Lecture : Water waves and Hamiltonian partial differential equations Walter Craig Department of Mathematics & Statistics Waves in Flows Prague Summer School 208 Czech Academy of Science August 30 208
Outline... the neverchanging everchanging water... Ulysses James Joyce Free surface water waves Symplectic forms and Hamiltonian PDEs Hamiltonian PDE - examples Quasilinear wave equations Nonlinear Schrödinger equations Shallow water equations Boussinesq equations Korteweg devries equations A Hamiltonian for overturning waves
Euler s equations and free surface water waves Fluid domain S(η) := {x R d, y ( h, η(x))}, d = 2, 3 Incompressibility and irrotationality therefore u = ϕ where u = 0, u = 0 ϕ = 0 On the solid bottom boundary of S(η) N u = 0 On the free surface Γ(η) := {y = η(x)} t η = y ϕ x η x ϕ t ϕ = gη 2 ϕ 2,
Figure : Great waves off the Oregon coast
kinetic and potential energy The energy H of the system of equations is η(x) η(x) H = K + P := R d 2 u 2 dydx + gy dydx h R d h η(x) = R d 2 ϕ 2 g dydx + h R d 2 η2 dx C, Rewrite the kinetic energy by integrating by parts η(x) K = R d 2 ϕ 2 dydx = h R d h + R d 2 ϕn ϕ ds bottom + η(x) 2ϕ ϕ dydx R d 2 ϕn ϕ ds free surface Define ξ(x) := ϕ(x, η(x)), the kinetic energy is therefore K = R d 2 ξn ϕ ds free surface = R d 2ξG(η)ξ dx where G(η) is the Dirichlet Neumann operator.
Dirichlet Neumann operator The Dirichlet Neumann operator G(η)ξ(x) = ( y x η(x) x )ϕ(x, η(x)) = R(N ϕ)(x, η(x)) where R = + x η 2 is a normalization factor so that G(η) is self-adjoint on L 2 (dx). The Hamiltonian H = R d 2 ξg(η)ξ + g 2 η2 dx Theorem (Zakharov (968)) The pair of functions (η(x), ξ(x)) are canonical variables for the water waves problem in which it can be written in Darboux coordinates, with Hamiltonian H(η, ξ).
Hamiltonian formulation Therefore the equations for water waves can be rewritten in Darboux coordinates; η = grad ξ H = G(η)ξ () ξ = grad η H = gη grad η K The expressions for K and grad η K involve derivatives of G(η) with respect to perturbations of the domain S(η). Proposition Let η C. Then G(η) satisfies:. G(η) is positive semidefinite. 2. It is self-adjoint (on an appropriately chosen domain). 3. G(η) maps H (R d ) to L 2 (R d ) continuously. 4. As an operator G(η) : H (R d ) L 2 (R d ) it is analytic in η B R (0) C (R d ).
ZCS system Rewriting the system for water waves in these coordinates t η = G(η)ξ t ξ = gη grad η K(η, ξ) Variations of the kinetic energy with respect to the free surface η(x) (the shape derivative) are given by grad η K(η, ξ) = ( 2( + x η 2 x ξ 2 (G(η)ξ) 2 ) 2( x η x ξ)g(η)ξ + ( x η 2 x ξ 2 ( x η x ξ) 2)) Variational formula of Hadamard (90), (96), related to the derivative of the Green s function with respect to the domain. Among other things, this is a useful formulation for numerical simulations
Simulations using the Dirichlet Neumann formulation 0.4 η/h I 0.2 0 0 0 20 30 40 50 60 0 20 40 70 60 80 80 t/(h/g)/2 00 20 90 40 60 x/h Figure : Head-on collision of two solitary waves, case S/h = 0.4 W. Craig, J. Hammack, D. Henderson, P. Guyenne & C. Sulem, Phys. Fluids 8, (2006)
proof by first principles of mechanics The Lagragian for surface water waves L := K P expressed in the tangent space variables (η, η). Use the kinematic condition η = y ϕ x η x ϕ = G(η)ξ The Lagrangian is thus L(η, η) = 2 R d ηg (η) η g 2 η2 dx The canonical conjugate variables are precisely those given by Zakharov. (η, η L) = (η, G (η) η) = (η, ξ)
Taylor expansion of G(η) The operator G(η)ξ(x) is analytic, given by the expression G(η)ξ = j 0 G (j) (η)ξ where each G (j) (η) is homogeneous of degree j in η Explicitely, for D x := i x G (0) ξ(x) = D x tanh(h D x )ξ(x) G () (η)ξ(x) = D x ηd x G (0) ηg (0) ξ(x) G (2) (η)ξ(x) = 2 (G(0) η 2 D 2 x + D 2 xη 2 G (0) 2G (0) ηg (0) ηg (0) )ξ(x) Accordingly the Hamiltonian is analytic, with expansion H(η, ξ) = R d 2 ξg(0) ξ + g 2 η2 dx + 2 j 3 R ξg(j 2) (η)ξ dx d = j 2 H (j) (η, ξ)
Hamiltonian PDE Phase space taken to be v M a Hilbert space with inner product X, Y M, for X, Y T(M) Symplectic form given by a two form ω(x, Y) = X, J Y M, J T = J for tangent vectors X, Y T(M). Antisymmetry J T = J Hamiltonian vector field X H defined through the relation dh(y) = ω(y, X H ) = grad v H, Y M for all Y T(M) t v = Jgrad v H(v), v(x, 0) = v 0 (x) The flow v(x, t) = ϕ t (v(x)), defined for v M 0 M Poisson brackets between H and other functions F are given by the expressions {H, F}(v) := d F(ϕ t (v)) = grad dt v H(v), Jgrad v F(v) M t=0
