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Traveling surface waves of moderate amplitude in shallow water

Anna Geyer

Universitat Autonòma de Barcelona, Spain

SIAM Conference on Analysis of PDE December 2013

joint work with

Armengol Gasull

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Setting

Water

is inviscid has constant density is incompressible.

Gravity water waves Irrotational flow Euler’s equation: ut + (u · ∇)u = −∇P Mass conservation Boundary conditions

u v y x 2a λ h0 Ω = {(x, y)| − h0 < y < η(x, t)} fluid domain free surface η

δ = h0

λ . . . shallowness,

ε = a

h0 . . . amplitude

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Shallow water waves

δ ≪ 1, ε = O(δ2): small amplitude. Korteweg–DeVries equation ut + uux + uxxx = 0

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Shallow water waves

δ ≪ 1, ε = O(δ2): small amplitude. Korteweg–DeVries equation ut + uux + uxxx = 0 δ ≪ 1, ε = O(δ): moderate amplitude. Camassa–Holm equation ut + utxx + 3uux + 2ωux = 2uxuxx + uuxxx

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Shallow water waves

δ ≪ 1, ε = O(δ2): small amplitude. Korteweg–DeVries equation ut + uux + uxxx = 0 δ ≪ 1, ε = O(δ): moderate amplitude. Camassa–Holm equation ut + utxx + 3uux + 2ωux = 2uxuxx + uuxxx Equation for the free surface1 ut + ux + 6uux − 6u2ux + 12u3ux − utxx + uxxx = −28uxuxx − 14uuxxx,

1 A. Constantin, D. Lannes, The hydrodynamical relevance of the

Camassa-Holm and Degasperis-Procesi equations, Arch. Ration.

• Mech. Anal. (2009).

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Traveling Wave Solutions

With the Ansatz u(x, t) = u(x − ct) we obtain traveling waves unidirectional propagation at constant speed c fixed shape. c − → A solitary traveling wave decays to a constant at infinity.

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Traveling Wave Solutions

With the Ansatz u(x, t) = u(x − ct) we obtain traveling waves unidirectional propagation at constant speed c fixed shape. c − → A solitary traveling wave decays to a constant at infinity. For traveling waves, our equation for surface waves is ((1 − c) + 6u − 6u2 + 12u3)u′ + (1 + c)u′′′ + 14uu′′′ + 28u′u′′ = 0 Goal: Existence and Properties of traveling waves

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Hamiltonian System

u′′ (u + 1 + c 14 ) + 1 2(u′)2 + K + (1 − c)u + 3u2 − 2u3 + 3u4 = 0.

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Hamiltonian System

u′′ (u + 1 + c 14 ) + 1 2(u′)2 + K + (1 − c)u + 3u2 − 2u3 + 3u4 = 0. An autonomous ODE of the form u′′ (u − ¯ u) + 1 2(u′)2 + F ′(u) = 0

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Hamiltonian System

u′′ (u + 1 + c 14 ) + 1 2(u′)2 + K + (1 − c)u + 3u2 − 2u3 + 3u4 = 0. An autonomous ODE of the form u′′ (u − ¯ u) + 1 2(u′)2 + F ′(u) = 0 can be written as the planar system      u′ = v v′ = −F ′(u) − 1

2 v2

u − ¯ u

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Hamiltonian System

u′′ (u + 1 + c 14 ) + 1 2(u′)2 + K + (1 − c)u + 3u2 − 2u3 + 3u4 = 0. An autonomous ODE of the form u′′ (u − ¯ u) + 1 2(u′)2 + F ′(u) = 0 is topologically equivalent to the system u′ = (u − ¯ u) v v′ = −F ′(u) − 1

2 v2

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Hamiltonian System

u′′ (u + 1 + c 14 ) + 1 2(u′)2 + K + (1 − c)u + 3u2 − 2u3 + 3u4 = 0. An autonomous ODE of the form u′′ (u − ¯ u) + 1 2(u′)2 + F ′(u) = 0 is topologically equivalent to the Hamiltonian system u′ = (u − ¯ u) v = Hv v′ = −F ′(u) − 1

2 v2

= −Hu with Hamiltonian H(u, v) = F(u) + 1 2 v2 (u − ¯ u) = h.

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Orbits in the phase plane

det(J) = F ′′(u)(u − ¯ u) u′ = (u − ¯ u) v v′ = −F ′(u) − 1

2 v2

• v2 = 2 h − F(u)

u − ¯ u F(u) ¯ u hs hp u u′ = v m uc s m s

F(u) = K u + 1 − c 28 u2 + 1 14u3 − 1 28u4 + 3 70u5, ¯ u = −1 + c 14 .

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Conditions for existence of solitary waves

Homoclinic orbits occur when there is 1 center and 1 saddle in the phase plane (u, v), i.e. when (C1) F has two distinct local extrema (C2) both lie to the left/right of ¯ u ⇒ F ′′(s) = 0 ⇒ s = ¯ u s c (C2) (C1)

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Solitary waves with compact support

On (C2): Existence time of the homoclinic orbit is finite: F(u) hs u ¯ u m

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Solitary waves with compact support

On (C2): Existence time of the homoclinic orbit is finite: F(u) hs u ¯ u m v2 = 2 hs − F(u) u − ¯ u = (u − ¯ u) p (u) T = m

¯ u

dr

• (r − ¯

u) p(r) < ∞. m ¯ u There exists a C2-extension to R: u = ¯ u for ξ ∈ R\(−T, T)

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Dependence of amplitude on speed

The amplitude a = m − s changes with speed c like d dc a = −6/7 F ′(m) F ′′(s)

• (s2 − m2) F ′′(s) + 2s F ′(m)
• Anna Geyer

(Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Dependence of amplitude on speed

The amplitude a = m − s changes with speed c like d dc a = −6/7 F ′(m) F ′′(s)

• (s2 − m2) F ′′(s) + 2s F ′(m)
• c

s

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Dependence of amplitude on speed

The amplitude a = m − s changes with speed c like d dc a = −6/7 F ′(m) F ′′(s)

• (s2 − m2) F ′′(s) + 2s F ′(m)
• decreasing

increasing decreasing increasing c s

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Peaked periodic waves

F(u) hp = F(¯ u) ¯ u v2 v1 m2 m1 v2

i = (u − m1)(u − m2) q(u)

T2 T1

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Peaked periodic waves

F(u) hp = F(¯ u) ¯ u v2 v1 m2 m1 v2

i = (u − m1)(u − m2) q(u)

Ti If T1 = T2: smooth periodic solutions undulating about u = ¯ u.

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Traveling Waves - Results

For the equation of surface waves of moderate amplitude we have studied traveling waves u(x − ct) and found that: Theorem (Gasull & G. ’13) Existence of smooth and peaked periodic waves Existence of solitary traveling waves

• f elevation and depression

with compact support

Properties:

Symmetry and exponential decay at infinity Monotonicity Amplitude increases/decreases with speed

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water

Traveling Waves - Results

For the equation of surface waves of moderate amplitude we have studied traveling waves u(x − ct) and found that: Theorem (Gasull & G. ’13) Existence of smooth and peaked periodic waves Existence of solitary traveling waves

• f elevation and depression

with compact support

Properties:

Symmetry and exponential decay at infinity Monotonicity Amplitude increases/decreases with speed

Thank you for your attention!

Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water