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A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times in the NLS regime provided the initial data is suitably close to a wave packet of sufficiently small amplitude in Sobolev spaces.


Introduction
The mathematical problem of two dimensional water waves concerns the evolution of an interface separating an inviscid, incompressible, irrotational fluid, under the influence of gravity, from a region of zero density (e.g., air) in two dimensional space. It is assumed that the fluid region lies below the air region. Assume the fluid is infinitely deep and has density 1, and that the gravitational field is g = (0, −1). At t ≥ 0, denote the fluid interface by Σ(t) and the fluid region by Ω(t). If surface tension is neglected, then the motion of the fluid is described by v t + v · ∇v = g − ∇p div v = 0, curl v = 0 on Ω(t), t ≥ 0 p = 0 on Σ(t) (v, 1) is tangent to the free surface (Σ(t), t) (1.1) where v is the fluid velocity, p is the fluid pressure.
Assume further that the interface Σ(t) is parametrized by z = z(α, t), where α ∈ R is the Lagrangian coordinate, i.e., z t (α, t) = v(z(α, t), t). Let a = − ∂p ∂n 1 |zα| , where n = izα |zα| is the unit outward normal of Ω(t). We know from [12] that (1.1) is equivalent to the following complex system on the interface: 3) where H is the Hilbert transform associated to the fluid region Ω(t): It is well-known (c.f. [9], [7]) that if one performs a multiscale analysis to determine modulation approximations to the finite or infinite depth 2D water wave equations, one should expect to find that the amplitude B is a profile that travels at the group velocity determined by the dispersion relation of the water wave equations over time intervals of length O(ǫ −1 ), and evolves according to a nonlinear Schrödinger equation (NLS) over time intervals of length O(ǫ −2 ). The first formal derivations of the NLS from the 2D water wave equations was obtained by Zakharov [15] for the infinite depth case, and by Hasimoto and Ono [6] for the finite depth case. In [4], Craig, Sulem and Sulem applied modulation analysis to the finite depth 2D water wave equation, derived an approximate solution of the form of a wave packet and showed that the modulation approximation satisfies the 2D finite depth water wave equation to leading order.
A rigorous justification of the NLS from the full water wave equations would bring us one step closer to understanding qualitative properties for wave packet-like solutions of the water wave equations from that of solutions to NLS on the appropriate time scales. Rigorous justifications of the KdV, KP, Boussinesq, shallow water and various other asymptotic models from the full water wave equations have been done in [5], [10], [1]. As was noted in [4], the reason that a justification for NLS has not been given is that the longest existence time in Sobolev spaces for the water waves equation demonstrated thus far have been on time scales of the order O(ǫ −1 ), for data with Sobolev norms of the order O(ǫ). However these times are too short to distinguish the NLS behavior of the wave packet from simple translation of the initial wave packet at group velocity. Since there is no existence result in Sobolev spaces on the necessary time scales, an attempt to justify NLS as a rigorous modulation approximation to the water wave system on that scale has not been made.
Let U g f = f • g, and for κ : R → R a diffeomorphism we introduce the notation [13], Wu showed that for any solution z of (1.2)-(1.3), the quantity Π := (I − H)(z − z) satisfies the equation π (D t ζ(α, t) − D t ζ(β, t))(ℑζ(α, t) − ℑζ(β, t)) |ζ(α, t) − ζ(β, t)| 2 ∂ β D t ζ(β, t)dβ and furthermore there is a coordinate change κ, such that in this coordinate system, the equation (1.6) contains no quadratic nonlinear terms. Using this favorable structure and the method of vector fields, Wu further proved the almost global well-posedness for the full water wave system (1.2)-(1.3) for data small in the generalized L 2 Sobolev spaces defined by the invariant vector fields. However, the wave packet data ǫB(ǫα)e ikα (for B sufficiently smooth and localized) has slow decay at infinity, and in terms of the generalized Sobolev norms used in [13] these are at least of size O(ǫ −1/2 ). In terms of the standard Sobolev norms they are of size O(ǫ 1/2 ). Therefore the standard L 2 Sobolev spaces suits our purposes better. As is suggested by the work of [8], in justifying the modulation approximation for a nonlinear system it is advantageous if the nonlinear system contains no quadratic nonlinear terms. We therefore use the equation (1.6) to perform the multiscale analysis. In fact, we will use a slightly different change of variables κ than that given in [13]. Upon performing this multiscale analysis, we derive an approximate wave packet-like solutionζ satisfies the transformed equations (see (2.7)-(2.8) below) with a residual of size O(ǫ 4 ). The special structure of (1.6) then allows us to obtain bounds for the error r = ζ −ζ between the true solution and the approximate solution on the appropriate time scale in Sobolev spaces.
We will see in the course of the multiscale analysis that the envelope of the leading term of ζ − α obeys a focusing cubic nonlinear Schrödinger equation which is globally well-posed in sufficiently regular Sobolev spaces. This implies that the approximate solutionζ is eternal. This fact, along with the a priori bounds on the remainder r, allows us to show existence and uniqueness of solutions of the system (1.2)-(1.3) on the proper O(ǫ −2 ) time scales, for initial data which is no more than O(ǫ 3/2 ) away from a wave packet ǫB(ǫα)e ikα in Sobolev spaces. A rigorous justification of wave packet approximations to solutions of the water wave system is then obtained in this special coordinate system κ. Upon changing variables, we obtain appropriate wave packet approximations to water waves in Lagrangian coordinates. Finally, by introducing some further restrictions on the initial data, we justify an Eulerian version of the asymptotics.

