Mathematical Aspects of Classical Water Wave Theory from the Past 20 Year

Transcription

1 T. Brooke Benjamin... a brilliant researcher... original and elegant applied mathematics... extraordinary physical insights... changed the way we now think about hydrodynamics... about 6ft tall... moved slowly... smiled slightly... often smoking a pipe... every word counted he and his wife Natalia were a joyous couple with many friends...

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5 Mathematical Aspects of Classical Water Wave Theory from the Past 20 Year - the fascination of what s difficult - Brooke Benjamin Lecture Oxford 26th November 2013

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7 Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface

8 Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface,

9 Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies

10 Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies Although there has been important recent progress on the initial-value problem for free boundaries and on fully 3-dimensional waves, that is not the concern here.

11 Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies Although there has been important recent progress on the initial-value problem for free boundaries and on fully 3-dimensional waves, that is not the concern here. 2-Dimensional Irrotational Water Waves:

12 Classical Water Waves What distinguishes the water wave problems from others in hydrodynamics is that the unknown is the region occupied by the fluid. It is referred to as the free surface For today Classical means two-dimensional irrotational flow under gravity of a heavy perfect liquid with a free surface, There is no surface tension and there are no floating bodies Although there has been important recent progress on the initial-value problem for free boundaries and on fully 3-dimensional waves, that is not the concern here. 2-Dimensional Irrotational Water Waves: In Eulerian co ordinates the velocity at a point (x, y, z) in the fluid at time t is given by the gradient of a scalar potential φ on R 2 v(x, y, z; t) = φ(x, y; t) which satisfy

13 Irrotational Water Waves Infinite depth Wave interior Ω = {(x, y) : y < η(x, t)} φ(x, y; t) = 0 φ(x, y; t) 0 as y

14 Irrotational Water Waves Boundary Conditions gravity g acts vertically down Wave Surface S = {(x,η(x, t)) : x R} φ t φ 2 + gy = 0 η t +φ x η x φ y = 0 } on S

15 This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh...

16 This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described

17 This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations

18 This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations... since no one had predicted the wave it remained for mathematicians to give an a priori demonstration a posteriori

19 This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations... since no one had predicted the wave it remained for mathematicians to give an a priori demonstration a posteriori Progress was very slow throughout the 19th century with approximation for certain scaling limits but the full equations remained essentially unexamined for more than one hundred years but progress had accelerated hugely in recent years

20 This is a very old problem, studied by Cauchy, Laplace, Lagrange, Poisson, Green, Airy, Stokes, Rayleigh... After he discovered experimentally the solitary wave, in 1844 John Scott Russell described the greater part of the investigations of Poisson and Cauchy under the name of wave theory are rather to be regarded as mathematical exercises, not physical investigations... since no one had predicted the wave it remained for mathematicians to give an a priori demonstration a posteriori Progress was very slow throughout the 19th century with approximation for certain scaling limits but the full equations remained essentially unexamined for more than one hundred years but progress had accelerated hugely in recent years Alex Craik, a PhD student of Brooke s, has written extensively on the history of the water wave problem

21 Energetics The total wave energy at time t is Kinetic + Potential: 1 2 η(x,t) φ(x, y; t)dy 2 dx + g η 2 (x; t)dx 2

22 Energetics The total wave energy at time t is Kinetic + Potential: 1 η(x,t) φ(x, y; t)dy 2 dx + g η 2 (x; t)dx 2 2 Given periodic functions η and Φ of the single variable x let Ω = {(x, y) : y < η(x)}

23 Energetics The total wave energy at time t is Kinetic + Potential: 1 η(x,t) φ(x, y; t)dy 2 dx + g η 2 (x; t)dx 2 2 Given periodic functions η and Φ of the single variable x let Ω = {(x, y) : y < η(x)} φ the solution of the corresponding Dirichlet problem φ(x, y) = 0 φ 0 as y φ(x,η(x)) = Φ(x) } on Ω

24 Energetics The total wave energy at time t is Kinetic + Potential: 1 η(x,t) φ(x, y; t)dy 2 dx + g η 2 (x; t)dx 2 2 Given periodic functions η and Φ of the single variable x let Ω = {(x, y) : y < η(x)} φ the solution of the corresponding Dirichlet problem φ(x, y) = 0 φ 0 as y φ(x,η(x)) = Φ(x) } on Ω let E(η,Φ) = 1 η(x) φ(x, y)dy 2 dx + gη 2 (x)dx 2

