ON THE PRECISE CUSPED BEHAVIOUR OF EXTREME SOLUTIONS TO WHITHAM-TYPE EQUATIONS

. We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni-and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the 120 ◦ interior angle at the crests of classical Stokes’ waves of greatest height. In particular, our results partially settle conjectures for such extreme waves posed in a series of recent papers [13, 15, 35]. Our methods may be generalised to solutions of other nonlocal equations, and can moreover be used to determine asymptotic behaviour of their derivatives to any order.


Introduction
The Whitham equation where ϕ represents the surface profile and is a fully dispersive variant of the classical Korteweg-de Vries (KdV) equation, originally proposed in [37].It features some properties that the KdV equation lacks, such as wave breaking [23,33], highest waves [14,15,35], and better high-frequency modelling [17].While Whitham added the dispersion in an ad hoc manner, the model has since been both justified experimentally and derived from the full water-wave problem in several ways.See for instance [9,24,27], in addition to the aforementioned [17].
Another water-wave model is similarly obtained by making the Boussinesq system -of which the KdV equation can be viewed as a unidirectional version -fully dispersive, so as to arrive at the Whitham-Boussinesq system also called the bidirectional Whitham equation.Here, ϕ again denotes the surface profile, v relates to the fluid velocity at the surface, and the convolution kernel K B is defined by its symbol Strictly speaking, there are several ways to make the Boussinesq system fully dispersive, but (1.2) represents one of the natural candidates that have been investigated in the literature, see e.g.[1,16,28,31].It is also currently the only of these fully dispersive systems that is known to admit highest steady waves [13].
Various steady solutions to the Whitham equations, both the uni-and bidirectional, have been found and studied.Of particular interest to us here are the global, locally analytic curves of periodic steady waves found in [13,15], bifurcating from the line of trivial waves and approaching a so-called limiting highest wave.These are waves whose height reach the maximal value of c/2 for the unidirectional Whitham equation, and c 2 /3 for the bidirectional Whitham equation, where c denotes the velocity of the wave.In the full water-wave problem, it is part of the famous Stokes' conjecture that the analogous highest Stokes' waves have angled crests, with interior angles of exactly 120 • .That is, a highest steady wave with a crest at the origin satisfies as x → 0. This was ultimately proved in [3,32].
For the Whitham equation (1.1), it was conjectured by Whitham 1 [36] that the local behaviour of an analogous highest wave should instead be the cusped variant as x → 0. The authors of [12,15] were able to determine that there indeed was a highest periodic wave φ for the Whitham equation.Furthermore, they showed that any bounded solution reaching that height must satisfy both but did not establish the full limit described in (1.4).More recently, the existence of full global curves of solitary waves up to a highest wave has also been proved [14,35].The same asymptotic estimates (1.5) from [15] apply equally well for these.Furthermore, there is also an innovative computer-assisted proof [18], where a highest periodic wave satisfying the limiting behaviour (1.4) is constructed.A form of local uniqueness, and the convexity of this highest wave is also obtained.The idea is to build an approximate ansatz for the solution using special functions, sufficiently good for a fixed-point argument to go through.A very large number of terms is required, as the map involved is just barely a contraction.A recent paper in the same direction for the Burgers-Hilbert equation is [10].
As for what concerns the bidirectional Whitham equation (1.2), it was shown in [13] that there exists a highest periodic wave φ with a corresponding v that satisfies lim sup The corresponding lower bound is stated, but a flaw in one of the preceding lemmas hinders a correct estimate.This is due to slightly subtle estimates where logarithmic factors are easily lost, making the proof more delicate than for the unidirectional Whitham equation.
The main purpose of this paper is to provide an analytic, and relatively transparent, argument establishing both the limit (1.4) for the Whitham equation, and the analogous result 1 With a minor error in the exact constant, which was pointed out in [15].
as x → 0, for the bidirectional Whitham equation.These results will follow from a somewhat more general method for calculating the local behaviour of solutions to two classes of nonlocal equations on the half-line.As the proofs are quite technical, we first provide some background; describing how these nonlinear waves are related to the more general formulation and results found in Sections 3 to 5.

