families of steady fully nonlinear shallow capillary ... · sir isaac newton (1642 – 1726) →...

Families of steady fully nonlinearshallow capillary-gravity waves

DENYS DUTYKH1

Charge de Recherche CNRS

1Universite Savoie Mont BlancLaboratoire de Mathematiques (LAMA)

73376 Le Bourget-du-Lac France

NUMERIWAVES SeminarBasque Center for Applied Mathematics (BCAM)

Otsaila 25, 2015

DENYS DUTYKH (CNRS – LAMA) Shallow capillary-gravity waves Bilbao, February 25, 2015 1 / 41

http://www.denys-dutykh.com/

Acknowledgements

Collaborators:

Didier CLAMOND: Professor (LJAD)Universite de Nice Sophia Antipolis

Andre GALLIGO: Emiritus ProfessorUniversite de Nice Sophia Antipolis


Outline

1 Epigraph

2 Physical problemGoverning equationsPhase-space analysis

3 Extended phase-space analysisCollision of rootsClassification of waves


EpigraphSir Isaac Newton (1642 – 1726) 7→ Gottfried Wilhelm Leibniz (1646 – 1716)

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 fluentesquantitae involvente fluxiones invenireet vice versa.”








Which can be translated into modern English as:

“It is useful to differentiate functionsand to solve differential equations.”








Which can be translated into modern English as:

“It is useful to differentiate functionsand to solve differential equations.”

We can study solutions without solving ODEs!


Outline

1 Epigraph




Capillary-gravity wavesDeep water case (for the sake of simplicity)

Consider the dispersion relation:

ω2 = k(

g +σ

ρk2

)

ω, k : wave frequency, wave numberg, σ: gravity acceleration, surface tension

Gravity wave regime (k ≪ 1):

ω2 = gk

Capillary wave regime (k ≫ 1):

ω2 =σ

ρk3

Capillary-gravity regime (k ∝ 1)



Phase velocity minimum:

λc = 2π√

σ

ρg

For the air-water interface:

λc ≈ 1.7 cm

In vicinity of λc both effects have to be taken into account!


Serre–(Green–Naghdi) equationsShallow 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]


Serre’s equations with surface tension1D case: the governing equations

Governing equations (mass conservation (⋆), momentum (∗)):

ht + [ h u ]x = 0, (⋆)

ut + u ux + g hx + 13 h−1 ∂x

[

h2 γ]

= τ[

hx

(

1 + h 2x

)

−1/2]

xx (∗)

Vertical acceleration:

γ ≡ h (u 2x − uxt − uuxx) = 2 h u 2

x − h [ ut + u ux ]x

Conservative form:

[ h u ]t +[

h u2 + 12g h2 + 1

3h2 γ − τ R]

x= 0

Surface tension (τ ≡ τρ ):

R = h hxx

(

1 + h 2x

)

−3/2+

(

1 + h 2x

)

−1/2,


Serre’s equations: the conservation laws

Momentum conservation:[

hu− 13(h

3ux)x]

t+[

hu2+ 12gh2− 1

32h3u 2x − 1

3h3uuxx −h2hx uux −R]

x= 0

Tangential velocity at the free surface:

[

u − (h3ux)x

3 h

]

t +[

12 u2 + gh − 1

2h2 u 2x − u (h3ux)x

3 h− τ hxx

(

1 + h 2x)3/2

]

x= 0

Energy conservation:[

12h u2 + 1

6h3 u 2x + 1

2g h2 + τ

√

1 + h 2x

]

t+

[

(12 u2+ 1

6h2 u 2x +gh+ 1

3hγ− τRh

)

h u+τ u√

1 + h 2x +

τ h hx ux√

1 + h 2x

]

x= 0


Serre–CG equations: travelling wavesFr 2

= c2/gd , Bo = τ/gd2, We = Bo / Fr = τ/c2d

Mass conservation: u = −cd / h, d =< h >= 12ℓ

∫ ℓ−ℓ h dx

Momentum conservations lead:Fr d

h+

h2

2 d2 +γh2

3gd2 − Bo hhxx(

1 + h 2x)

32

− Bo(

1 + h 2x)

12

= Fr +12− Bo + K1

Tangential velocity:Fr d2

2 h2 +hd+

Fr d2 hxx

3 h− Fr d2 h 2

x

6 h2 − Bo d hxx(

1 + h 2x)

32

=Fr2

+ 1 +Fr K2

2

γ/g = Fr d3hxx/h2 − Fr d3h2x/h3

Integration constants:

K2 =⟨ (3 + h 2

x ) d2

3 h2 − 1⟩

K1 + 12 + Fr − Bo =

⟨ Fr d2

h2 +12

− Bo d /h(

1 + h 2x)

12

⟩/

⟨ dh

⟩

.