25 With this functional E(η,Φ) := 1 η(x) φ(x, y)dy 2 dx + gη 2 (x)dx 2 and with the variational derivatives E E and Φ η

26 With this functional E(η,Φ) := 1 η(x) φ(x, y)dy 2 dx + gη 2 (x)dx 2 and with the variational derivatives E E and Φ η Zakharov (1968) observed that solutions (η,φ) of η t = E Φ (η,φ); Φ t = E η (η,φ) yields a water wave

27 Benjamin & Olver (1982) studied η t = E Φ (η,φ); Φ t = E η (η,φ) as a Hamiltonian system of classical type ẋ = J E(x), x = (η,φ), J = ( 0, I I, 0 η, φ being the infinite dimensional canonical variables which they referred to as coordinates and momentum and in an Appendix gives the Hamiltonian formulation independent of coordinates. Both Zakharov and Benjamin was conscious of the implications of the Hamiltonian formulation for stability ),

28 Spatially Periodic Waves Normalised Period 2π

29 Spatially Periodic Waves Normalised Period 2π Based on conformal mapping theory any rectifiable periodic Jordan curve S = {(x, η(x)) : x R} can be re-parametrised as S = {( ξ Cw(ξ), w(ξ)) : ξ R} where Cw is the Hilbert transform of a periodic function w: Cw(ξ) = pv 1 2π π π w(σ) dσ tan 1 (ξ σ) 2

30 Spatially Periodic Waves Normalised Period 2π Based on conformal mapping theory any rectifiable periodic Jordan curve S = {(x, η(x)) : x R} can be re-parametrised as S = {( ξ Cw(ξ), w(ξ)) : ξ R} where Cw is the Hilbert transform of a periodic function w: Cw(ξ) = pv 1 2π π π w(σ) dσ tan 1 (ξ σ) 2 Dyachenko, Kuznetsov, Spector & Zakharov (1996) used complex function theory in a deep and beautiful way to reduce Zakharov s awkward system to the following simple system: ẇ(1+cw ) Cϕ w Cẇ = 0 C ( w ϕ ẇϕ +λww ) +( ϕ+λw)(1+cw ) ϕ Cẇ = 0 = / t, = / x w = wave height ϕ = potential at surface: 0 < λ = gravity after normalisating the wavelength as 2π

31 Spatially Periodic Waves Normalised Period 2π Based on conformal mapping theory any rectifiable periodic Jordan curve S = {(x, η(x)) : x R} can be re-parametrised as S = {( ξ Cw(ξ), w(ξ)) : ξ R} where Cw is the Hilbert transform of a periodic function w: Cw(ξ) = pv 1 2π π π w(σ) dσ tan 1 (ξ σ) 2 Dyachenko, Kuznetsov, Spector & Zakharov (1996) used complex function theory in a deep and beautiful way to reduce Zakharov s awkward system to the following simple system: ẇ(1+cw ) Cϕ w Cẇ = 0 C ( w ϕ ẇϕ +λww ) +( ϕ+λw)(1+cw ) ϕ Cẇ = 0 = / t, = / x w = wave height ϕ = potential at surface: 0 < λ = gravity after normalisating the wavelength as 2π But it does not look like a Hamiltonian system any more!

32 A Symplectic Form For (w,ϕ) M := W 1,2 2π W 1,2 2π let ω (w,ϕ) ( (w1,ϕ 1 ),(w 2,ϕ 2 ) ) = π (1+Cw )(ϕ 2 w 1 ϕ 1 w 2 ) π + w ( ) ϕ 1 Cw 2 ϕ 2 Cw 1 ϕ ( w 1 Cw 2 w 2 Cw 1 ) dξ

33 A Symplectic Form For (w,ϕ) M := W 1,2 2π W 1,2 2π let ω (w,ϕ) ( (w1,ϕ 1 ),(w 2,ϕ 2 ) ) = π (1+Cw )(ϕ 2 w 1 ϕ 1 w 2 ) π + w ( ) ϕ 1 Cw 2 ϕ 2 Cw 1 This is a skew-symmetric bilinear form with where ω is the 1-form ϖ ϕ,w (ŵ, ˆϕ) = π π ω = dϖ ϕ ( w 1 Cw 2 w 2 Cw 1 ) dξ { ϕ(1+cw )+C ( ϕw )} ŵdξ