Background and overview
The Whitham equation (1.1) is a prototypical example of a more general family of nonlocal, nonlinear shallow-water wave models of form where K ∈ L 1 (R) is an even, positive integral kernel that is convex on R + := (0, ∞).Generally, this kernel will arise from a Fourier multiplier symbol K(ξ) of negative order.We shall here consider the orders −1 and −1/2, appearing in the bi-and unidirectional gravity water wave problems, respectively [25].The decay and smoothness of the symbol is realised as a corresponding singularity at the origin of an otherwise smooth kernel K of exponential decay; see [21,34].
Seeking steady solutions ϕ(t, x) = φ(x − ct) to (2.1), one arrives at for some constant A ∈ R after integration.Under quite general conditions, equations like (2.1) have only symmetric solitary waves of elevation [5,7].A similar statement is true for steady periodic waves under an additional reflection assumption [8].This property is inherited via the maximum principle for the elliptic convolution operator.Naturally, this leads to the study of a possible maximal height φ(0) for solutions of (2.2).Supposing now that f is increasing to the left of a nondegenerate local maximum at t = γ, and is sufficiently smooth, we can write with g(0) = 0. Thus, if φ is a solution to (2.2) that achieves φ(0) = γ from below, then vanishing at the origin.Motivated by this computation, we therefore consider the condensed equation where K again has the properties as described after (2.1), and n(0) = 0. We see that any pointwise solution will necessarily have to satisfy u(0) = 0. Equations similar to the one in (2.3) also appear in a plethora of other contexts: examples include harmonic, functional and stochastic analysis.Finally, note further that (2.3) is equivalent to the equation for even functions, where we have conveniently recognised the second-order central difference in the integrand.Whereas the first-order difference in (2.3) is very useful when one wants to establish global estimates for u, (2.4) is able to take direct advantage of the convexity of K.It is therefore especially well adapted for studying u precisely at x = 0. We will consider (2.4) under general assumptions, but first formally outline the theory below for kernels capturing the same singular behaviour.The exact assumptions and rigorous statements follow in Section 3.

Homogeneous singularity (Whitham).
If we replace K in (2.4) with the homogeneous, but merely locally integrable for s ∈ (0, 1), and let n = 0, we obtain the toy equation which in fact has an explicit unbounded solution.It is convenient to introduce as a solution for every s ∈ (0, 1), where B denotes the beta function.
Proof.This is an immediate consequence of the identity which can most easily be seen for s ∈ (0, 1/2) by splitting the integral according to where all but the first term will cancel.Indeed, observe that through the change of variables τ → (τ + 1) and integration by parts.It follows that whence (2.9) holds.Finally, analytic continuation yields (2.9) also for s ∈ [1/2, 1).□ In particular, it is reasonable to expect that well-behaved solutions to (2.4) should still satisfy when K behaves like H 1/2 near the origin; which is the case for a scaled version of the Whithamkernel K W .Under mild conditions, equations such as (2.3) have the feature that solutions are smooth away from where they vanish.This comes from a general "off-diagonal" convolution property for pseudo-differential operators [34], and can be seen as in [15].The behaviour of a solution in the vicinity of the origin arises from a balancing act between the square on the left-hand side, and the asymptotics of the second difference (2.5) as x → 0. As the square root is not regular, one consequently faces an upper threshold on the regularity of u.Simplifying to (2.7), an essential part of the argument in [15] relies on first bootstrapping global C 1/2− -regularity, and then noting that for all α ∈ (0, 1/2) and x ̸ = 0, where (2.12) If we now, for the sake of argument, assume that the supremum of the left-hand side in (2.11) is always achieved for |x| ≤ 1, then we obtain whereupon we can let α → 1/2.
A curious thing about this calculation is that if Φ had been non-negative, then (2.13) would have immediately yielded by (2.9), which would be optimal.Similarly, if one knew that the limit of u(x)/|x| 1/2 existed as x → 0, one could have chosen α = 1/2 in (2.11) and let x → 0 to find (2.10) by dominated convergence.In reality, however, Φ changes from negative to positive at a point τ 0 ∈ (0, 1), as seen in Figure 1; and the existence of a limit is exactly what is difficult to show.To establish (2.10), we therefore identify in (2.9) the significance of the points τ ∈ {τ 0 , 1}, and write (2.4) as x K(y)u(y) dy for 0 < x < ν, or, in essence, under appropriate assumptions.The constant ν > 0 is used to single out a small interval where u has desirable properties, but can otherwise be made arbitrarily small.Its exact value is therefore not important to the theory.Because ν/x → ∞ as x ↘ 0, the last integral will vanish in the limit.
The remaining integrals are less straightforward, and the main obstacle in their treatment is the limited information about monotonicity or the existence of the limit.Our trick here is to consider sequences realising , in a strategic manner.As Φ changes signs at τ 0 , we are thereby able to make the estimates by taking limits in (2.14).This system of inequalities will have solutions described by Figure 2. In addition to the expected solution, which is isolated, there is also a wedge-like set of unwanted solutions for which M > m.A refinement is made to the second inequality of (2.15) to exclude this area, yielding the desired conclusion that m = M = π/2.The shape of the curves in Figure 2 is naturally determined by integrals involving Φ s , but there is some leeway.Therefore, this method works essentially unmodified for a range of homogeneous singularities; not only when s = 1/2.In fact, there is some s 0 ≈ 1/3 such that it works for s ∈ (s 0 , 1), but fails for s ∈ (0, s 0 ).The reason why it breaks down is that the expected solution m = M = β s from Lemma 2.1 stops being isolated, even with the refined inequality.A new idea would therefore be required to proceed past this value.Highest Hölder and Lipschitz waves have been constructed in a number of settings [2,4,6,20,22,26,30], and we expect a similar approach to go through for many such equations.
Logarithmic singularity (Bidirectional Whitham).For order −1, the singular behaviour of the kernel is instead captured by so that homogeneity is replaced by additivity.One finds that differs substantially from (2.8), in that does not depend on x at all.This leads to an entirely different set of estimates, and, in turn, changes the relative importance of the integrals appearing in governing equation.The qualitative behaviour of Λ is still the same as Φ in Figure 1, but in the logarithmic case the contribution of the entire interval (0, x) turns out to be negligible in the limit.In fact, the final estimate hinges only on an integral over (x, ν).Explicitly, writing (2.4) as x K(y)u(y) dy, we see that the analogue of (2.14) becomes The first integral is killed in the limit when u(x)/(x log(1/x)) is bounded, and the third integral is still negligible as before.The limit "should" therefore be in this case.Contrary to what we saw for a homogeneous singularity, it is possible to obtain (2.18) directly using the aforementioned approach with sequences.There is no need to thereafter go through a system of inequalities.