Serre–CG equations: travelling wavesParticular emphasis on solitary waves

Combination of two equations:

Fr d2 h

− h2

2 d2 − Fr d h 2x

6 h− Bo

(

1 + h 2x)

12

+(Fr + 2 + Fr K2) h

2 d= Const

Consider solitary waves (K1 = K2 ≡ 0):

F (h, h′) ≡ Fr h′2

3+

2 Bo h/d(

1 + h′2)

12

− Fr +(2Fr + 1 − 2Bo ) h

d

− (Fr + 2) h2

d2 +h3

d3 = 0

Property (a discrete symmetry): h(x) ≡ h(−x)

At the crest of a regular wave: h(0) = d + a, h′(0) = 0Fr = 1 + a /d


Serre–CG equations: travelling wavesParticular emphasis on solitary waves

Combination of two equations:

Fr d2 h

− h2

2 d2 − Fr d h 2x

6 h− Bo

(

1 + h 2x)

12

+(Fr + 2 + Fr K2) h

2 d= Const

Consider solitary waves (K1 = K2 ≡ 0):

F (h, h′) ≡ Fr h′2

3+

2 Bo h/d(

1 + h′2)

12

− Fr +(2Fr + 1 − 2Bo ) h

d

− (Fr + 2) h2

d2 +h3

d3 = 0

We are interested in solutions: h(∞) = d , h′(∞) = 0


Implicit Ordinary Differential EquationsSome basic notions and theoretical facts [6]

Consider an implicit ODE (surface M ⊆ R3):

F (x , y , p) = 0, p :=dydx

.

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 x0

if and only ify(x)− z(x) = o

(

|x − x0|k)

Consider the projection map:

π : M → R2 (x , y , p) 7→ (x , y)

Definition:

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 OrdinaryDifferential Equations (p. 351). New York: Springer–Verlag.


Phase portrait of implicit ODEsThe set of critical points

Criminant and discriminant curves:


Normal form of an implicit ODE

Regular points (F ′

p(x0, y0, p0) 6= 0):

The normal form is simply: y ′ = f (x , y).




p(x0, y0, p0) 6= 0):


Theorem ([6])

Let (x0, y0, p0) be a regular critical pointa of the equationF (x , y , p) = 0. Then, there exists a diffeomorphism of aneighbourhood of the point (x0, y0) ∈ R

2 on a neighbourhood of thepoint (X ,Y ) = (0, 0) transforming locally the equation F (x , y , p) = 0 toP2 = X, where P = dY

dX .

arankD(F ,F ′

p)

D(x,y,p) = 2




p(x0, y0, p0) 6= 0):


Theorem ([6])

Let (x0, y0, p0) be a regular critical pointa of the equationF (x , y , p) = 0. Then, there exists a diffeomorphism of aneighbourhood of the point (x0, y0) ∈ R

2 on a neighbourhood of thepoint (X ,Y ) = (0, 0) transforming locally the equation F (x , y , p) = 0 toP2 = X, where P = dY

dX .

arankD(F ,F ′

p)

D(x,y,p) = 2

Remarks:

Rene THOM found “only” the normal form P2 = X E(X ,Y )

Yu. A. BRODSKY improved his proof to obtain P2 = X

=⇒ Integral curves are locally diffeomorphic to Y = 23X 3/2 + C


Phase portrait of implicit ODEs - IIBehaviour of integral curves near a regular critical point

Normal form analysis suggests:


Implicit ODEsA simple example - I

Consider the following implicit ODE:

F (x , y , p) = p2 − 1 =(dy

dx

)2− 1 = 0

It is equivalent to a union of two ODEs: dydx = −1 and dy

dx = 1

Remark:

We have two integral curves passingthrough any point of the phase plane!

To be compared with classical(explicit) ODEs


Implicit ODEsA simple example - II


F (x , y , p) = p2 − 1 =(dy

dx

)2− 1 = 0

Can we have non-smooth solutions?


Implicit ODEsA simple example - II


F (x , y , p) = p2 − 1 =(dy

dx

)2− 1 = 0

Can we have non-smooth solutions?

No, since there are no critical points!

F (x , y , p) = p2 − 1 = 0 =⇒ p = ±1

∂F ′

p(x , y , p) = 2p = 0


Serre–CG equation: integrability caseIntroduce a simplification to equations

Small slope approximation [7]:(

1 + h′2)

−12 ≈ 1 − 1

2 h′2 + . . .