34 A Symplectic Form For (w,ϕ) M := W 1,2 2π W 1,2 2π let ω (w,ϕ) ( (w1,ϕ 1 ),(w 2,ϕ 2 ) ) = π (1+Cw )(ϕ 2 w 1 ϕ 1 w 2 ) π + w ( ) ϕ 1 Cw 2 ϕ 2 Cw 1 This is a skew-symmetric bilinear form with where ω is the 1-form ϖ ϕ,w (ŵ, ˆϕ) = π π ω = dϖ Hence ω is exact (and so closed) ϕ ( w 1 Cw 2 w 2 Cw 1 ) dξ { ϕ(1+cw )+C ( ϕw )} ŵdξ From Riemann-Hilbert theory it is non-degenerate

35 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain:

36 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain: S = {(x,η(x, t)) : x R} η(x + 2π) = η(x, t) = η(x, t + 2π) } φ t φ 2 +λy = 0; on S φ t +φ x η x φ y η y = 0 For (x, y; t) with y < η(x, t) φ(x, y; t) = 0 φ(x, y; t) 0 as y φ(x, y; t) = φ(x + 2π, y; t)

37 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain: S = {(x,η(x, t)) : x R} η(x + 2π) = η(x, t) = η(x, t + 2π) } φ t φ 2 +λy = 0; on S φ t +φ x η x φ y η y = 0 For (x, y; t) with y < η(x, t) φ(x, y; t) = 0 φ(x, y; t) 0 as y φ(x, y; t) = φ(x + 2π, y; t) is equivalent to:

38 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain: S = {(x,η(x, t)) : x R} η(x + 2π) = η(x, t) = η(x, t + 2π) } φ t φ 2 +λy = 0; on S φ t +φ x η x φ y η y = 0 is equivalent to: two equations For (x, y; t) with y < η(x, t) φ(x, y; t) = 0 φ(x, y; t) 0 as y φ(x, y; t) = φ(x + 2π, y; t)

39 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain: S = {(x,η(x, t)) : x R} η(x + 2π) = η(x, t) = η(x, t + 2π) } φ t φ 2 +λy = 0; on S φ t +φ x η x φ y η y = 0 For (x, y; t) with y < η(x, t) φ(x, y; t) = 0 φ(x, y; t) 0 as y φ(x, y; t) = φ(x + 2π, y; t) is equivalent to: two equations each with quadratic nonlinearities

40 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain: S = {(x,η(x, t)) : x R} η(x + 2π) = η(x, t) = η(x, t + 2π) } φ t φ 2 +λy = 0; on S φ t +φ x η x φ y η y = 0 For (x, y; t) with y < η(x, t) φ(x, y; t) = 0 φ(x, y; t) 0 as y φ(x, y; t) = φ(x + 2π, y; t) is equivalent to: two equations each with quadratic nonlinearities for two real-valued functions

41 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain: S = {(x,η(x, t)) : x R} η(x + 2π) = η(x, t) = η(x, t + 2π) } φ t φ 2 +λy = 0; on S φ t +φ x η x φ y η y = 0 For (x, y; t) with y < η(x, t) φ(x, y; t) = 0 φ(x, y; t) 0 as y φ(x, y; t) = φ(x + 2π, y; t) is equivalent to: two equations each with quadratic nonlinearities for two real-valued functions of one space and one time variable

42 Thus the Water-Wave Problem involving a PDE with nonlinear boundary conditions on an unknown domain: S = {(x,η(x, t)) : x R} η(x + 2π) = η(x, t) = η(x, t + 2π) } φ t φ 2 +λy = 0; on S φ t +φ x η x φ y η y = 0 For (x, y; t) with y < η(x, t) φ(x, y; t) = 0 φ(x, y; t) 0 as y φ(x, y; t) = φ(x + 2π, y; t) is equivalent to: two equations each with quadratic nonlinearities for two real-valued functions of one space and one time variable on a fixed domain:

43 The Hamiltonian System defined by E(w,ϕ) = 1 2 with the skew form ( ω (w,ϕ) (w1,ϕ 1 ),(w 2,ϕ 2 ) ) = π π π ˆϕCϕ +λw 2 (1+Cw ) dξ (1+Cw )(ϕ 2 w 1 ϕ 1 w 2 ) π + w ( ) ϕ 1 Cw 2 ϕ 2 Cw 1 ϕ ( w 1 Cw 2 w 2 Cw 1 ) dξ for x-periodic functions (φ(x, t), w(x, t)) of real variables leads to the equations ẇ(1+cw ) Cϕ w Cẇ = 0 C ( w ϕ ẇϕ +λww ) +( ϕ+λw)(1+cw ) ϕ Cẇ = 0 a tidy version of the water-wave problem!!

