An Existence Theory for Gravity–Capillary Solitary Water Waves

In the applied mathematics literature solitary gravity–capillary water waves are modelled by approximating the standard governing equations for water waves by a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). These formal arguments have been justified by sophisticated techniques such as spatial dynamics and centre-manifold reduction methods on the one hand and variational methods on the other. This article presents a complete, self-contained account of an alternative, simpler approach in which one works directly with the Zakharov–Craig–Sulem formulation of the water-wave problem and uses only rudimentary fixed-point arguments and Fourier analysis.


The Main Results
The classical water-wave problem concerns the two-dimensional, irrotational flow of a perfect fluid of unit density subject to the forces of gravity and surface tension. We use dimensionless variables, choosing h as length scale, (h/g) 1 2 as time scale and introducing the Bond number β = σ/gh 2 , where h is the depth of the water in its undisturbed state, g is the acceleration due to gravity and σ > 0 is the coefficient of surface tension. The fluid thus occupies the domain D η = {(x, y) : x ∈ R, y ∈ (0, 1 + η(x, t))}, where (x, y) are the usual Cartesian coordinates and η > −1 is a function of the spatial coordinate x and time t, and the mathematical problem is In memory of Walter Craig.
For each sufficiently small value of ε > 0 there exists a symmetric solitary-wave solution of (1.1)-(1.4) whose free surface is given by uniformly over x ∈ R.
(ii) Suppose that β < 1 3 and c 2 = c 2 0 (1−ε 2 ), where c 0 = c(ω) is the global minimum of the linear dispersion relation (see Fig. 1 (right)). For each sufficiently small value of ε > 0 there exist two symmetric solitary-wave solutions of (1.1)-(1.4) whose free surfaces are given by This article presents an alternative, simpler proof of Theorem 1.1 in which one works directly with the Zakharov-Craig-Sulem formulation of the travelling waterwave equations (see below) and uses only rudimentary fixed-point arguments and Fourier analysis. Some intermediate results are special cases of more general theorems available elsewhere; their proofs have been included here for the sake of a complete, self-contained exposition.

