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 Euler’s equation: is inviscid u t + ( u · ∇ ) u = −∇ P has constant density is incompressible. Mass conservation Gravity water waves Boundary conditions Irrotational flow y free surface η 2 a λ h 0 fluid domain v Ω = { ( x , y ) | − h 0 < y < η ( x , t ) } u x δ = h 0 ε = a λ . . . shallowness, h 0 . . . amplitude Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water
Shallow water waves δ ≪ 1 , ε = O ( δ 2 ) : small amplitude . Korteweg–DeVries equation u t + uu x + u xxx = 0 Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water
Shallow water waves δ ≪ 1 , ε = O ( δ 2 ) : small amplitude . Korteweg–DeVries equation u t + uu x + u xxx = 0 δ ≪ 1 , ε = O ( δ ) : moderate amplitude . Camassa–Holm equation u t + u txx + 3 uu x + 2 ω u x = 2 u x u xx + uu xxx Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water
Shallow water waves δ ≪ 1 , ε = O ( δ 2 ) : small amplitude . Korteweg–DeVries equation u t + uu x + u xxx = 0 δ ≪ 1 , ε = O ( δ ) : moderate amplitude . Camassa–Holm equation u t + u txx + 3 uu x + 2 ω u x = 2 u x u xx + uu xxx Equation for the free surface 1 u t + u x + 6 uu x − 6 u 2 u x + 12 u 3 u x − u txx + u xxx = − 28 u x u xx − 14 uu xxx , 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 c − → at constant speed c fixed shape. 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 c − → at constant speed c fixed shape. A solitary traveling wave decays to a constant at infinity. For traveling waves, our equation for surface waves is (( 1 − c ) + 6 u − 6 u 2 + 12 u 3 ) u ′ + ( 1 + c ) u ′′′ + 14 uu ′′′ + 28 u ′ 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 + 3 u 2 − 2 u 3 + 3 u 4 = 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 + 3 u 2 − 2 u 3 + 3 u 4 = 0 . An autonomous ODE of the form u ) + 1 u ′′ ( u − ¯ 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 + 3 u 2 − 2 u 3 + 3 u 4 = 0 . An autonomous ODE of the form u ) + 1 u ′′ ( u − ¯ 2 ( u ′ ) 2 + F ′ ( u ) = 0 can be written as the planar system u ′ = v v ′ = − F ′ ( u ) − 1 2 v 2 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 + 3 u 2 − 2 u 3 + 3 u 4 = 0 . An autonomous ODE of the form u ) + 1 u ′′ ( u − ¯ 2 ( u ′ ) 2 + F ′ ( u ) = 0 is topologically equivalent to the system � u ′ = ( u − ¯ u ) v v ′ = − F ′ ( u ) − 1 2 v 2 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 + 3 u 2 − 2 u 3 + 3 u 4 = 0 . An autonomous ODE of the form u ) + 1 u ′′ ( u − ¯ 2 ( u ′ ) 2 + F ′ ( u ) = 0 is topologically equivalent to the Hamiltonian system � u ′ = ( u − ¯ u ) v = H v v ′ = − F ′ ( u ) − 1 2 v 2 = − H u with Hamiltonian H ( u , v ) = F ( u ) + 1 2 v 2 ( u − ¯ u ) = h . Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water
Orbits in the phase plane � u ′ = ( u − ¯ u ) v � det ( J ) = F ′′ ( u )( u − ¯ v 2 = 2 h − F ( u ) u ) v ′ = − F ′ ( u ) − 1 2 v 2 u − ¯ u u ′ = v F ( u ) m h s h p s u ¯ u s u c m F ( u ) = K u + 1 − c 28 u 2 + 1 14 u 3 − 1 28 u 4 + 3 u = − 1 + c 70 u 5 , ¯ 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 ⇒ F ′′ ( s ) = 0 (C2) both lie to the left/right of ¯ u ⇒ s = ¯ u s c (C1) (C2) 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 ) h s u ¯ m u 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: v 2 = 2 h s − F ( u ) F ( u ) u − ¯ u = ( u − ¯ u ) p ( u ) h s � m d r T = < ∞ . � ( r − ¯ u ) p ( r ) ¯ u m u ¯ m u ¯ u There exists a C 2 -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 − 6 / 7 d � ( s 2 − m 2 ) F ′′ ( s ) + 2 s F ′ ( m ) � d c a = F ′ ( m ) F ′′ ( 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 − 6 / 7 d � ( s 2 − m 2 ) F ′′ ( s ) + 2 s F ′ ( m ) � d c a = F ′ ( m ) F ′′ ( s ) s c 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 − 6 / 7 d � ( s 2 − m 2 ) F ′′ ( s ) + 2 s F ′ ( m ) � d c a = F ′ ( m ) F ′′ ( s ) s c decreasing decreasing increasing increasing Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water
Peaked periodic waves ¯ u F ( u ) h p = F (¯ u ) v 2 i = ( u − m 1 )( u − m 2 ) q ( u ) v 2 v 1 T 1 m 2 m 1 T 2 Anna Geyer (Universitat Autonòma de Barcelona) Traveling surface waves of moderate amplitude in shallow water
Peaked periodic waves ¯ u F ( u ) h p = F (¯ u ) v 2 i = ( u − m 1 )( u − m 2 ) q ( u ) v 2 v 1 T i m 2 m 1 If T 1 = T 2 : 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 of 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 of 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
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