44 Standing Waves S π π

45 Standing Waves S π π The first person to consider the initial-value problem for water waves was Siméon Denis Poisson ( ) In the process he considered the standing waves le clapotis he called them which offer a good example of how the Hamiltonian approach helps organise a fiendishly difficult problem

46 Linearized Standing Wave Problem Standing Waves have normalised spatial period 2π and temporal period T

47 Linearized Standing Wave Problem Standing Waves have normalised spatial period 2π and temporal period T The velocity potential φ on the lower half plane {(x, t) : R 2 : y < 0}: 2 φ x + 2 φ = 0, x, t R, y < 0, 2 y2

48 Linearized Standing Wave Problem Standing Waves have normalised spatial period 2π and temporal period T The velocity potential φ on the lower half plane {(x, t) : R 2 : y < 0}: 2 φ x + 2 φ = 0, x, t R, y < 0, 2 y2 Boundary Conditions φ(x + 2π, y; t) = φ(x, y; t) = φ(x, y; t + T), x, t R, y < 0, 2 φ t 2 + g φ y = 0, y = 0 φ( x, y; t) = φ(x, y; t) = φ(x, y; t), x, t R, y < 0 φ(x, y; t) (0, 0), y

49 Linearized Standing Wave Problem Standing Waves have normalised spatial period 2π and temporal period T The velocity potential φ on the lower half plane {(x, t) : R 2 : y < 0}: 2 φ x + 2 φ = 0, x, t R, y < 0, 2 y2 Boundary Conditions φ(x + 2π, y; t) = φ(x, y; t) = φ(x, y; t + T), x, t R, y < 0, 2 φ t 2 + g φ y = 0, y = 0 φ( x, y; t) = φ(x, y; t) = φ(x, y; t), x, t R, y < 0 φ(x, y; t) (0, 0), The wave Elevation η: gη(x, t) = φ (x, 0, t) t y

50 In 1818 Poisson observed that when λ := gt 2 /2πΛ is irrational there are no non-constant solutions

51 In 1818 Poisson observed that when λ := gt 2 /2πΛ is irrational there are no non-constant solutions However, when λ Q, for every m, n with n2 m = λ is a solution φ(x, y, t) = sin ( 2nπt T ) ( 2mπx ) cos exp Λ ( 2mπy ) Λ

52 In 1818 Poisson observed that when λ := gt 2 /2πΛ is irrational there are no non-constant solutions However, when λ Q, for every m, n with n2 m = λ is a solution φ(x, y, t) = sin ( 2nπt T ) ( 2mπx ) cos exp Λ ( 2mπy ) The eigenvalues λ of the linearized problem are Q [0, ) which is dense and each eigenspace is infinite-dimensional Λ

53 In 1818 Poisson observed that when λ := gt 2 /2πΛ is irrational there are no non-constant solutions However, when λ Q, for every m, n with n2 m = λ is a solution φ(x, y, t) = sin ( 2nπt T ) ( 2mπx ) cos exp Λ ( 2mπy ) The eigenvalues λ of the linearized problem are Q [0, ) which is dense and each eigenspace is infinite-dimensional This is called complete resonance. It is a huge nightmare because inversion of the linearised problem involves small-divisor problems when attacking the nonlinear problem. Λ

54 In 1818 Poisson observed that when λ := gt 2 /2πΛ is irrational there are no non-constant solutions However, when λ Q, for every m, n with n2 m = λ is a solution φ(x, y, t) = sin ( 2nπt T ) ( 2mπx ) cos exp Λ ( 2mπy ) The eigenvalues λ of the linearized problem are Q [0, ) which is dense and each eigenspace is infinite-dimensional This is called complete resonance. It is a huge nightmare because inversion of the linearised problem involves small-divisor problems when attacking the nonlinear problem. However following work by Amick and others, in the past decade this Hamiltonian formulation combined with the Nash-Moser approach (from differential geometry) has led to non-trivial solutions of the full nonlinear problem for a measurable set of λ which is dense at 1 (Plotnikov & Iooss) Λ