Derivation of the Main Equations
In this section we introduce our notation as well as collect for future reference the main equations and formulas from [13] that we will use. We first recall the definition of the Hilbert transform H γ associated to the interface determined by a curve parametrization γ(α) : R → C : We adopt the following notations for Hilbert transforms associated to specific curves: H is the Hilbert transform associated to z already defined, H is the Hilbert transform associated to ζ, and H 0 is the flat Hilbert transform associated to the line γ(α) = α. In general, the Hilbert transform H γ satisfies the convention H γ 1 = 0 and the identity H 2 γ = I in L 2 . Let Ω be a domain in R 2 , with ∂Ω parametrized by γ(α), α ∈ R, oriented clock-wisely. We know f (·) = F (γ(·)) ∈ L 2 (R) is the trace of a holomorphic function F in Ω if and only if (I − H γ )f = 0. The celebrated result of [3] (see Theorem B.1) states that H γ is bounded on L 2 provided that γ satisfies the chord-arc condition: There exist constants ν, N > 0 so that ν|α − β| ≤ |γ(α) − γ(β)| ≤ N|α − β| for all α, β ∈ R.
(2) For all f, g ∈ L 2 (∂Ω), [f, With these preparations, we give the change of variables used to convert (1.2)-(1.3) into a more suitable equation for our purposes. Originally, in [13], the change of variables κ was introduced using a Riemann map Φ(z, t) : Ω(t) → P − which for each t mapped the fluid region Ω(t) to the lower half plane, and then defined by α → z(α, t) + z(α, t) − h(α, t), where h was taken to be α → Φ(z(α, t), t).
However, the only property of h that was used was that it was a real-valued trace of a holomorphic function defined on Ω(t). This idea was already used in the 3D setting to prove global existence of solutions to the 3D water wave problem [14]. We use it here by choosing to set h(α, t) = z(α, t) − 1 2 (I + H)(I + K) −1 (z(α, t) − z(α, t)) , where K = ℜH is the double layer potential operator associated to the curve z. It is easy to see from the definition that h is a real-valued trace of a holomorphic function in Ω(t). Then the change of variables is defined by κ(α, t) = z(α, t) + z(α, t) − h(α, t) = z(α, t) + 1 2 (I + H)(I + K) −1 (z(α, t) − z(α, t)) (2.3) Our choice of κ then gives us the crucial identity 4) and from this it follows immediately in the new coordinates that ξ := ζ − α, the perturbation of ζ from the rest state α. Then from (1.6) and (2.4) we have that solutions z also satisfy the system We will also need the equations corresponding to the time derivative, which by virtue of (1.3) and a derivative D t to (2.7) are given by We also have the following formulas for b and A in terms of ζ (c.f. Proposition 2.4 of [13] for a proof. From the proof, it is clear that (2.8) and (2.11) together implies (2.13) and (2.14).): 14) The commutator in the right hand side of (2.10) can be rewritten using We also record Proposition 2.7 of [13]: To estimate terms involving time derivatives of singular integral operators we record the following Then Denote the Fourier transform on R bŷ For s ∈ R we have the usual Sobolev spaces and the homogeneous Sobolev spaceṡ A well-known consequence of the Sobolev embedding theorem is that H s is continuously embedded in W s−1,∞ for s ≥ 1. Given a Banach space X, let C([0, T ]; X) be In the rest of the paper, we make the following A Priori Assumption. Let s ≥ 6, and let ζ be a solution to the water wave system (2.7)- First we choose δ > 0 sufficiently small so that ζ satisfies the chord-arc condition (2.2) and A ≥ 1/2 (c.f. [13]). In the course of the paper we will need to choose δ smaller still.
In order to use the formulas (2.13), (2.14), (2.16) to get estimates for b, A and U −1 κ (a t /a) in H s we use the following lemma, whose proof is essentially that of Lemma 3.8 and Lemma 3.15 of [13]: Let s ≥ 4, and suppose that ζ satisfies (2.18). Then there exists a constant C depending on S(T 0 ), so that for all real-valued f we have the following estimates: ( The goal of this section is to derive a formal solution to the system (2.7)-(2.8) which is to leading order a wave packet. Since we want our approximation to remain bounded for times on the order O(ǫ −2 ), we calculate this formal solution using a multiscale analysis. As mentioned in the introduction, we expect from similar formal derivations of modulation approximations to the water wave equations that the amplitude of the wave packet is a profile which travels at the group velocity of the water wave operator, and evolves according to a nonlinear Schrödinger equation.
To effect this multiscale analysis, we must first formally expand the Hilbert transform H appearing in the water wave equations. In particular, we must intepret how the flat Hilbert transform H 0 acts on multiple scale functions of the form F (ǫα)e ikα for k = 0.
3.1. Formal Expansion of the Hilbert Transform. Understanding the system (2.7), (2.8) depends on understanding the Hilbert Transform H. Since our first goal is to seek a perturbation expansion we must find a corresponding development of H into a formal power series To predict what the terms of this series ought to be, we heuristically expand the kernel of H in a formal power series as follows: Equating like powers of ǫ on the right hand side of this last expression suggests the following formulas for H 1 : (1) , H 0 ]f α and for H 2 : and so we define the approximate Hilbert Transform IfH acts on a multiple scale function f (α 0 , α 1 ) = f (α, ǫα), then we have the expansioñ where 3) Later we will need to estimate the operator where Hζ is the Hilbert transform associated to the curve given by the approximationζ. We will see that for our purposes it suffices to develop the approximate solutionζ to the third order:ζ (α, t) = α + ǫζ (1) (α, t) + ǫ 2 ζ (2) (α, t) + ǫ 3 ζ (3) (α, t) Hence we record the following formula as a first step towards analyzing Hζ −H: Lemma 3.1. (Hζ −H)f can be written as the following finite sum of singular integrals: where S = {(n 1 , n 2 , m, p 1 , p 2 ) : n 1 p 1 + n 2 p 2 + m ≥ 3, 0 ≤ p 1 + p 2 ≤ 2, 0 ≤ n 1 , n 2 , m ≤ 3} and C p 1 ,p 2 are constants depending only on p 1 , p 2 .
Proof. First observe that with an integration by parts we have the formulas and Now we repeatedly apply the identity so as to arrive at the identity The last of these terms is of size O(ǫ 3 ). As for the rest, if we arrangeζ β (β)/ ζ (α) −ζ(β) in powers of ǫ up through ǫ 2 , we see that All of the terms here up through order O(ǫ 2 ) precisely compriseH, and so vanish upon subtractingH. The remaining O(ǫ 3 ) terms consists of a finite number of terms which can be written explicitly in the form where S = {(n 1 , n 2 , m, p 1 , p 2 ) : n 1 p 1 + n 2 p 2 + m ≥ 3, 0 ≤ p 1 + p 2 ≤ 2, 0 ≤ n 1 , n 2 , m ≤ 3} and C p 1 ,p 2 are constants depend only on p 1 , p 2 .