Poisson brackets and conservation laws Express a conservation law using the Poisson bracket t K(η(t, ), ξ(t, )) = {H, K} := grad η K grad ξ H grad ξ K grad η H dx Conservation of mass M(η) = η dx {H, M} = grad η M grad ξ H grad ξ M grad η H dx = G(η)ξ dx = G(η) ξ dx = 0 Momentum I(η, ξ) = η x ξ dx t I = {H, I} = 0 Energy H(η, ξ) t H = {H, H} = 0
example 2: the quasilinear wave equation Quasilinear wave equations 2 t u u + t G 0 ( t u, x u) + Σ j xj G j ( t u, x u) = 0 (2) with initial data u(0, x) = f (x) and t u(0, x) = g(x) This equation can be written as a Hamiltonian PDE. The Lagrangian is L = 2 u2 2 xu 2 + G( u, x u) dx Use the Legendre transform to change to canonical conjugate variables p := u L = p( u, x u) The Hamiltonian is now H(u, p) := 2 p2 + 2 xu 2 + R(p, x u) dx where R(p, x u) = G( u, x u) for u = u(p, x u)
quasilinear wave equation (cont.) Equation (2) can be rewritten as t u = grad p H(u, p) = p + p R t p = grad u H(u, p) = u Σ j xj uxj R(p, x u) which is in Darboux coordinates. Theorem (local existence theorem) Systems of equations of this form are symmetrizable hyperbolic systems. Therefore for Sobolev data of sufficient smoothness r > 0 a solution exists locally in time (u 0 (x), p 0 (x)) H r (R d ) (u(t, x), p(t, x)) = ϕ t (u 0, p 0 ) C([ T, +T] : H r (R d )) := M 0 L 2 (R d )
Example 3: NLS Nonlinear Schrödinger equation On R d or on a domain T d = R d /Γ, for period lattice Γ i t u 2 xu + Q(x, u, u) = 0 (3) with Hamiltonian H NLS (u) = Rewritten 2 u 2 + G(x, u, u) dx, u G = Q t u = i grad u H NLS In many cases the Schrödinger equation admits a gauge symmetry under phase translation, in which case G = G(x, u 2 )
example 4: Shallow water equations Shallow water equations Model equations for water waves t η = x ((h + η) x ξ ) (4) t ξ = gη 2 xξ 2 This is a Hamiltonian PDE with Hamiltonian H SW = 2 (h + η(x)) x ξ 2 + gη 2 dx The mathematically rigorous study of the shallow water limit of the equations of water waves was initiated by T. Kano & T. Nishida (979). The work of D. Lannes (2000) extended their results from the analytic category to to Sobolev spaces, and to more subtle limiting situations such as the Green - Naghdi system
example 5: Boussinesq systems Boussinesq system on spatial domain D R The Hamiltonian is H Boussinesq (p, q) = T 2 p2 + g 2 q2 ± 2 ( xp) 2 + G(x, q) dx The symplectic form is given by J Boussinesq = ( 0 ) x x 0 ṗ = x grad q H Boussinesq (q, p) = x (gq + q G) (5) q = x grad p H Boussinesq (q, p) = x (p 6 2 x p + p G) The sign is badly ill-posed (McKean, the bad Boussinesq), while the + sign is well posed (the good Boussinesq). Completely integrable nonlinear cases include G = p 3 (Zakharov) and G = 2 qp2 (Kaup, Sachs)
Example 6: the generalized KdV Korteweg de Vries equation t r = 6 3 x r x ( r G(x, r)), x T (6) The Hamiltonian is H KdV (r) = T 2 ( xr) 2 + G(x, r) dx Rewritten t r = J grad r H KdV, where J = x Completely integrable cases are G = r 3 and G = r 4 Rigorous studies of water waves in the KdV scaling regime and limit are given in Craig (985), T. Kano & T. Nishida (986) and others
Water waves Hamiltonian in general coordinates Following a question of T. Nishida in 206 Fluid domain Ω(t) R 2 with free surface γ(t, s). Evolution determined by free surface conditions T txy N txy = 0, p(t, γ(t, s)) = 0 Energy = kinetic + potential H = K + P, K = 2 Potential energy P = Ω V dydx, Ω ϕ(x, y) 2 dydx V = (0, gy2 2 ) Dirichlet Neumann operator, with ϕ(γ(s)) = ξ(s) G(γ)ξ(s) := N ϕ(γ(s)), K = 2 ξ(s)g(γ)ξ(s) ds γ γ
Legendre transform Lagrangian L = K P The kinematic boundary condition states that N γ = N ϕ(γ) = G(γ)ξ Decmpose the vector field γ(s) along the curve γ = (γ (s), γ 2 (s)) in terms of its Frenet frame (T(s), N(s)) n(t, s) = N γ(t, s), τ(t, s) = T γ(t, s) then the Lagrangian is L = 2 n(t, s)g (γ)n(t, s) ds γ The Legendre transform γ δ γ L = G (γ)n(t, s) = ξ(s), H = 2 ξ(s)g(γ)ξ(s) ds γ + γ γ γ V N ds γ g 2 γ2 2(s) sγ (s) ds γ s γ
Hamilton s canonical equations variations δγ and δξ of the Hamiltonian N γ = n = δ ξ K = G(γ)ξ the kinematic boundary conditions Decomposing boundary variations δγ = (N δγ) N + (T δγ) T variations of the potential energy δ γ P δγ = δ N δγ P δγ = gγ 2 (s) N δγ ds γ Finally δ γ K has normal and tangential components, with the result that ( t ξ = gγ 2 2 s γ 2 ( sξ) 2 (G(γ)ξ) 2 ) s γ sξτ The tangential component τ = T γ depends upon the manner in which the surface is parametrized, which in most cases imposes a holonomic constraint γ
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