Master equation becomes:

(Fr 2

3− Bo h

d

)

h′2 = Fr 2 − (2Fr2 + 1) hd

+(Fr 2 + 2) h2

d2 − h3

d3

Change of independent variables:

dξ = | 1 − 3 We h /d |−12 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 (⋆)


Asymptotic analysisFollowing MCCOWAN (1891) [8]

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 = 1

3 Fr

Algebraic decay:

h(x) ∼ d + a(κx)−α, x → +∞, α > 1

Then necessarily =⇒ Fr ≡ 1α = 1 =⇒ Bo 6= 1

3 , α > 2 =⇒ Bo = 13


Phase-space analysis: local behaviourNonlinear autonomous ODE analysis

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

FFr ,Bo (h, p) :=Fr3

p2 + 2Bo h

(1 + p2)12

+

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

With asymptotic behaviour at x → ∞ (p := h′):h(∞) = 1, p(∞) = 0

Typical workflow with a parametrized curve:

Find multiple points (with horizontal tangent): ∂pFFr ,Bo = 0

Find points with vertical tangent: ∂hFFr ,Bo = 0

Decompose it into oriented branches (p ≷ 0, h րց 0)

Study the family of algebraic curves FFr ,Bo (h, p) ∈ R2 with

certified topology methods [9]


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


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

−60 −40 −20 0 20 40 60

−1

−0.8

−0.6

−0.4

−0.2

0

x

η(x,t)

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

Free surface

SW amplitude

Bottom



−60 −40 −20 0 20 40 60

−1

−0.8

−0.6

−0.4

−0.2

0

x

η(x,t)


Free surface

SW amplitude

Bottom


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


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


Outline

1 Epigraph




Phase space for polynomial rootsGeometric interpretation for quadratic equations

Polynomial quadratic equation: x2 + ax + b = 0

Coefficients as parameters:

P : R3 7→ R

(a, b, x) 7→ x2 + ax + b

Consider an algebraic surface S ⊆ R3:

(a, b, x) ∈ S ⇐⇒ P(a, b, x) = x2 + 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.


Corresponding algebraic surfaceQuadratic polynomial: x2

+ ax + b = 0


Cubic equationsGeometric interpretation for cubic equations

Consider a generic cubic polynomial equation:

x3 + rx2 + sx + t = 0

Obtain the reduced form by substitution x → x − 13 r

x3 + ax + b = 0,

a := s − r2

3, b :=

2r3

27− sr

3+ t .

By following Girolamo CARDANO (1501 – 1576), the discriminant is

D2 :=b2

4+

a3

27

See Lodovico FERRARI (1522 – 1565) for the discriminant ofquartic polynomials.


Corresponding algebraic surfaceCubic polynomial: x3

+ ax + b = 0


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

Consider two polynomials P,Q ∈ R[x ]:

deg P = n, deg Q = m

P(x) = a0xn + . . . , Q(x) = b0xm + . . .

Resultant of two polynomials is

R(P,Q) := am0 bn

0

n∏

i=1

m∏

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)n(n−1)

2 R(P,P ′)


Some illustrative examplesThe concept of the discriminant for polynomials with one variable

Quadratic polynomials:

D(a2x2 + a1x + a0) = a21 − 4a2a0

Cubic polynomials:

D(a3x3 + a2x2 + a1x + a0) = a21a2

2 − 4a31a3−

4a0a32 − 27a2

0a23 + 18a0a1a2a3

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: Birkhauser.


Phase space analysis: global behaviourDetection of multiple points

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

The 2nd equation can be solved analytically:p = 0Fr (p2 + 1)3 = 9Bo 2h2

To avoid cubic roots, change of variables:p2 = y2 − 1, y ≥ 1Wave height can be expressed as h = Fr

3Bo Y 3

Polynomial equation in y :

f := Fr 2y9 − (3Fr − 2)Fr Bo y6+

9Bo 2(1 + 2Fr − 2Bo )y3 + 27Bo 3y2 − 36Bo 3

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


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


Phase-space analysis: weakly singular solitary waveA particular example for Fr = 1.01, Bo = 0.3334121494 > 1/3


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


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


Conclusions & Perspectives

Conclusions:

Capillary-gravity solitary waves were analyzed in shallow waterregimeFully 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-wavesGo to 3D !

Compute fully nonlinearlump-solitary waves


Thank you for your attention!




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 lescanaux.

La Houille blanche, 8:830–872, 1953.

C. H. Su and C. S. Gardner.

KdV equation and generalizations. Part III. Derivation of Korteweg-deVries 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.


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.

V. I. Arnold.

Geometrical Methods in the Theory of Ordinary Differential Equations.

Springer-Verlag, New York, 1996.

F. Dias and P. Milewski.

On the fully-nonlinear shallow-water generalized Serre equations.

Phys. Lett. A, 374(8):1049–1053, 2010.

J. McCowan.

On the solitary wave.

Phil. Mag. S., 32(194):45–58, 1891.


References III

L. Gonzalez-Vega and I. Necula.

Efficient topology determination of implicitly defined algebraic planecurves.

Computer Aided Geometric Design, 19(9):719–743, December 2002.

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky.

Discriminants, resultants and multidimensional determinants.

Birkhauser, Boston, 1994.


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