3.2.
The Action of H 0 on Multiscale Functions. As we saw in the last section, the operators appearing in the power series expansion of the Hilbert Transform of the interface can be written in terms of the flat Hilbert transform It is known that H 0 is a Fourier multiplier with Fourier symbolĤ 0 (ξ) = − sgn(ξ). However, it still remains to be seen how to interpret the action of H 0 on a multiscale function f = f (α, ǫα) as a multiscale function.
Since we are interested in the modulation approximation of the water wave problem, we will choose the leading order of our approximation to be a wave packet of the form B(ǫα)e ikα for k > 0. Hence the formal calculation depends upon understanding the action of H 0 on such wave packets. Since the amplitude of B(ǫα)e ikα is slowly varying for small ǫ, we heuristically expect for k = 0 that where ∼ indicates an error depending on ǫ. The following result confirms this intuition. We adopt the usual practice of assuming, unless otherwise stated, that a constant C may denote different constants in the process of deriving an inequality.  Proof. It suffices to consider the case k > 0, since the case k < 0 follows by complex conjugation and the fact that H 0 = −H 0 . We first derive a bound for ∂ n α (I − H 0 )f (ǫα)e ikα L 2 . We calculate that But since ǫ ≤ 1, we have for any m ≥ 0 that 2 As a consequence we may freely assume in the multiscale calculation that H 0 formally treats the amplitude of the wave packet B(ǫα)e ikα as a constant when k = 0. However, note that in the case k = 0 we can at best say that and so these must be retained as functions of the slow variable α 1 = ǫα whenever they occur in the multiscale calculation.
We record an immediate consequence of this result that will be used frequently in the multiscale calculation.
Corollary 3.1. Let s ≥ 1, m ≥ 0, ǫ ≤ 1 and f, g ∈ H s+m (R) and suppose that k, l are given so that l = 0, −k, and sgn(l) = sgn(k + l). Then We are now prepared to find an approximate solutioñ ζ to the four equations (2.7)-(2.11) which is to leading order a wave packet, where G is given by (2.9). Our approach will be to derive an approximate solution to the system (2.7)-(2.8) having residual O(ǫ 4 ) with a multiscale analysis and then verify that this approximate solution also satisfies (2.10)-(2.11) up to a residual of size O(ǫ 4 ). We begin by seeking a perturbative ansatz for (2.7)-(2.8) In order to construct an expansion that is valid on times on the order O(ǫ −2 ), we introduce multiple scales t 0 = t, t 1 = ǫt, t 2 = ǫ 2 t, α 0 = α, α 1 = ǫα and so we seek a solution of the form which formally satisfies the original equations up to terms of size O(ǫ 4 ). Before we begin solving these equations, we expand the auxiliary quantities and operators in powers of ǫ. In particular we must determine the expansions in ǫ of the quantities Notice that since b and A − 1 are of quadratic order and G is of cubic order, it follows that We will also show in the sequel that A 2 = 0 and b 2 = b 2 (α 1 , t 1 , t 2 ); the linear operator associated to the water wave equation thus has the multiscale expansion Recall that we also have the formulas for the multiscale expansion of the Hilbert Transform given by (3.3) We will find explicit formulas for b 2 , b 3 and G 3 in the course of the analysis. In what follows we will repeatedly use the fact, justified by the last section, that and hence that We are now ready to expand (2.7)-(2.8) in powers of ǫ. Collecting like terms yields a hierarchy of systems that allow us to successively solve for the holomorphic trace 1 2 (I+H 0 )ζ (n) and the antiholomorphic trace 1 2 (I − H 0 )ζ (n) of the ζ (n) 's in the lower half plane. The terms of order O(ǫ) in (2.7)-(2.8) yield the system Because we are interested in solutions which to leading order are given by wave packets, we assume an ansatz concentrated in Fourier space about the fixed wave number k > 0: Injecting the above ansatz into (3.10) forces B − = 0 by (3.7). Similarly substituting this ansatz into (3.9) yields the condition (∂ 2 t 0 + k)B + = 0, which implies that B + (α 1 , t 0 , t 1 , t 2 ) = B(α 1 , t 1 , t 2 )e iωt 0 , where we have introduced the wave frequency ω which satisfies the water wave dispersion relation Thus we take as our solution where we have introduced the phase φ := kα 0 + ωt 0 . Moving to the O(ǫ 2 ) terms from (2.7), we have by (3.12) and using (3.8) that where ω ′ = dω/dk is the group velocity of the wave packet. If we want (I − H 0 )ζ (2) to be uniformly bounded for all time we must insist that the right hand side of (3.13) be equal to zero in order to avoid secular terms. Therefore we choose where ω ′ = dω/dk is the group velocity. The O(ǫ 2 ) terms from (2.8) yield the equation An obvious choice seems to be ζ (2) = ik|B| 2 + B 2 (α 1 , t 1 , t 2 )e iφ . However such choice leads to unavoidable secular growth in the O(ǫ 3 ) level. Instead, we find that taking ζ (2) so that (I − H 0 )ζ (2) = 0 avoids such secular growth. Hence we take Before we move on to the O(ǫ 3 ) system, we must first derive formulas for b 2 , A 2 and G 3 . Substituting the expansion of ζ into the formula (2.13) we see immediately upon collecting like powers of ǫ that b 0 = b 1 = 0. Therefore we have and so since b 2 is real-valued we conclude that Similarly, using (2.14) we have immediately that A 1 = 0 and that Finally we derive from (2.9) a formula for G 3 : β 0 (β)dβ := I 1 + I 2 Using (3.7) and (3.8) yields Similarly, we simplify In summary, We can now arrange the O(ǫ 3 ) terms of (2.7), and using (3.12) and (3.16) along with (3.17), (3.18) and (3.8) arrive at the equation where ω ′′ = d 2 ω/dk 2 . To supress secular growth we now insist that the amplitude B satisfy the focusing cubic nonlinear Schrödinger equation 1 With this choice made we solve (3.19) by taking ( Finally, the O(ǫ 3 ) terms from (2.8) yields the equation Now that we have constructed an approximate solutionζ to the equations (2.7)-(2.8), we claim thatζ also solves the system (2.10)-(2.11) up to an O(ǫ 4 ) residual. First we notice that (2.10) is obtained by applying a derivative D t to (2.7), therefore it is clear thatζ solves (2.10) up to an O(ǫ 4 ) residual. Now we consider (2.11). By (2.8) we have that Hence to show thatζ satisfies (2.11) up to an O(ǫ 4 ) residual it suffices to show that our approximation of b satisfies (2.13) up to a residual of size O(ǫ 4 ). Hence we need only choose b 3 so that In summary, we have shown that the equations (2.7)-(2.8)-(2.10)-(2.11) are satisfied up to a residual of size O(ǫ 4 ) by the approximatioñ From (3.23), enforcing the reality condition on b 3 yields We also defineÃ : 27) Corresponding to this approximate solution (3.24) we introducẽ ξ :=ζ − α as well asD We then have the formulas for the difference as well as for For future reference we also include the following calculation.
Proposition 3.2. Letζ,b,Ã be as above. Then Proof. The first statement is straightforward. For the second, observe that by (1) which vanishes by virtue of the above observation, (3.12), (3.14), and (3.16). Finally, the O(ǫ 3 ) terms are given by For the same reasons as for the O(ǫ 2 ) terms all of the above are immediately seen to vanish except for the last, which by (3.17) is given by Now we have shown that the approximationζ depends on B and B X , where B satisfies the NLS equation (3.20). To be certain that the forthcoming objects are well-defined, we appeal to the following global well-posedness result for NLS: [11]) Let m ≥ 1 be given, and suppose that B 0 ∈ H m is given. Then there exists a unique solution Fix s ≥ 6, T > 0. For the rest of the paper we assume that B 0 ∈ H s+7 , and hence by the above theorem that If we calculateζ andD tζ from B ∈ H s+7 through (3.24), we see by counting the maximum number of derivatives that fall on B that we have the bound ξ ,D tζ ,D 2 For the rest of the paper, we choose ǫ < ǫ 0 for ǫ 0 ≤ 1 sufficiently small depending on B 0 so thatζ satisfies the chord-arc condition (2.2). Along with the a priori assumption (2.18) using an appropriately small choice of δ > 0 , this implies that the singular integrals in the next section are well-defined.