55 Stokes Waves

56 Stokes Waves With φ(x, t) = φ(x ct) and w(x, t) = w(x ct) the system simplifies dramatically to φ = cw and an equation for w only: Cw = λ ( w + wcw +C(ww ) ) ( ) here the wave speed c has been absorbed in λ

57 Stokes Waves With φ(x, t) = φ(x ct) and w(x, t) = w(x ct) the system simplifies dramatically to φ = cw and an equation for w only: Cw = λ ( w + wcw +C(ww ) ) ( ) here the wave speed c has been absorbed in λ Note that w Cw is first-order, non-negative-definite, self-adjoint which is densely defined on L 2 2π by C(e ik ) = k e ik, k Z

58 Stokes Waves With φ(x, t) = φ(x ct) and w(x, t) = w(x ct) the system simplifies dramatically to φ = cw and an equation for w only: Cw = λ ( w + wcw +C(ww ) ) ( ) here the wave speed c has been absorbed in λ Note that w Cw is first-order, non-negative-definite, self-adjoint which is densely defined on L 2 2π by Therefore C(e ik ) = k e ik, k Z Cw = 2 w ξ 2 behaves like an elliptic differential operator but significantly it lacks a maximum principle

59 Stokes Waves With φ(x, t) = φ(x ct) and w(x, t) = w(x ct) the system simplifies dramatically to φ = cw and an equation for w only: Cw = λ ( w + wcw +C(ww ) ) ( ) here the wave speed c has been absorbed in λ Note that w Cw is first-order, non-negative-definite, self-adjoint which is densely defined on L 2 2π by Therefore C(e ik ) = k e ik, k Z Cw = 2 w ξ 2 behaves like an elliptic differential operator but significantly it lacks a maximum principle Note also that ( ) is the Euler-Lagrange equation of π π wcw λ ( w 2 (1+Cw ) ) dξ,

60 The Stokes Wave Equation Equation ( ) is quadratic in w and has beautiful structure:

61 The Stokes Wave Equation Equation ( ) is quadratic in w and has beautiful structure: apart from being variational it can be re-written (1 2λw)Cw = λ(w +C(ww ) wcw )

62 The Stokes Wave Equation Equation ( ) is quadratic in w and has beautiful structure: apart from being variational it can be re-written (1 2λw)Cw = λ(w +C(ww ) wcw ) where miraculously C(ww ) wcw is always negative and always smoother than w.

63 The Stokes Wave Equation Equation ( ) is quadratic in w and has beautiful structure: apart from being variational it can be re-written (1 2λw)Cw = λ(w +C(ww ) wcw ) where miraculously C(ww ) wcw is always negative and always smoother than w. When 1 2λw > 0 it becomes ( w +C(ww Cw ) wcw ) = λ 1 2λw and an elementary application of bifurcation theory yields small amplitude waves with λ close to 1 This problem vexed Lord Rayleigh and was finally settled by Nekrasov and Levi Civita in the 1920s

64 The Stokes Wave Equation Equation ( ) is quadratic in w and has beautiful structure: apart from being variational it can be re-written (1 2λw)Cw = λ(w +C(ww ) wcw ) where miraculously C(ww ) wcw is always negative and always smoother than w. When 1 2λw > 0 it becomes ( w +C(ww Cw ) wcw ) = λ 1 2λw and an elementary application of bifurcation theory yields small amplitude waves with λ close to 1 This problem vexed Lord Rayleigh and was finally settled by Nekrasov and Levi Civita in the 1920s The only difficulty arises when 1 2λw has zeros

65 The Stokes Wave Equation Equation ( ) is quadratic in w and has beautiful structure: apart from being variational it can be re-written (1 2λw)Cw = λ(w +C(ww ) wcw ) where miraculously C(ww ) wcw is always negative and always smoother than w. When 1 2λw > 0 it becomes ( w +C(ww Cw ) wcw ) = λ 1 2λw and an elementary application of bifurcation theory yields small amplitude waves with λ close to 1 This problem vexed Lord Rayleigh and was finally settled by Nekrasov and Levi Civita in the 1920s The only difficulty arises when 1 2λw has zeros This corresponds to Stokes wave of extreme form

66 History For small-amplitude travelling waves ( ) was derived by Babenko (1987) who was realised that its variational structure was important.