Methodology
We proceed by formulating the water-wave problem (1.1)-(1.4) in terms of the variables η and = ϕ| y=1+η (see Zakharov [22] and Craig and Sulem [7]). The Zakharov-Craig-Sulem formulation of the water-wave problem is where the velocity potential ϕ is recovered as the (unique) solution of the boundaryvalue problem and the Dirichlet-Neumann operator G(η) is given by Travelling waves are solutions of the form η( It is possible to reduce Eqs. (1.11), (1.12) to a single equation for η. Using (1.11), one finds that = −cG(η) −1 η x , and inserting this formula into (1.12) yields the equation (1.14) Note the equivalent definition where ϕ is the solution of the boundary-value problem (which is unique up to an additive constant). We proceed by defining the Fourier transformû = F[u] of a function u of a real variable by the formulaû and using the notation m(D) with D = −i∂ x for the Fourier multiplier-operator with symbol m, so that m(D)u = F −1 [mû]. The Ansätze (1.5) and (1.9) suggest that the Fourier transform of a solitary wave is concentrated near the points k = ±ω (which coincide at k = 0 when β > 1 3 ). Indeed, writing c 2 = c 2 0 (1 − ε 2 ), one finds that the linearisation of (1.13) at ε = 0 is with equality precisely when k = ±ω (so that g(ω) = g (ω) = 0 and g (ω) > 0). We therefore decompose η into the sum of functions η 1 and η 2 whose Fourier transformŝ η 1 andη 2 are supported in the region S = (−ω − δ, −ω + δ) ∪ (ω − δ, ω + δ) (with Fig. 3 (a) The support ofη 1 is contained in the set S, where S = (−δ, δ) for β > 1 3 (left) and δ ∈ (0, 1 3 )) and its complement (see Fig. 3), so that η 1 = χ(D)η, η 2 = (1 − χ(D))η, where χ is the characteristic function of the set S. Decomposing (1.13) into one finds that the second equation can be solved for η 2 as a function of η 1 for sufficiently small values of ε > 0; substituting η 2 = η 2 (η 1 ) into the first yields the reduced equation Finally, the scaling (1.20) transforms the reduced equation into for β > 1 3 , while the scaling transforms the reduced equation into for β < 1 3 ; here χ 0 is the characteristic function of the set (−δ, δ), the symbol D now means −i∂ X and precise estimates for the remainder terms are given in Sect. 4. Eqs. (1.21) and (1.23) are termed full dispersion versions of (perturbed) stationary Korteweg-de Vries and nonlinear Schrödinger equations since they retain the linear part of the original equation (1.13); the fully reduced model equations (1.6) and (1.9) are recovered from them in the formal limit ε → 0.
Variational versions of this reduction procedure have previously been given by Groves and Wahlén [12]. Starting with the observation that (1.13) is the Euler-Lagrange equation for the functional they use the decomposition η = η 1 +η 2 (η 1 ) and scaling of η 1 described above to derive reduced variational functionals for ρ and ζ whose Euler-Lagrange equations are given to leading order by (1.6) and (1.9). Critical points of the reduced functionals (and hence solitary-wave solutions of the reduced equations) are found by the direct methods of the calculus of variations. In the present paper we apply a more direct perturbative approach introduced by Stefanov and Wright [21] for another full dispersion Kortewegde Vries equation, namely, the Whitham equation (see Ehrnström et al. [9] for a variational treatment of this equation). The travelling-wave Whitham equation is Noting that its linear dispersion relation has a unique global maximum at k = 0 (with c(0) = 1), one writes c = 1 + ε 2 and seeks solitary waves of the form which can be rewritten as a fixed-point equation of the form (1.24) In the formal limit ε → 0 we recover the stationary Korteweg-de Vries equation its (unique, symmetric) solitary-wave solution w is nondegenerate in the sense that the only bounded solution of its linearisation at w is w X . Restricting to spaces of symmetric functions eliminates this antisymmetric solution of the linearised equation and a solution to (1.24) can be constructed as a perturbation of w using the implicitfunction theorem. In Sect. 5 we apply the above argument to (1.21) and (1.23), first reformulating them as fixed-point equations. The functions ρ and ±ζ are nondegenerate solutions of (1.6) and (1.9) in the sense that the only bounded solutions of their linearisations at ρ and ±ζ are respectively ρ X and ±ζ X , ±iζ . Observe that equation (1.13) is invariant under the reflection η(x) → η(−x), and the reduction procedure preserves this property: the reduced equation for η 1 is invariant under the reflection η 1 (x) → η 1 (−x), so that (1.21) and (1.23) are invariant under respectively ρ(x) → ρ(−x) and ζ(x) → ζ(−x). Restricting to spaces of symmetric functions thus eliminates the antisymmetric solutions ρ and ±ζ X , ±iζ of the linearised equations, and solutions to (1.21) and (1.23) can be constructed as perturbations of ρ and ±ζ using an appropriate version of the implicit-function theorem.

Function Spaces
We study the equation are the usual Bessel-potential spaces. The decomposition η = η 1 + η 2 , where )η, is accommodated by writing X as the direct sum of X 1 = χ(D)X and X 2 = (1 − χ(D))X , where X 1 and X 2 are equipped with respectively the scaled norm The following proposition yields in particular the estimate for the supremum norm of η 1 ∈ X 1 . We can also estimate higher-order derivatives of η 1 ∈ X 1 using the fact that the support ofη 1 is contained in the fixed bounded set S, so that, for example

Proposition 1.2 The estimate
Proof This estimate follows from the calculation It is also helpful to use the larger space is analytic at the origin and deduce that K, L map the open neighbourhood Moreover, we take advantage of the estimate for η ∈ H 2 (R) to obtain estimates for K and L which are necessary for the reduction procedure described above (see Sect. 3). Note, however, that in the entirety of the existence theory we work in the fixed subset U of H 2 (R) (whose elements are 'wellbehaved' functions).

Analyticity
In this section, we study the operator K given by (1.16) using basic results from the theory of analytic functions in Banach spaces (see the treatise by Buffoni and Toland [6] for a complete account). In particular, we present an elementary proof that , are analytic at the origin (see Sect. 1.3 above). A more comprehensive treatment of the analyticity of operators of Dirichlet-Neumann and Neumann-Dirichlet type in water-wave problems is given by Lannes [18, Ch. 3 and Appendix A] (see also Nicholls and Reitich [19] and Hu and Nicholls [14]). We begin with the boundary-value problem (1.17)-(1.19), which is handled using the change of variable We discuss (2.1)-(2.3) using the standard Sobolev spaces H n ( ), n ∈ N, together with H n+1 ( ), n ∈ N, which is defined as the completion of

Proposition 2.1
For each F 1 , F 2 ∈ H n ( ) and ξ ∈ H n+1/2 (R), n ∈ N, the boundaryvalue problem admits a unique solution u = S(F 1 , F 2 , ξ) in H n+1 ( ) given (with a slight abuse of notation, in that derivatives should be taken) by the explicit formula