Estimates of the Remainder
Now that we have derived a formal approximation of the solution ζ to the system (2.7)-(2.8)-(2.10)-(2.11), we can consider the size of the remainder r = ζ −ζ. Our basic approach is to expand the known equations for ζ and formulas for quantities defined in terms of ζ given in §2 by writing ζ = r +ζ and thereby find the appropriate governing equations from which we will derive energy estimates for r.
In §4.1 we derive from (2.7)-(2.8)-(2.10)-(2.11) new equations in terms of quantities related to r. Many functions and operators will arise in these equations that we need to study before we can estimate them appropriately. In particular we devote §4.2 to studying the remainder between the true and approximate Hilbert transforms introduced in §3.1.
To clearly describe the respects in which we consider quantities to be small, we adopt the following terminology: we say a term is of nth order (with linear, quadratic, cubic having the typical meaning) if the term consists of n small factors.
Alternately, given a Banach space X with norm · X , we say that a term f ∈ X as being O(ǫ n ) in X when there exists a constant C so that f X ≤ Cǫ n . If we use the notation O(ǫ n ) without mentioning a norm explicitly, we mean size in the physical sense O(ǫ n ) as we have used in §3. Since we ultimately seek bounds in Sobolev spaces H s , we introduce the special notation that f ∈ H s is O(ǫ n ), which means that f is O(ǫ n ) in H s where the index s will be clear from context.
We ultimately plan to control all of our quantities in terms of r α and D t r in Sobolev space, and so we need some idea of how large we expect r α and D t r to be in terms of ǫ. Since we are only interested in the leading term of the approximation, it is a suitable goal to seek a remainder which is of physical size O(ǫ 2 ), and in the L 2 sense to be O(ǫ 3/2 ). Therefore, we expect here that r α and D t r should be O(ǫ 3/2 ).
In §4.3 we bound in H s the remaining quantities appearing in the cubic nonlinearities of the equations of §4.1 by terms involving the quantity which we expect to be O(ǫ 3 ). We will then show that for ǫ < ǫ 0 with ǫ 0 chosen sufficiently small, the quantity E s is bounded above by the quantity which in turn is bounded above by the energy E for the remainder.
We then use these estimates to show that the cubic nonlinearities of the remainder equations of §4.1 are O(ǫ 7/2 ). Having done so, we derive in §4.5 an energy inequality which roughly reads dE/dt ≤ O(ǫ 5 ). Heuristically, an inequality of this type is suitable since on time scales on the order O(ǫ −2 ) this implies E s is of size O(ǫ 3 ), as we would like. We then go on to rigorously derive a priori bounds of E s on O(ǫ −2 ) time scales.

The Derivation of the Equations for the Remainder.
Here we derive the equations governing the evolution of the quantities Our goal in this section is to manipulate the nonlinearities of these equations so that they will be in a suitable form for showing they are of size O(ǫ 7/2 ). For example, from (2.11) we have ). Hence the right hand side of (4.2) is of size O(ǫ 5/2 ). We now give the equation for the remainder corresponding to (2.7). In decomposing the right hand side of this equation, we keep two goals in mind. First, we must split the terms in such a way as to arrive atG so as to cancel the O(ǫ 3 ) contribution from G. Next, we must whenever possible avoid estimating terms formed by P acting on complicated terms, so as to reduce all estimates whenever possible to those already derived. Specifically we expand using Proposition 2.1 as follows: where ǫ 4 R :=G −P(I −H)ξ is the residual arising from the approximate equation corresponding to (2.7). Note that at most five 2 derivatives of B are taken in R through the term ∂ 2 t H 2 ζ (3) , and so R ∈ H s provided B ∈ H s+5 . Similarly, at most seven derivatives of B are taken inD t R through the term As in the calculation (2.13) et. seq. of [13], we exploit the fact that D tξ is almost holomorphic. Using (2.11) and Proposition 2.2 allows us to rewrite this term as To see that the last of these terms is acceptably small, we again apply Proposition 2.2 to see that which is now easily seen to be O(ǫ 7/2 ) by (4.2). Thus our equation for ρ is now Note that the terms on the right hand side of (4.4) are cubic, and so a priori there may be contributions of size O(ǫ 5/2 ). However, we will show later that all such contributions arise as terms depending only onξ and ǫ of physical size O(ǫ 3 ); moreover, these putative terms will be shown to vanish by multiscale calculations. Next we derive the evolution equation for σ. First we calculate that Similarly we have Subtracting these two equations then gives the desired evolution equation for σ: The right hand side of (4.5) is O(ǫ 7/2 ) provided we can show that the right hand side of (4.5) is O(ǫ 7/2 ). The formula (2.16) implies that the third term on the right hand side of (4.5) is of size O(ǫ 7/2 ). Before we can show that the rest of the terms are appropriately small, we must study the quantities appearing on the right hand side of these equations further. We will see that estimates for these quantities presuppose a satisfactory bound for the difference H −H, and so estimating this operator in Sobolev space is our first task.

4.2.
Estimates for the Difference Operator H −H. While the operatorH is well suited for multiscale calculation, it remains to be seen howH compares to our original Hilbert Transform H corresponding to the true solution ζ of the water wave system. To do so, we will bound the operator H −H in H s . This entails decomposing it as where Hζ is the Hilbert transform corresponding to the approximate interfaceζ. If we apply Proposition B.1 to the formula of Lemma 3.1 we arrive at The analogous result for the first sum in the decomposition is Let s ≥ 4 be given, and suppose (2.18) holds. Then for all t ≤ T 0 , Proof. We use the fact that this operator can be written in two different ways using integration by parts: Now consider the nth derivative of the first formula. If all n derivatives fall on f , then we can pass to an integral of the second form above via integration by parts. Such an integral can then be bounded in L 2 by either If at least one derivative falls on the logarithm, then we have a kernel of the form This yields a singular integral which can be bounded in H n by either The proposition follows by summing these bounds n = 0, 1, . . . , s.
Combining these lemmas yields the We will also need to estimate the operator D t (H −H). To do so, it will suffice to consider the commutator [D t , H −H].
Proof. We decompose H −H = (H − Hζ) + (Hζ −H) and estimate each term separately. We begin with the latter operator and apply Lemma 2.1 to (3.4). Using the product rule, this results in a sum of singular integrals whose numerators are products of differences involving the functionsξ, ζ (n) , D tξ , D t ζ (n) , n = 1, 2, 3. Then using the identity D t g = (b −b)g α +D t g, we can further split these terms until we arrive at a sum of kernels whose numerators are products of differences involving the functions In order to estimate the terms (b −b)g that arise here for g =ξ, ζ (n) , notice that (2.18), along with (2.13) and Lemma 2.2, shows that The resulting kernels have the properties that (1) each has at least three factors in its numerator of size at most O(ǫ) in the sense of L ∞ , (2) each has the same number of factors in the numerator as in the denominator. In estimating this sum of singular integrals we always estimate f in L 2 so as not to lose any half-powers of ǫ. In doing so, the largest number of derivatives of B that appears is inD tζ ; a time derivative will fall on B X in the formula for ζ Next, using Lemma 2.1, we explicitly write the kernel and appealing to the crude bound of b −b above now implies the proposition.