67 History For small-amplitude travelling waves ( ) was derived by Babenko (1987) who was realised that its variational structure was important. But he did not grasped its global significance and was unaware of the connection to Dyachenko, Kuznetsov, Spector & Zakharov

68 History For small-amplitude travelling waves ( ) was derived by Babenko (1987) who was realised that its variational structure was important. But he did not grasped its global significance and was unaware of the connection to Dyachenko, Kuznetsov, Spector & Zakharov Independently Plotnikov (1992) and Balk (1996) rediscovered ( ) for travelling waves

69 History For small-amplitude travelling waves ( ) was derived by Babenko (1987) who was realised that its variational structure was important. But he did not grasped its global significance and was unaware of the connection to Dyachenko, Kuznetsov, Spector & Zakharov Independently Plotnikov (1992) and Balk (1996) rediscovered ( ) for travelling waves Plotnikov introduced Morse index calculations in his brilliant study of the non-uniqueness question for solitary waves

70 History For small-amplitude travelling waves ( ) was derived by Babenko (1987) who was realised that its variational structure was important. But he did not grasped its global significance and was unaware of the connection to Dyachenko, Kuznetsov, Spector & Zakharov Independently Plotnikov (1992) and Balk (1996) rediscovered ( ) for travelling waves Plotnikov introduced Morse index calculations in his brilliant study of the non-uniqueness question for solitary waves Buffoni & Séré and Dancer added a great deal to existence theory using this variational structure. However there remain many open questions.

71 History For small-amplitude travelling waves ( ) was derived by Babenko (1987) who was realised that its variational structure was important. But he did not grasped its global significance and was unaware of the connection to Dyachenko, Kuznetsov, Spector & Zakharov Independently Plotnikov (1992) and Balk (1996) rediscovered ( ) for travelling waves Plotnikov introduced Morse index calculations in his brilliant study of the non-uniqueness question for solitary waves Buffoni & Séré and Dancer added a great deal to existence theory using this variational structure. However there remain many open questions. In the last ten years the theory has been extended to a very general class of free boundary problems Shargorodsky

72 History For small-amplitude travelling waves ( ) was derived by Babenko (1987) who was realised that its variational structure was important. But he did not grasped its global significance and was unaware of the connection to Dyachenko, Kuznetsov, Spector & Zakharov Independently Plotnikov (1992) and Balk (1996) rediscovered ( ) for travelling waves Plotnikov introduced Morse index calculations in his brilliant study of the non-uniqueness question for solitary waves Buffoni & Séré and Dancer added a great deal to existence theory using this variational structure. However there remain many open questions. In the last ten years the theory has been extended to a very general class of free boundary problems Shargorodsky Remarkably, despite its very special form nothing has emerged that makes Stokes Waves special in that much wider class of free-boundary problems

73 Numerical Evidence Suggests a Global Bifurcation Picture Like This: λ = 1 Moreover: energy oscillates maximum slope oscillates number of inflection points increases without bound

74 Numerical Evidence Suggests a Global Bifurcation Picture Like This: Stokes Wave of Extreme Form (1880) λ = 1 Moreover: energy oscillates maximum slope oscillates number of inflection points increases without bound as the extreme wave ((1 2λw = 0) is approached

75 Morse Index M(w) The Morse index M(w) of a critical point w is the number of eigenvalues µ < 0 of D 2 J(w): D 2 J(w)φ = µφ, µ < 0 where D 2 J[w] is the linearisation of ( ) at (λ, w): D 2 J(w)φ = Cφ λ ( φ+φcw + wcφ +C(wφ) )

76 Morse Index M(w) The Morse index M(w) of a critical point w is the number of eigenvalues µ < 0 of D 2 J(w): D 2 J(w)φ = µφ, µ < 0 where D 2 J[w] is the linearisation of ( ) at (λ, w): D 2 J(w)φ = Cφ λ ( φ+φcw + wcφ +C(wφ) ) The Morse Index may be infinite if 1 2λw has zeros (corresponding to an extreme wave)

77 Morse Index M(w) The Morse index M(w) of a critical point w is the number of eigenvalues µ < 0 of D 2 J(w): D 2 J(w)φ = µφ, µ < 0 where D 2 J[w] is the linearisation of ( ) at (λ, w): D 2 J(w)φ = Cφ λ ( φ+φcw + wcφ +C(wφ) ) The Morse Index may be infinite if 1 2λw has zeros (corresponding to an extreme wave) Plotnikov showed that this is equivalent to a simpler eigenvalue problem Cψ q[λ, w]ψ = µψ