Lemma 2.2 For each ξ ∈ H 3/2 (R) and each sufficiently small
and note that the solutions of (2.1)-(2.3) are precisely the zeros of T (·, η, ξ). Using the estimates and are analytic at the origin; it follows that T is also analytic at the origin. Furthermore T (0, 0, 0) = 0 and (because S is linear and F 1 , F 2 are linear in their second arguments), so that d 1 T [0, 0, 0] = I is an isomorphism. By the analytic implicit-function theorem there exist open neighbourhoods N 1 and N 2 of the origin in Z and H 3/2 (R) and an analytic function v : Since v is linear in ξ one can take N 2 to be the whole space H 3/2 (R).
is analytic at the origin.

Corollary 2.4
The formulae (1.14), (1.15) define functions U → L 2 (R) which are analytic at the origin and satisfy Proof This result follows from Corollary 2.3 and the facts that where u j is homogeneous of degree j in η and linear in ξ , and where K j (η), K j (η) and L j (η) are homogeneous of degree j in η (and we accordingly abbreviate K 0 (η) to K 0 ).

Remark 2.5 Note that
We examine the first few terms and in the Maclaurin expansions of K and L in more detail since they play a prominent role in our subsequent calculations. We begin by computing explicit expressions for K 0 , K 1 and K 2 .

Lemma 2.6
(i) The operators K 0 and K 1 are given by the formulae The operator K 2 is given by the formula under the additional regularity hypothesis that η ∈ H 3 (R) and ξ ∈ H 5/2 (R).

Proof (i) The solution to the boundary-value problem
while the solution to the boundary-value problem (ii) Supposing that ξ ∈ H 5/2 (R), so that u 0 ∈ H 3 ( ), and η ∈ H 3 (R), so that u 1 ∈ H 3 ( ) (see Proposition 2.1), we find that the solution u 2 ∈ H 4 ( ) to the boundary-value problem It follows that

Remark 2.7
Explicit expressions for K 3 , K 4 , . . . can be computed in a similar fashion. However, computing an expansion in terms of Fourier-multiplier operators in this fashion leads a loss of one derivative at each order. It is therefore necessary to compensate by increasing the regularity of ξ and η by one derivative at each order.

Corollary 2.8
(i) The function L 2 is given by the formula The function L 3 is given by the formula which is helpful when performing calculations, and some straightforward estimates for the higher-order parts of K and L.

Proposition 2.10
(i) The quantities Proof These estimates follow from the explicit formulae (2.5), (2.6), together with the calculations

Reduction
In this section, we reduce the equation to a locally equivalent equation for η 1 . Clearly η ∈ U satisfies (3.1) if and only if and these equations can be rewritten as We proceed by writing (3.3) as a fixed-point equation for η 2 using Proposition 3.1, which follows from the fact that g(k) |k| 2 for k / ∈ S, and solving it for η 2 as a function of η 1 using Theorem 3.2, which is proved by a straightforward application of the contraction mapping principle. Substituting η 2 = η 2 (η 1 ) into (3.2) yields a reduced equation for η 1 . Note that the reduced equation is invariant under the reflection η 1 (x) → η 1 (−x), which is inherited from the invariance of (3.1) under the reflection η(x) → η(−x) (see below). Theorem 3.2 Let X 1 , X 2 be Banach spaces, X 1 , X 2 be closed, convex sets in, respectively, X 1 , X 2 containing the origin and G : X 1 × X 2 → X 2 be a smooth function. Suppose that there exists a continuous function r : for each x 2 ∈B r (0) ⊆ X 2 and each x 1 ∈ X 1 . Under these hypotheses there exists for each x 1 ∈ X 1 a unique solution is a smooth function of x 1 ∈ X 1 and in particular satisfies the estimate

Strong Surface Tension
Suppose that β > 1 3 . We write (3.3) in the form and apply Theorem 3.2 with the function G is given by the right-hand side of (3.4). Using Proposition 1.2 one can guarantee that η 1 L 1 (R 2 ) < 1 2 M for all η 1 ∈ X 1 for an arbitrarily large value of R 1 ; the value of R 2 is constrained by the requirement that η 2 2 < 1 2 M for all η 2 ∈ X 2 .