4.3.
Formulas for Remainders of b and A. Applying the energy method to the remainder equations (4.4)-(4.5), we expect to obtain bounds on the quantity: However in (4.4)-(4.5), the quantities b −b, A −Ã, etc., arise as coefficients of the operators P −P and D t (P −P). Moreover, such energy estimates would give bounds on the quantities D t ∂ n α ρ and D t ∂ n α σ, not directly on the quantities r α and D t r. So in the following subsections we must perform the following tasks: (1) Bound b −b in terms of E s and ǫ.
(2) Bound D t (b −b) in terms of E s , ǫ, and a small multiple of D 2 t r.
Observe that in the last step we have relaxed the estimate so that every term is of the optimal size O(ǫ 5/2 ).
It now suffices to consider the difference Estimating each of these terms in H s using Proposition B.1, we sum the bounds under the assumption of (2.18) to find by Corollary 4.1 that for s ≥ 4: and so choosing ǫ 0 and δ so that the coefficient C(ǫ + δ) of b −b H s on the right hand side is less than 1 2 for all ǫ < ǫ 0 yields the bound where the constant C = C(S(T 0 ), B H s+4 ). From this bound and (3.25) we also have b H s ≤ C E 1/2 s + ǫ 3/2 (4.8) Step 2. Controlling D t (b −b) by E s , ǫ, and a small multiple of D 2 t r. To control D t (b −b), we write Step 1 and Corollary 4.1 we have that where the constant C depends only on S(T 0 ) and B H s+4 . To estimate the remaining terms we appeal to the formula (2.17): By a multiscale calculation, the term (I −H)D tb has the property that the residual quantity . Therefore it suffices to estimate the difference between each term in (2.17) with its approximate analogue. We may estimate the first such difference crudely, since by Step 1 we have that where the constant C depends only on S(T 0 ) and B H s+4 , and where we estimated the commutator [D t ζ, H]b α ζα term-by-term. The estimate of the difference proceeds by decomposing in the same manner as in Step 1, and yields the bound C(E s + ǫE 1/2 s + ǫ 5/2 ). Next, by writing D t ζ = D t r + (b −b)ζ α +D tζ , the remaining singular integrals Again decomposing in the fashion of Step 1, we arrive at a sum of commutators all controlled in H s by C(E s + ǫE 1/2 s + ǫ 5/2 ) except for two commutators. The first is which has been expanded using (3.30), and is controlled in H s by Summing all of these estimates, we therefore have for δ chosen sufficiently small that Step 3. Controlling A −Ã in terms of E s , ǫ, and a small multiple of D 2 t r. SinceÃ = 1 by (3.26), it suffices to control A − 1 in H s . The right hand side of the formula (3.26) consists of terms that are almost the same as those in the formula (2.17) for D t b, and so the same methods of estimation will apply. However, from §3.3 we know that A 2 = 0, and so we will want to decompose the right hand side of the formula (2.14) so that it is easily seen that the pure O(ǫ 2 ) contribution vanishes. From (2.14) we have Decomposing the difference corresponding to I 2 as in Step 2, we have The difference corresponding to I 1 is decomposed as follows: Note that in the expressionD 2 tζ , five derivatives fall on B through ζ (3) , and so we need five extra derivatives on B to boundD 2 tζ in H s . Using Step 1 and Corollary 4.1 then gives Now since a multiscale calculation shows that the function consists only of terms of order O(ǫ 3 ), the highest number of derivatives appearing is through the term H 2 ∂ 2 t ζ (3) which contains five derivatives of B. This residual is thus controlled in H s by C( B H s+5 )ǫ 5/2 . Combining these estimates, we can choose ǫ 0 and δ sufficiently small so as to arrive at the following estimate for A −Ã: Step 2 and possibly choosing δ and ǫ 0 smaller still allows us to give the following preliminary bound for A −Ã: Step 4. Bounding D 2 t r in terms of E s , ǫ, and a small multiple of A−Ã. We start by deriving a formula for D 2 t r. Changing variables via U κ −1 in (1.2) yields the equation Pζ = −i and so decomposing as ξ =ξ + r yields By Proposition 3.2 we have Pξ H s ≤ Cǫ 5/2 with the constant depending on S(T 0 ) and B H s+5 . Next, using Step 1, (3.30) and (4.10) gives We also have Finally we have from (4.10) that Combining these estimates through (4.11) gives Hence we can choose ǫ 0 and δ sufficiently small so that where the constant C depends only on S(T 0 ) and B H s+5 . Then we immediately have by virtue of Step 3, as well as from Step 2. From this last inequality we have Note that from (4.11), applying (4.13), (4.15), we also have the estimate and hence if we choose δ and ǫ 0 sufficiently small, we conclude that Step 5. Showing that D t ρ, σ and D t σ are equivalent to r α and D t r. In the sequel we will show that the energy constructed from the equations of §4.1 is bounded below by the sum Therefore, this energy will control E s provided we can show that E s is bounded above by this sum. We will show that this is the case with the following three claims.
which by virtue of (4.2) and the above bound on D t r − I yields Hence for sufficiently small δ and ǫ 0 the claim follows.2 Claim 2. Given s ≥ 4, then for δ and ǫ < ǫ 0 chosen sufficiently small we have for all n = 0, 1, . . . , s that Proof of Claim 2. First note that for every n = 0, 1, . . . , s we have The latter term can be easily estimated by C(E 1/2 s + ǫ 3/2 ) 2 using the product rule, Claim 1, and Step 1. Therefore it suffices to bound D 2 t r − D t σ in H s . Again denote I := The last two terms are controlled by C(E s +ǫE 1/2 s +ǫ 5/2 ) by Step 1, (4.14) and by a multiscale calculation. Using Proposition 2.1 we can write t ξ Now we effect the usual decomposition of all of these terms. The terms which are not of size O(ǫ 5/2 ) are This completes the estimate of the term D 2 t r − D t I , and hence the claim.2 Claim 3. Given s ≥ 4, for δ and ǫ < ǫ 0 chosen sufficiently small, we have for all n = 0, 1, . . . , s that Proof of Claim 3. First observe that we can write and thus Taking the L 2 norm of this equation, using (4.2) and summing over n = 0, 1, . . . , s yields and so the claim follows.2 Summary of Estimates. Hence we have shown that for s ≥ 4, there exists an ǫ 0 > 0 and a δ > 0 so that if (2.18) holds, then for all 0 < ǫ < ǫ 0 , the quantity b is bounded in H s by C(E 1/2 s + ǫ 3/2 ), and the quantities are bounded in H s by C E s + ǫE 1/2 s + ǫ 5/2 , where the constant C depends only on S(T 0 ) and B H s+7 . It is also useful to note that under the same conditions, Finally, from step 4 we have that for δ and ǫ 0 sufficiently small, from Step 5 and (4.16) we have that for δ and ǫ 0 sufficiently small, 4.4. The Estimates of the Cubic Nonlinearities in the Equations for the Remainder. Now that we have satisfactory estimates of the remainders of the auxiliary quantities, we can show that the right hand sides of (4.4) and (4.5) are sufficiently small to provide suitable energy estimates. We begin by controlling the quantities appearing in the right hand side of (4.4).
Proposition 4.2. Let s ≥ 4 be given. Then there exist ǫ 0 , δ so that if (2.18) holds, then for all ǫ < ǫ 0 , Proof. By (4.4) we must estimate the terms We estimate these terms in steps. We make the blanket assumption that all constants C may depend on S(T 0 ) and B H s+7 .
Step 1. We collect in this step terms with immediate bounds. We have already seen though Proposition 3.2 that [P,H]ξ H s ≤ Cǫ 7/2 . We also have by Corollary 4.1 that By (3.31) and the estimates we obtained in Section 4.3, we have Next, where as usual we estimated the former term with Proposition B.1 and the latter term crudely in H s . By (4.2), (4.3) and Corollary 4.1 we have Step 2. Next we consider the integral Since this integral is cubic, the only way it will contribute a term larger than O(ǫ 7/2 ) is if it contributes a term independent of r of order O(ǫ 3 ). To see that this does not occur, we decompose the integral in the same way as in Step 2 of §4.3. Decomposing the differences in the numerator of the integrand by writing yields a sum of integrals depending on r or b −b which are controlled in H s by as well as the following integral: Next, decomposing the differences in the denominator of this integral via the identity yields a sum of integrals controlled in H s by ExpandingD tζ andξ α in powers of ǫ and collecting like powers yields a sum of integrals controlled by Cǫ 7/2 except for the leading term of size O(ǫ 3 ) given by the integral which is also controlled by Cǫ 7/2 by Corollary 3.1.
Step 3. We turn to the term Decomposing the differences in the numerator of the integral as in Step 2 yields a sum of singular integrals. All but one of these singular integrals depends on r and are controlled in H s by C ǫE s + ǫ 2 E 1/2 s + ǫ 7/2 The remaining singular integral is given by of which the leading term is isolated by expandingζ = α + ǫζ (1) + ǫ 2 ζ (2) + ǫ 2 ζ (3) , yielding By the same calculation in §3. 3 showing that the I 1 term of G 3 vanished, we see that this leading term is actually O(ǫ 4 ) by Corollary 3.1. Therefore only terms of size O(ǫ 4 ) appear, and so we have that Similarly, the same method of decomposition allows us to expand G in the same way, until the leading term of the part of the decomposition that is independent of r is apparent. However, this leading term of size O(ǫ 3 ) is by construction equal toG, with which it cancels. Therefore G −G and hence the whole right hand side of (4.4) is bounded in H s by C(E Next we consider the right hand side of (4.5).
Proof. It suffices to show that the following terms are O(ǫ 7/2 ): Clearly I 7 H s ≤ Cǫ 7/2 and I 5 H s ≤ Cǫ 7/2 . By (3.31), and the estimates of §4.3 we have that Using Lemma 2.1 along with Proposition B.1, we can decompose D t G into a sum of singular integrals as in Step 2 of §4.3. Each of these integrals can be bounded by C(E 3/2 s +ǫE s +ǫ 2 E 1/2 s ) except forD tG , which has leading term of size O(ǫ 3 ); but then I 6 is O(ǫ 7/2 ). Similarly, if we effect the usual decomposition on the right hand side of the formula (2.16), we see that the only term not of size O(ǫ 5/2 ) is the term The only term left to estimate is I 1 . We first write and by Step 5 of §4.3 we have that the latter term is bounded by Finally, we have by (4.2) that Therefore (I − H)D 2 t r H s ≤ C(E s + ǫE 1/2 s + ǫ 5/2 ), from which the Proposition follows.