78 Morse Index M(w) The Morse index M(w) of a critical point w is the number of eigenvalues µ < 0 of D 2 J(w): D 2 J(w)φ = µφ, µ < 0 where D 2 J[w] is the linearisation of ( ) at (λ, w): D 2 J(w)φ = Cφ λ ( φ+φcw + wcφ +C(wφ) ) The Morse Index may be infinite if 1 2λw has zeros (corresponding to an extreme wave) Plotnikov showed that this is equivalent to a simpler eigenvalue problem Cψ q[λ, w]ψ = µψ Moreover the Plotnikov potential q[λ, w] becomes singular as min{1 2λw} 0

79 Plotnikov s Theorem Suppose that a sequence {(λ k, w k )} of solutions of ( ) has 1 2λ k w k 0 and the Morse indices {M(w k )} are bounded.

80 Plotnikov s Theorem Suppose that a sequence {(λ k, w k )} of solutions of ( ) has 1 2λ k w k 0 and the Morse indices {M(w k )} are bounded. Then for some α > 0 1 2λ k w k α, k N None of these solutions is close to being an extreme wave. As the extreme Stokes wave is approached it becomes more and more unstable in the sense of Morse indices

81 Plotnikov s Theorem Suppose that a sequence {(λ k, w k )} of solutions of ( ) has 1 2λ k w k 0 and the Morse indices {M(w k )} are bounded. Then for some α > 0 1 2λ k w k α, k N None of these solutions is close to being an extreme wave. As the extreme Stokes wave is approached it becomes more and more unstable in the sense of Morse indices Recently Shargorodsky has quantified the relation between the size of the Morse index and the size of α

82 Primary Branch λ = 1

83 Primary Branch Stokes Wave of Extreme Form (1880) λ = 1 Moreover: energy oscillates maximum slope oscillates number of inflection points increases without bound Morse index grows without bound as the extreme wave is approached

84 Open Question By methods of topological degree theory the global branch of Stokes waves terminates at Stokes extreme wave. So Plotnikov s result implies that there are solutions arbitrarily large Morse index Despite this and the attractive form of J, a global variational theory of existence remains undiscovered The question is: For all large n N does there exist a solution with Morse index n?

85 What Does Plotnikov s Result Mean?

86 What Does Plotnikov s Result Mean? In abstract terms Babenko s equation for travelling waves is where Φ = 1 2 Cw = λ Φ(w) ( ) π π { } w 2 (1+Cw ) dx and it can be proved that there is a curve of solutions {(λ s, w s ) : s [0, )} with M(w s ) as s. Moreover the parametrisation s (λ s, w s ) is real-analytic.

87 What Does Plotnikov s Result Mean? In abstract terms Babenko s equation for travelling waves is where Φ = 1 2 Cw = λ Φ(w) ( ) π π { } w 2 (1+Cw ) dx and it can be proved that there is a curve of solutions {(λ s, w s ) : s [0, )} with M(w s ) as s. Moreover the parametrisation s (λ s, w s ) is real-analytic. Suppose that the Morse Index changes as s passes through s

88 What Does Plotnikov s Result Mean? In abstract terms Babenko s equation for travelling waves is where Φ = 1 2 Cw = λ Φ(w) ( ) π π { } w 2 (1+Cw ) dx and it can be proved that there is a curve of solutions {(λ s, w s ) : s [0, )} with M(w s ) as s. Moreover the parametrisation s (λ s, w s ) is real-analytic. Suppose that the Morse Index changes as s passes through s The variational structure of ( )

89 What Does Plotnikov s Result Mean? In abstract terms Babenko s equation for travelling waves is where Φ = 1 2 Cw = λ Φ(w) ( ) π π { } w 2 (1+Cw ) dx and it can be proved that there is a curve of solutions {(λ s, w s ) : s [0, )} with M(w s ) as s. Moreover the parametrisation s (λ s, w s ) is real-analytic. Suppose that the Morse Index changes as s passes through s The variational structure of ( ) The fact that λ is a multiplier on the right

90 What Does Plotnikov s Result Mean? In abstract terms Babenko s equation for travelling waves is where Φ = 1 2 Cw = λ Φ(w) ( ) π π { } w 2 (1+Cw ) dx and it can be proved that there is a curve of solutions {(λ s, w s ) : s [0, )} with M(w s ) as s. Moreover the parametrisation s (λ s, w s ) is real-analytic. Suppose that the Morse Index changes as s passes through s The variational structure of ( ) The fact that λ is a multiplier on the right The Morse Index changes as s passes through s