Lemma 3.3 The estimates
and using Propositions 2.9 and 2.10(i), one finds that part (i) follows from these estimates and inequality (1.25). Parts (ii) and (iii) are obtained in a similar fashion. (3.4) has a unique solution η 2 ∈ X 2 which depends smoothly upon η 1 ∈ X 1 and satisfies the estimates

Weak Surface Tension
Suppose that β < 1 3 . Since χ(D)L 2 (η 1 ) = 0 the nonlinear term in (3.2) is at leading order cubic in η 1 , so that this equation may be rewritten as To compute the reduced equation for η 1 we need an explicit formula for the leadingorder quadratic part of η 2 (η 1 ), which is evidently given by It is convenient to write η 2 = F(η 1 ) + η 3

Proposition 3.5 The estimates
Proof This result follows from the formula

Remark 3.6 Noting that
and that m(u 1 , v 1 ) has compact support for all u 1 , v 1 ∈ X 1 , one finds that K 0 F(η 1 ) satisfies the same estimates as F(η 1 ).

Lemma 3.7 The quantity
satisfies the estimates for each η 1 ∈ X 1 and η 3 ∈ X 3 .

Proof We estimate
and its derivatives, which are computed using the chain rule, using Propositions 2.9 and 3.5 and inequality (1.25).

Lemma 3.8 The quantity
satisfies the estimates for each η 1 ∈ X 1 and η 3 ∈ X 3 .
Proof We estimate N 2 and its derivatives, which are computed using the chain rule, using Propositions 2.10(i) and 3.5 and inequality (1.25).
Altogether we have established the following estimates for G and its derivatives (see Remark 3.6 and Lemmata 3.7 and 3.8).

Derivation of the Reduced Equation
In this section we compute the leading-order terms in the reduced equations (3.5) and (3.11) and hence derive the perturbed full dispersion Korteweg-de Vries and nonlinear Schrödinger equations announced in Sect. 1. The main steps are approximating the Fourier-multiplier operators appearing in lower-order terms by constants, estimating higher-order terms and performing the scalings (1.20) and (1.22). It is convenient to introduce some additional notation to estimate higher-order 'remainder' terms.

Strong Surface Tension
The leading-order terms in the reduced equation derived in Sect. 3.1 are computed by approximating the operators ∂ x and K 0 in the quadratic part of the equation by constants.

Proposition 4.2 The estimates
the corresponding estimates for their derivatives are trivially satisfied since the operators are linear. The quantity to be estimated in (iii) is quadratic in η 1 ; it therefore suffices to estimate the corresponding bilinear operator. The argument used above yields for each u 1 , v 1 ∈ X 1 , where we have also used Young's inequality.

Lemma 4.3 The estimate
holds for each η 1 ∈ X 1 .
Proof Using Proposition 2.9 and Theorem 3.4, one finds that and because of (2.7) and Proposition 4.2.

Lemma 4.4 The estimate
Proof This result follows from Proposition 2.10(i) and Theorem 3.4.
We conclude that the reduced equation for η 1 is the perturbed full dispersion Korteweg-de Vries equation and applying Proposition 4.2, one can further simplify it to Finally, we introduce the Korteweg-de Vries scaling where R > 0 and ε is chosen small enough that ε 3/2 R ≤ R 1 , solves the equation (note that |||η||| = ε 3/2 ρ 1 , the change of variable from x to X = εx introduces an additional factor of ε 1/2 in the remainder term and the symbol D now means −i∂ X ). The invariance of the reduced equation under η 1 (x) → η 1 (−x) is inherited by (4.1), which is invariant under the reflection ρ(X ) → ρ(−X ).

Weak Surface Tension
In this section we compute the leading-order terms in the reduced equation (3.11) derived in Sect. 3.2. To this end, we write where η + 1 = χ + (D)η 1 and η − 1 = η + 1 , so that η + 1 satisfies the equation (and η − 1 satisfies its complex conjugate). We again begin by showing how Fouriermultiplier operators acting upon the function η 1 may be approximated by constants.

Lemma 4.5 The estimates
and iterating this argument yields (ii); moreover The corresponding estimates for their derivatives are trivially satisfied since the operators are linear. Notice that the quantities to be estimated in (iv)-(vii) are quadratic in η 1 ; it therefore suffices to estimate the corresponding bilinear operators. To this end we take u 1 , v 1 ∈ X 1 . The argument used for (iii) above yields where we have also used Young's inequality. Turning to (v), we note that Estimates (vi) and (vii) are obtained in the same fashion.