4.5.
Construction of the Energy for the Remainder. In this section we construct the energy corresponding to the equations (4.4) and (4.5). We then show that this energy obeys a differential inequality which yields a priori bounds on a O(ǫ −2 ) time scale. The energy so constructed will control the quantity D t r 2 H s + D 2 t r 2 H s , and hence by (4.16) it follows that for sufficiently small energies also yields suitable bounds on E s .
Bounds on the Equations for the Derivatives. We must first show that the nonlinearities in the corresponding equations for the derivatives are appropriately bounded in L 2 .
where C depends only on S(T 0 ) and B H s+7 .
Proof. Let Θ = ρ, σ as above. Observe that for any n ≥ 1 we can write we rewrite as we have by the product rule, Steps 2 and 3 of §4.3, (4.18) and Proposition 4.2 that for all 1 ≤ n ≤ s, where the last inequality follows from Step 5 of §4.3.
Construction of the Energy and the Energy Inequality. Now that we have shown that the equations for the derivatives of the quantities in (4.4) and (4.5) also have O(ǫ 7/2 ) nonlinearities, we can construct the energies corresponding to these equations. Since the principal operator of (4.4) and (4.5) is P, we can use the same construction given by Lemma 4.1 of [13] to construct our energy; we record this lemma here for convenience.
Moreover if Θ is the trace of a holomorphic function on Ω(t) c , i.e., if Θ = 1 2 (I − H)Θ, then For brevity, we introduce the quantities ρ (n) := ∂ n α ρ and σ (n) := ∂ n α σ (4.23) We cannot use the second part of Proposition 4.5 directly for n > 0 since ρ (n) and σ (n) need not be the trace of a holomorphic function on Ω(t) c . Hence we further introduce the notation Consider now the case 0 ≤ n ≤ s. Define and We must show that the parts contributed by dEn dt to the energy inequality by the parts of these terms that are antiholomorphic in Ω(t) c are at most of size O(ǫ 5 ).
Observe first that we can write and so R (n) is bounded by C(E s + ǫE Differentiating this with respect to t and integrating by parts yields We want to show that the right hand side of this inequality is O(ǫ 5 ). By Proposition 4.4 and (2.16) it is clear that the first integral is O(ǫ 5 ), and so it suffices to show that the second integral is of size O(ǫ 5 ). The arguments for handling the first two terms rely on the fact that φ (n) and R (n) are almost orthogonal in L 2 , and so the arguments showing these terms are small are similar to each other, so we will only consider the term R and since the latter term is O(ǫ 5/2 ), it suffices to consider only the former term. Likewise, recalling that the adjoint 3 H * of the Hilbert transform satisfies the identity H * f = −ζ α H(f /ζ α ), the identity [H, ∂ α /ζ α ] = 0 of Proposition 2.1 implies that ∂ α H = −H * ∂ α , and so we can write φ But now, using the usual L 2 pairing 4 , , we have 1 4 and so these integrals are bounded by From (4.27), estimating as usual gives bound of R (n) s ) and C(E s + ǫE 1/2 s + ǫ 5/2 ), respectively. Summing these bounds, we have that (4.28) reads If we try to apply the same argument to F n as we just did to E n , we find that dFn dt has an extra half-derivative than can be controlled by the energy, since F n consists of quantities with one time derivative more than the quantities comprising E n . Now since which by Step 5 of §4.3 and Proposition 4.4 implies Hence we need only show that the quantity F n itself is bounded below by D t σ (n) 2 L 2 up to a term of size O(ǫ 5 ), for n = 0, . . . , s. By writing σ (n) = ψ (n) + S (n) we can use Proposition 4.5 to estimate The adjoint T * of a linear operator T : L 2 → L 2 is defined by f T * (g) dα = g T (f ) dα for all f, g ∈ L 2 . 4 Here we use the real inner product f, g = f g dα for f, g ∈ L 2 . Now, as with R (n) , we can rewrite From the above formula for ψ (n) we see that ψ (n) α has one more spatial derivative than the energy provides. However, if we integrate by parts and use Step 5 of §4.3, we can estimate and if we choose δ sufficiently small, then we have by (4.20) that Thus if we choose ǫ 0 and δ still smaller, we have from the inequality that the following lemma is demonstrated: Let E be defined as in (4.29). Then there exists an ǫ 0 > 0 and a δ > 0 so that if (2.18) holds, then there is a constant C = C(ǫ 0 , δ) so that for all ǫ < ǫ 0 , where the constants C depend only on ǫ 0 and δ.