91 What Does Plotnikov s Result Mean? In abstract terms Babenko s equation for travelling waves is where Φ = 1 2 Cw = λ Φ(w) ( ) π π { } w 2 (1+Cw ) dx and it can be proved that there is a curve of solutions {(λ s, w s ) : s [0, )} with M(w s ) as s. Moreover the parametrisation s (λ s, w s ) is real-analytic. Suppose that the Morse Index changes as s passes through s The variational structure of ( ) The fact that λ is a multiplier on the right The Morse Index changes as s passes through s means that

92 at least one of two things happens

93 at least one of two things happens * (λ s, w s ) turning point

94 at least one of two things happens * (λ s, w s ) * (λ s, w s ) turning point crossing

95 at least one of two things happens * (λ s, w s ) * (λ s, w s ) turning point crossing The numerical evidence is that the first happens each time the Morse index changes

96 at least one of two things happens * (λ s, w s ) * (λ s, w s ) turning point crossing The numerical evidence is that the first happens each time the Morse index changes We do not have a proof This seems to be a very hard problem

97 at least one of two things happens * (λ s, w s ) * (λ s, w s ) turning point crossing The numerical evidence is that the first happens each time the Morse index changes We do not have a proof This seems to be a very hard problem But there s more we can say...

98 .. if you look at things differently

99 .. if you look at things differently The solutions {(λ s, w s ) : s [0, )} with M(w s ) as s are 2π-periodic.

100 .. if you look at things differently The solutions {(λ s, w s ) : s [0, )} with M(w s ) as s are 2π-periodic. So they are also 2nπ-periodic for any n

101 .. if you look at things differently The solutions {(λ s, w s ) : s [0, )} with M(w s ) as s are 2π-periodic. So they are also 2nπ-periodic for any n Let M n (w) denote their Morse index in that new setting

102 .. if you look at things differently The solutions {(λ s, w s ) : s [0, )} with M(w s ) as s are 2π-periodic. So they are also 2nπ-periodic for any n Let M n (w) denote their Morse index in that new setting For p sufficiently large prime M p (w s ) changes as s passes through s p s close to s

103 .. if you look at things differently The solutions {(λ s, w s ) : s [0, )} with M(w s ) as s are 2π-periodic. So they are also 2nπ-periodic for any n Let M n (w) denote their Morse index in that new setting For p sufficiently large prime M p (w s ) changes as s passes through s p s close to s Thus near s there is a bifurcation from the primary branch of solution with minimal period 2pπ * (λ s, w s )

104 .. if you look at things differently The solutions {(λ s, w s ) : s [0, )} with M(w s ) as s are 2π-periodic. So they are also 2nπ-periodic for any n Let M n (w) denote their Morse index in that new setting For p sufficiently large prime M p (w s ) changes as s passes through s p s close to s Thus near s there is a bifurcation from the primary branch of solution with minimal period 2pπ s = s p 1 * (λ s, w s )

105 .. if you look at things differently The solutions {(λ s, w s ) : s [0, )} with M(w s ) as s are 2π-periodic. So they are also 2nπ-periodic for any n Let M n (w) denote their Morse index in that new setting For p sufficiently large prime M p (w s ) changes as s passes through s p s close to s Thus near s there is a bifurcation from the primary branch of solution with minimal period 2pπ s = s p 1 * (λ s, w s ) s = s p 2

106 That period-multiplying (sub-harmonic) bifurcation occurs near the turning points on the primary branch was observed numerically by Chen & Saffman (1980)

107 That period-multiplying (sub-harmonic) bifurcation occurs near the turning points on the primary branch was observed numerically by Chen & Saffman (1980) Brooke knew their result and would I think have loved this subtle proof

108 That period-multiplying (sub-harmonic) bifurcation occurs near the turning points on the primary branch was observed numerically by Chen & Saffman (1980) Brooke knew their result and would I think have loved this subtle proof but alas once again the mathematicians had discovered an a priori proof a postiori

109 That period-multiplying (sub-harmonic) bifurcation occurs near the turning points on the primary branch was observed numerically by Chen & Saffman (1980) Brooke knew their result and would I think have loved this subtle proof but alas once again the mathematicians had discovered an a priori proof a postiori I think I had better stop here Thank You