Proposition 4.6
The estimate holds for each η 1 ∈ X 1 .
Proof Using equation (2.7) and the expansions given in Lemma 4.5, we find that It follows that because of Lemma 4.5(vi), (vii) and the facts that χ(D)L 2 (η 1 ) = 0 and

Proposition 4.7 The estimate
holds for each η 1 ∈ X 1 .
Proof Observe that in which we have used the calculations and it follows from (2.7) and Lemma 4.5 that

Proposition 4.8 The estimates
Proof Using the estimates for F(η 1 ) and η 3 (η 1 ) given in Proposition 3.5 and Theorem 3.10, we find that and (because of equation (2.5)). It similarly follows from the formula (see equation (2.6) and Remark 2.5) that and using Corollary 2.8(ii) twice yields and

Proposition 4.9 The estimates
Proof This result follows from Propositions 2.10(ii) and 3.5 and Theorem 3.10.

Proposition 4.10
The estimate holds for each η 1 ∈ X 1 .
Proof Using Proposition 3.5 and Theorem 3.10 we find that and furthermore (see Lemma 2.6(i)).
We conclude that the reduced equation for η 1 is the perturbed full dispersion nonlinear Schrödinger equation and applying Lemma 4.5(iii), one can further simplify it to Finally, we introduce the nonlinear Schrödinger scaling where R > 0 and ε is chosen small enough that ε 1/2 R ≤ 2R 1 , solves the equation (note that |||η 1 ||| = ε 1/2 ζ 1 , the change of variable from x to X = εx introduces an additional factor of ε 1/2 in the remainder term and the symbol D now means −i∂ X ). The invariance of the reduced equation under , which is invariant under the reflection ζ(X ) → ζ(−X ).

Solution of the Reduced Equation
In this section, we find solitary-wave solutions of the reduced equations and noting that in the formal limit ε → 0 they reduce to respectively the stationary Korteweg-de Vries equation and the stationary nonlinear Schrödinger equation Theorem 5.1 Let X be a Banach space, X 0 and 0 be open neighbourhoods of respectively x in X and the origin in R and G : X 0 × 0 → X be a function which is differentiable with respect to x ∈ X 0 for each λ ∈ 0 . Furthermore, suppose that There exist open neighbourhoods X of x in X and of 0 in R (with X ⊆ X 0 , ⊆ 0 ) and a uniquely determined mapping h : → X with the properties that

Strong Surface Tension
The first step in the proof of Theorem 5.2 is to write (5.1) as the fixed-point equation and use the following result to 'replace' the nonlocal operator with a differential operator.

Proof Clearly
and g(s) s 2 , s ∈ R.
Using the above proposition, one can write equation (5.5) as It is convenient to replace this equation with (here ε is replaced by |ε| so that G(ρ, ε) is defined for ε in a full neighbourhood of the origin in R). Observe that (1 + |k| 2 )|û| 2 dk and that H 3/4 (R) is a Banach algebra, we find that has the (unique) nontrivial solution ρ in H 1 e (R) and it remains to show that is an isomorphism. Noting that ρ ∈ S(R), we obtain the following result by a familiar argument (see Kirchgässner [17,  This proposition implies in particular that d 1 G[ρ , 0] is a Fredholm operator with index 0. Its kernel coincides with the set of symmetric bounded solutions of the ordinary differential equation 6) and the next proposition shows that this set consists of only the trivial solution, so that d 1 G[ρ , 0] is an isomorphism.

Proposition 5.5
Every bounded solution to the equation (5.6) is a multiple of ρ X and is therefore antisymmetric.

Weak Surface Tension
Theorem 5. 6 For each sufficiently small value of ε > 0 equation (5.2) has two small- We again begin the proof of Theorem 5.6 by 'replacing' the nonlocal operator in the fixed-point formulation of equation (5.2) with a differential operator.
Using the above proposition, one can write equation ( The previous proposition implies in particular that G 1 , G 2 are Fredholm operators with index 0. The kernel of G 1 coincides with the set of symmetric bounded solutions of the ordinary differential equation − a 1 ζ 1X X + a 2 ζ 1 − 3a 3 ζ 2 ζ 1 = 0, (5.7) while the kernel of G 2 coincides with the set of antisymmetric bounded solutions of the ordinary differential equation − a 1 ζ 1X X + a 2 ζ 1 − a 3 ζ 2 ζ 1 = 0, (5.8) and the next proposition shows that these sets consists of only the trivial solution, so that G 1 , G 2 and hence d 1 G[ζ , 0] are isomorphisms.

Proposition 5.9
(i) Every bounded solution to the equation (5.7) is a multiple of ζ X and is therefore antisymmetric. (ii) Every bounded solution to the equation (5.8) is a multiple of ζ and is therefore symmetric.
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