ( ) What is the most important Newton s discovery? DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
5 Epigraph Sir Isaac Newton ( ) Gottfried Wilhelm Leibniz ( ) What is the most important Newton s discovery? Newton communicated his discovery to Leibniz (via Oldenburg) in 1677: 6accdae13eff7i319n4o4qrr4s8t12ux DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
6 Epigraph Sir Isaac Newton ( ) Gottfried Wilhelm Leibniz ( ) What is the most important Newton s discovery? Newton communicated his discovery to Leibniz (via Oldenburg) in 1677: 6accdae13eff7i319n4o4qrr4s8t12ux This anagram encodes: Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
7 Epigraph Sir Isaac Newton ( ) Gottfried Wilhelm Leibniz ( ) What is the most important Newton s discovery? Newton communicated his discovery to Leibniz (via Oldenburg) in 1677: 6accdae13eff7i319n4o4qrr4s8t12ux This anagram encodes: Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. Which can be translated into modern English as: It is useful to differentiate functions and to solve differential equations. DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
8 Epigraph Sir Isaac Newton ( ) Gottfried Wilhelm Leibniz ( ) What is the most important Newton s discovery? Newton communicated his discovery to Leibniz (via Oldenburg) in 1677: 6accdae13eff7i319n4o4qrr4s8t12ux This anagram encodes: Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. Which can be translated into modern English as: It is useful to differentiate functions and to solve differential equations. We can study solutions without solving ODEs! DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
9 Outline 1 Epigraph 2 Physical problem Governing equations Phase-space analysis 3 Extended phase-space analysis Collision of roots Classification of waves DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
10 Capillary-gravity waves Deep water case (for the sake of simplicity) Consider the dispersion relation: ω 2 = k (g + σ ρ k 2) ω, k: wave frequency, wave number g, σ: gravity acceleration, surface tension Gravity wave regime (k 1): ω 2 = gk Capillary wave regime (k 1): Capillary-gravity regime (k 1) ω 2 = σ ρ k 3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
11 Capillary-gravity waves Deep water case (for the sake of simplicity) DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
12 Capillary-gravity waves Deep water case (for the sake of simplicity) Phase velocity minimum: For the air-water interface: σ λ c = 2π ρg λ c 1.7 cm In vicinity of λ c both effects have to be taken into account! DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
![(1876) [1] (only steady version) F. SERRE (1953) [2] C. SU & C. GARDNER (1969) [3] A. GREEN & P. NAGHDI (1976) [4] E.](/docs-images/82/85632205/images/13-2.jpg "PELINOVSKY & ZHELEZNYAK (1985) [5] DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, 2015 7")
13 Serre (Green Naghdi) equations Shallow water equations Main constitutive assumptions: Free surface is a graph: y = η(x, t) Long wave (or equivalently the water is shallow): λ d Nonlinearity is finite: ε a/d O(1) 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] DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
14 Serre s equations with surface tension 1D case: the governing equations Governing equations (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 (τ τ ρ ): R = h h xx (1+h 2 x ) 3/2 + ( 1+h 2 x) 1/2, DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
15 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 Bilbao, February 25, / 41
16 Serre CG equations: travelling waves Fr 2 = 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 Bilbao, February 25, / 41
17 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 = Const 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 Property (a discrete symmetry): h(x) h( x) 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 Bilbao, February 25, / 41
18 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 = Const 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 Bilbao, February 25, / 41
19 Implicit Ordinary Differential Equations Some basic notions and theoretical facts [6] Consider an implicit ODE (surface M R 3 ): F(x, y, p) = 0, p := dy dx. Coordinates (x, y, p) live in the 1-jet space: Two smooth functions y(x) and z(x) belong to the same k-jet in x 0 if and only if y(x) z(x) = o ( x x 0 k) Consider the projection map: Definition: π : M R 2 (x, y, p) (x, y) A point (x, y, p) M is regular if π is a diffeomorphism in this point. Reference [6]: Arnold, V. I. (1996). Geometrical Methods in the Theory of Ordinary Differential Equations (p. 351). New York: Springer Verlag. DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
20 Phase portrait of implicit ODEs The set of critical points Criminant and discriminant curves: DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
21 Normal form of an implicit ODE Regular points (F p(x 0, y 0, p 0 ) 0): The normal form is simply: y = f(x, y). DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
22 Normal form of an implicit ODE Regular points (F p(x 0, y 0, p 0 ) 0): The normal form is simply: y = f(x, y). Theorem ([6]) Let (x 0, y 0, p 0 ) be a regular critical point a of the equation F(x, y, p) = 0. Then, there exists a diffeomorphism of a neighbourhood of the point (x 0, y 0 ) R 2 on a neighbourhood of the point (X, Y) = (0, 0) transforming locally the equation F(x, y, p) = 0 to P 2 = X, where P = dy dx. a rank D(F,F p ) D(x,y,p) = 2 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
23 Normal form of an implicit ODE Regular points (F p(x 0, y 0, p 0 ) 0): The normal form is simply: y = f(x, y). Theorem ([6]) Let (x 0, y 0, p 0 ) be a regular critical point a of the equation F(x, y, p) = 0. Then, there exists a diffeomorphism of a neighbourhood of the point (x 0, y 0 ) R 2 on a neighbourhood of the point (X, Y) = (0, 0) transforming locally the equation F(x, y, p) = 0 to P 2 = X, where P = dy dx. a rank D(F,F p ) D(x,y,p) = 2 Remarks: René THOM found only the normal form P 2 = X E(X, Y) Yu. A. BRODSKY improved his proof to obtain P 2 = X = Integral curves are locally diffeomorphic to Y = 2 3 X 3/2 + C DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
24 Phase portrait of implicit ODEs - II Behaviour of integral curves near a regular critical point Normal form analysis suggests: DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
25 Implicit ODEs A simple example - I Consider the following implicit ODE: F(x, y, p) = p 2 1 = It is equivalent to a union of two ODEs: dy dx ( dy ) 2 1 = 0 dx = 1 and dy dx = 1 Remark: We have two integral curves passing through any point of the phase plane! To be compared with classical (explicit) ODEs DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