Long time existence of wave packet-like solutions
We would like to show that for wave packet-like data, the solution of the water wave system (1.2)-(1.3) exists on the O(ǫ −2 ) time scale, and is well approximated by the wave packet whose modulation evolves according to NLS. Thus far we have found a globally existing approximationζ, as well as a suitable a priori bound on the energy of the remainder r for O(ǫ −2 ) time scales. Sinceζ does not in general satisfy the water wave system, the wave packet data (ζ(0),D tζ (0),D 2 tζ (0)) cannot be taken as the initial data for the water wave system (1 .2)-(1.3).
In what follows, we will show that there is data for the water wave system that is within O(ǫ 3/2 ) to the wave packet (ζ(0),D tζ (0),D 2 tζ (0)). Moreover for all such data, the solution of the system (1.2)-(1.3) exists on the O(ǫ −2 ) time scale. The a priori bound on r gives the estimate of the error between ζ and the wave packetζ on the order O(ǫ 3/2 ) for time on the O(ǫ −2 ) scale. The appropriate wave packet approximation to z is then obtained upon changing coordinates back to the Lagrangian variable.
To prove Part 2, we note that since ξ 0 H s+6 and ξ (0) H s+6 do not exceed δ, we can estimate that from which Part 2 follows. Since the construction of v 0 is determined by ζ 0 , Part 3 is now shown in the same way as was Part 2 once we observe thatD tζ (0) ∈ H s+4 and H ζ 0 is bounded from H s+4 to H s+4 . We now prove Part 4. By the definition of w 0 we have Since we are assuming ξ 0 and v 0 are constructed as above, we can in the above formula for A 0 − 1. As usual, we can isolate the O(ǫ 2 ) leading term and see that it vanishes by a multiscale calculation, and what remains gives us the estimate But then we have by the above that w 0 − ǫ(iω) 2 ζ (1) (0) H s+4 ≤ Cǫ 3/2 for a sufficiently small choice of ǫ 0 .
Definition. We call (ξ 0 , v 0 , w 0 ) a B 0 -admissible initial data if (ξ 0 , v 0 , w 0 ) ∈ A s and there is a constant C depending only on B 0 H s+7 so that It is clear that for B 0 -admissible initial data, we have

5.2.
Long-Time Existence of ζ and z. In this section we will make rigorous the existence and uniqueness of the solutions z and ζ on the appropriate O(ǫ −2 ) time scales. We begin with the following local well-posedness (c.f., [12], [13]): Theorem 5.1. Let n ≥ 5 be given. Suppose that initial data ξ 0 , v 0 , w 0 are given so that satisfy the water wave system: i.e.v 0 = H z(0)v0 , and w 0 = ia 0 ∂ α z(0) − i for some real valued function a 0 . Suppose further that z(0) = α + ξ 0 (α), α ∈ R defines a chord-arc curve: i.e. there exists ν > 0, such that Then there exists a T 0 > 0 so that the system (1.2)-(1.3) with initial data z(0) = ξ 0 + α, z t (0) = v 0 , z tt (0) = w 0 has a unique solution z(α, t) for t ∈ [0, T 0 ] with the property that there exist constants C = C(T, ∂ α ξ 0 H n−1/2 , v 0 H n+1/2 , w 0 H n , ν) and µ > 0, Moreover, if T * is the supremum over all such T 0 , then either T * = ∞ or Given this result, we take any B 0 -admissible initial data (ξ 0 , v 0 , w 0 ) ∈ A s and use Theorem 5.1 to construct a solution z = z(α, t) on the time interval [0, T 0 ] with (z α (t)−1, z t (t), z tt (t)) ∈ H s × H s+1 × H s+1/2 . Using this solution we construct the change of variables on [0, T 0 ] as in §2. In order to use this change of variables to control ζ in terms of z, we need the following elementary calculus lemma.
Lemma 5.2. Let n ≥ 3, let f ∈ H n , and let γ ∈ H n be given with γ ′ (α) ≥ c 0 > 0 for all α ∈ R and γ ′ − 1 H n−1 ≤ M. Then Proof. First we have which proves (1). To prove (2), first observe that Now let n ≥ 3 and let 2 ≤ j ≤ n be an integer. By the chain and product rules there exist polynomials p l,j (γ ′ , . . . , γ (j−1) ) of total degree 5 at most j such that The lemma follows upon estimating the first term with f ′ • γ in L ∞ and ∂ (j−1) α (γ ′ − 1) in L 2 , and the remaining terms with f (l) • γ in L 2 by (1) and p l,j in L ∞ .
To use this Lemma to change from the ζ quantites back to the z quantities, we need control of κ α − 1 in terms of z α − 1 in H s .
Proof. Differentiating (2.3) with respect to α we get z α Now the Lemma follows from Lemma 2.2 and the H n boundedness of the operator H.
Since we have chosen initial data that is O(ǫ 1/2 ), by Theorem 5.1 there is an interval [0, T 0 ], such that for all t ∈ [0, T 0 ], both z α (t) − 1 H s and z t (t) H s are of order O(ǫ 1/2 ). Also by Lemma 5.3 we have that κ α − 1 H s is of order O(ǫ 1/2 ). Then for ǫ < ǫ 0 we can choose ǫ 0 > 0 so small that κ α − 1 L ∞ ≤ 1 2 and κ α − 1 H s ≤ 1. Applying Lemma 5.2, we can choose ǫ 0 sufficiently small so that for all times t ∈ [0, T 0 ], where δ is the quantity required by (2.18). This now justifies the a priori bound (2.18) on [0, T 0 ]. Since we now legitimately have such a bound, all of the work through Proposition 4.6 now holds on [0, T 0 ] for δ and ǫ 0 chosen sufficiently small. We are now ready to prove the main Theorem 5.2. Let s ≥ 6 and k > 0 be given. Let B 0 ∈ H s+7 , and T > 0 be given. Denote by B(X, T ) the solution of (3.20) with initial data B(0) = B 0 , and let ζ (1) be defined as in (3.12). Then there exists an ǫ 0 = ǫ 0 ( B 0 H s+7 ) > 0 so that for all ǫ < ǫ 0 the following holds: there exists initial data (ξ 0 , v 0 , w 0 ) ∈ A s for the system (1.2)-(1.3) satisfying and for all such initial data, there exists a possibly smaller Proof. Given our initial data, we have shown that there is some time interval [0, T 0 ] on which a solution to (1.2)-(1.3) exists with that initial data. We have also shown that for sufficiently small ǫ 0 the a priori bound ( In particular, this estimate holds with a constant C independent of T * .
In order to use this bound on ζ to in turn control z, we would like to show that the change of variables κ can be constructed in terms of ζ so that it is controlled independently of T * . This will imply that there are similar a priori estimates for z, and so the long-time existence with appropriate regularity will then follow from the blow-up criterion of Theorem 5.1.
We know κ(α, t) satisfies κ t (α, t) = b(κ(α, t), t) κ(α, 0) = α (5.7) 45 with b determined through (2.13). Writing (5.7) in integral form, differentiating with respect to α, and using Lemma 5.2 then gives the bound Taking the supremum over 0 ≤ t ≤ T * and choosing ǫ 0 sufficiently small then yield where the constant C depends on T , and is independent of T * . Now on [0, T * ], we have that ζ(κ(α, t), t) = z(α, t). Hence if we apply Lemma 5.2 we have for t ∈ [0, T * ], where the constants C are independent of T * . Thus it follows by the blow-up criteria given in Theorem 5.1 that we can continue the solution z to t ∈ [0, T 1 ] for some T 1 > T * . On the other hand, we can choose ǫ 0 so small that for ǫ < ǫ 0 , the bounds Cǫ 1/2 in (5.6) and (5.8) are small enough that there exist T * < T 2 < T 1 , so that on [0, T 2 ], κ α − 1 L ∞ ≤ 1 2 , κ α − 1 H s ≤ 1 and the a priori estimate (2.18) holds. This contradicts the maximality of T * . Therefore we must have T * = T ǫ −2 and the long time existence of z follows. The error estimate (5.5) then follows from (5.3) and Proposition 4.6.
There is still the matter of interpreting this result in more familiar coordinates. Changing variables by κ, we can convert the estimates of the above theorem into estimates in Lagrangian coordinates: Calculating the asymptotic expansion of z α − 1, z t , z tt now depends on understanding κ − α. From (5.7) we have that Using our estimate of κ α −1 H s ≤ Cǫ 1/2 and writing the integrand as b = (b−b)+ǫ 2 b 2 +ǫ 3 b 3 yields the following leading order expression: From (5.10), we can obtain and justify asymptotics for ∂ α ℑz, z t and z tt without any additional restriction on the initial data. However, justifying the asymptotics for ℜz α −1 requires an understanding of the asymptotic for κ α up to order O(ǫ 3/2 ), which is not available merely from the estimates given in Theorem 5.2. We therefore leave open the justification of the modulation approximations for ℜz α − 1. Note that the leading term of the right hand side of (5.10) can be as large as O(1) on times of order O(ǫ −2 ), and so would contribute corrections to the asymptotic formula for ℜz α − 1.