26 Implicit ODEs A simple example - II Consider the following implicit ODE: ( dy ) 2 F(x, y, p) = p 2 1 = 1 = 0 dx Can we have non-smooth solutions? DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
27 Implicit ODEs A simple example - II Consider the following implicit ODE: F(x, y, p) = p 2 1 = Can we have non-smooth solutions? No, since there are no critical points! ( dy ) 2 1 = 0 dx F(x, y, p) = p 2 1 = 0 = p = ±1 F p(x, y, p) = 2p = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
28 Serre CG equation: integrability case Introduce a simplification to equations Small slope approximation [7]: ( ) 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 Bilbao, February 25, / 41
29 Asymptotic analysis Following MCCOWAN (1891) [8] 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 Bilbao, February 25, / 41
30 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, p) := Fr Bo h 3 p (1+p 2 ) 1 2 Bo +(2Fr 2Bo + 1)h (Fr + 2)h 2 + h 3 = 0 With asymptotic behaviour at x (p := h ): h( ) = 1, p( ) = 0 Typical workflow with a parametrized curve: Find multiple points (with horizontal tangent): p F Fr,Bo = 0 Find points with vertical tangent: h F Fr,Bo = 0 Decompose it into oriented branches (p 0, h րց 0) Study the family of algebraic curves F Fr,Bo (h, p) R 2 with certified topology methods [9] DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
31 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 Bilbao, February 25, / 41
32 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 Bilbao, February 25, / 41
33 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 Bilbao, February 25, / 41
34 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 Bilbao, February 25, / 41
35 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 Bilbao, February 25, / 41
36 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 Bilbao, February 25, / 41
37 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 Bilbao, February 25, / 41
38 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 Bilbao, February 25, / 41
39 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 Bilbao, February 25, / 41
40 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 Bilbao, February 25, / 41
41 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 Bilbao, February 25, / 41
42 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 Bilbao, February 25, / 41
43 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 Bilbao, February 25, / 41
44 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 Bilbao, February 25, / 41
45 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 Bilbao, February 25, / 41
46 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 Bilbao, February 25, / 41
47 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 Bilbao, February 25, / 41
48 Outline 1 Epigraph 2 Physical problem Governing equations Phase-space analysis 3 Extended phase-space analysis Collision of roots Classification of waves DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
49 Phase space for polynomial roots 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 Bilbao, February 25, / 41
50 Corresponding algebraic surface Quadratic polynomial: x 2 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
51 Corresponding algebraic surface Quadratic polynomial: x 2 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
52 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 See Lodovico FERRARI ( ) for the discriminant of quartic polynomials. DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
53 Corresponding algebraic surface Cubic polynomial: x 3 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
54 Corresponding algebraic surface Cubic polynomial: x 3 + ax + b = 0 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
55 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 Bilbao, February 25, / 41
56 Some illustrative examples The concept of the discriminant for polynomials with one variable Quadratic polynomials: D(a 2 x 2 + a 1 x + a 0 ) = a1 2 4a 2a 0 Cubic polynomials: D(a 3 x 3 + a 2 x 2 + a 1 x + a 0 ) = a1 2 a2 2 4a3 1 a 3 4a 0 a2 3 27a2 0 a a 0a 1 a 2 a 3 Discriminant depends on the degree bound! Reference [10]: Gelfand, I. M., Kapranov, M. M., & Zelevinsky, A. V. (1994). Discriminants, resultants and multidimensional determinants (p. 516). Boston: Birkhäuser. DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
57 Phase space analysis: global behaviour Detection of multiple points Points with horizontal tangent satisfy: F Fr,Bo (p, h) = 0 p F Fr,Bo (p, h) = 0 The 2 nd equation can be solved analytically: p = 0 Fr(p 2 + 1) 3 = 9Bo 2 h 2 To avoid cubic roots, change of variables: p 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 Bilbao, February 25, / 41
58 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 Bilbao, February 25, / 41
59 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 Bilbao, February 25, / 41
60 Phase-space analysis: weakly singular solitary wave A particular example for Fr = 1.01, Bo = > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
61 Phase-space analysis: weakly singular solitary wave A particular example for Fr = 1.01, Bo = > 1/3 DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
62 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 Bilbao, February 25, / 41
63 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 Bilbao, February 25, / 41
64 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 Bilbao, February 25, / 41
65 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 Bilbao, February 25, / 41
66 Thank you for your attention! DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
67 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 Bilbao, February 25, / 41
68 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, V. I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York, F. Dias and P. Milewski. On the fully-nonlinear shallow-water generalized Serre equations. Phys. Lett. A, 374(8): , J. McCowan. On the solitary wave. Phil. Mag. S., 32(194):45 58, DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
69 References III L. Gonzalez-Vega and I. Necula. Efficient topology determination of implicitly defined algebraic plane curves. Computer Aided Geometric Design, 19(9): , December I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, resultants and multidimensional determinants. Birkhäuser, Boston, DENYS DUTYKH (CNRS LAMA) Shallow capillary-gravity waves Bilbao, February 25, / 41
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