Justification of an Eulerian Version
By imposing some additional mild restrictions on the initial data, we are able to obtain justifications of the derivative in the space variable of the interface and the trace of the velocity field on the interface in Eulerian coordinates. With further restrictions on the initial data, we are able to justify the asymptotics for the profile itself. All these reduce to obtaining an appropriate bound and, in the latter case, asymptotics for ℜζ(α, t) − α in C([0, T ǫ −2 ]; L 2 ), which can be achieved by introducing another quantity as follows.
Following the proof of Proposition 2.3 of [13], we introduce the velocity potential Φ(x, t) of the fluid in the domain Ω(t) that satisfies ∇Φ = v. Let ψ(α, t) = Φ(z(α, t), t) be the trace of Φ on the interface Σ(t). If we write Ψ = ψ • κ −1 , then the time derivative of the quantity λ := (I − H)Ψ is comparable to the imaginary part of ζ through the identity (c.f. (2.46) of [13]): We also know by Proposition 2.3 of [13] that λ satisfies an evolution equation of the form Since G λ is of third order and depends only on ζ α − 1, D t ζ, D 2 t ζ, we expect that we can construct an energy from this equation that allows us to bound D t λ by Cǫ 1/2 , provided the initial energy is bounded by Cǫ. This is enough to control ℜζ(·, t) − α L 2 and justify an Eulerian version of Theorem 5.2. The details are given in Section 6.1 below.
However, with further restrictions on the initial data we can justify asymptotics for the profile itself, and we will devote the remainder of Section 6 to this task. Specifically, we will develop an approximate solutionλ to (6.2) to the desired order O(ǫ 4 ) and thereby construct an energy for the remainder l = λ −λ. As was the case with the quantities D t ρ and D t σ, such an energy will bound the L 2 norm of D t l for O(ǫ −2 ) times. This will allow us to justify asymptotics for the profile under reasonable restrictions on the initial profile and the initial velocity potential restricted to the initial interface.
Proof. We begin by deriving conditions under which ℜ(ζ) − α is controlled in L 2 . We can construct the energy corresponding to (6.2): Since λ is the trace of a holomorphic function on Ω(t) c , we have by Proposition 4.5 that D t λ 2 L 2 ≤ CL(t). Formula (6.1) provides the estimate Clearly we also have (I − H)ℑζ L 2 ≤ C ξ L 2 . However, by (2.8) and Lemma 2.2, we conversely have that Hence it suffices to show that L(t) is O(ǫ) whenever t ≤ T ǫ −2 . Now by Proposition 4.5 and Theorem 5.2 the energy L satisfies Solving this inequality gives us that sup 0≤t≤T ǫ −2 Hence the question now reduces to asking which conditions on the initial data imply that L(0) is O(ǫ). We first have that and so to control this part of L(0) it suffices to take ξ 0 L 2 ≤ Cǫ 1/2 . The second part of L(0) takes more work. Recall that our parametrization for the initial data was chosen so that ζ(0) = z(0). Let ψ 0 , λ 0 , etc., be the initial values of ψ, λ, etc., respectively. To estimate the second part of L(0), we follow the discussion of initial data in section 5.1 of [13]. Observe that we can choose a function Ξ 0 holomorphic in Ω(0) for which ℜ(Ξ 0 ) • ζ 0 = Ψ 0 , specifically Ξ 0 • ζ 0 = (I + H ζ 0 )(I + K ζ 0 ) −1 Ψ 0 ; such a function will satisfy ∂ z Ξ 0 = v 0 . Since we have the operator identity 6.3. Estimates of the Remainder of λ. Our goal here is to construct an energy from an evolution equation for l = λ −λ (6.12) This will enable us to show that the quantity D t l = D t (λ −λ) is bounded in L 2 . In turn, we will control r in L 2 for O(ǫ −2 ) times. is of size at most Cǫ 3/2 in L 2 . Hence, using (6.1), we have that D t l = −(I −H)ℑ(r)+O(ǫ 3/2 ). But then this implies the bound 1 C r L 2 − Cǫ 3/2 ≤ D t l L 2 ≤ C r L 2 + Cǫ 3/2 (6.14)

Appendix A. Glossary of Symbols
We collect the commonly used notations and symbols. References such as (1.1) refer to the equation in which the symbol was introduced, whereas p. 1 refers to the page number in which the symbol is first used.

N
The set of nonnegative integers R The set of real numbers C The set of complex numbers w The complex conjugate of w ∈ C ℜ(w) The real part of w ∈ C ℑ(w) The imaginary part of w ∈ C

[F, G]
The commutator F G − GF of the operators F and G U g Precomposition by g, p. 2

·, ·
The real inner product on L 2 , p. 38 The new water wave interface variable, (1.5) ξ The perturbation of ζ from the still water solution, p. 5 H The Hilbert transform associated to ζ, p. 3 K The double layer potential operator associated to the curve ζ, p. 51 H γ The Hilbert transform associated to the curve γ, p. 3 K γ The double layer potential operator associated to the curve γ H 0 The Hilbert transform associated to the curve α, p. 3 The wave number of the wave packet approximation to ζ, p. 12 ω The wave frequency of the wave packet approximation to ζ, p. 12 φ The phase of the wave packet approximation to ζ, p. 12 A s The manifold of admissible initial conditons for (z, z t ), p. 41 + c.c.
Adds the complex conjugate of the preceding term, p. 52