Setup
We have seen that, after an appropriate change of variables, the highest waves of both (1.1) and (1.2) satisfy an equation of the form (2.4).The following assumptions are made on the objects involved, where R + 0 := R + ∪ {0}: Assumption 2. The kernel K ∈ L 1 (R) is even, positive, and convex on R + .Moreover, it admits a decomposition K = S + R, where the singular part S is of the form , and the regular part R has a weak second derivative R ′′ ∈ L 1 (R).
The regularity of n in Assumption 1 is only needed for the limit of the derivative in our main result, not for the behaviour of u(x) itself.One similarly would need to demand higher regularity of n in order to prove asymptotics for higher order derivatives of u.The assumption of continuous differentiability of u close to the origin in Assumption 3 is in fact redundant under the other properties, see Section 5.All of the assumptions may be further weakened, but as added generality would come at the expense of clarity, we shall not push this question further here.
Our main result is the following.
Main Theorem.Suppose that the above Assumptions 1 to 3 hold.If the singular part of K takes the logarithmic form L(x) = log(1/|x|), then the solution u admits the limits while if it takes the homogeneous form H(x) = |x| −1/2 , then u admits the limits This theorem combines Propositions 4.2, 4.4, 5.5 and 5.7, of which the first two are proved in Section 4 and the latter two in Section 5.These results are in turn employed in Section 4.2 to both establish the limit (1.6) and global regularity for the bidirectional highest waves obtained in [13], and in Section 5.2 to prove the limit (1.4) for the unidirectional highest waves obtained in [15,35].Furthermore, immediately preceding Section 5.2, we outline how one would determine the asymptotic behaviour of derivatives to any order.
For reference, we include the following corollary, which lists the implied asymptotic behaviour for the highest waves in the Whitham equations.The precise details are, as explained above, presented in Sections 4.2 and 5.2.

Corollary 3.1 (Whitham equations, abridged). Let φ denote the surface profile of a highest wave, with a peak at zero, of the Whitham equation. Then
The corresponding limit for a highest wave in the bidirectional Whitham equation is 3.1.Preliminaries.We will here list a few useful properties of the kernel K = S + R that follow from Assumption 2. The first lemma shows that the tail of δ 2 x K is both nonnegative and small, which will later ensure that it may be disregarded when analysing the local behaviour of u near the origin.Introducing the antiderivative for the kernel will occasionally be useful.
Proof.For 0 ≤ x < y, the convexity of K on R + immediately yields and by virtue of (3.1), we get for all 0 ≤ x < ν.Here we have used that −K ′ is nonincreasing on R + .□ The following lemma likewise demonstrates that δ 2 x R is small, and so it too will have a negligible effect on the local behaviour of u.

Lemma 3.3. The second difference δ 2
x R is integrable and satisfies Proof.By Assumption 2, R ′′ is integrable, and so for all x ≥ 0. For the second part, we note that R = K − S is necessarily even; and so R ′ is odd.Since R ′ is also absolutely continuous, we therefore conclude that R ′ (x) = x 0 R ′′ (y) dy, which gives the desired bound.□

Logarithmic kernel
In this section, we adopt Assumptions 1 to 3, and specifically assume that the singular part of the kernel K is of the form S(x) = L(x) = log(1/|x|).Additionally, we restrict x to an interval (0, ν] throughout, for some 0 < ν ≪ 1 such that u is continuously differentiable and increasing on (0, ν].This is possible due to Assumption 3.
Since we will prove the first two limits of Main Theorem here, we naturally introduce the shorthand and which is well-defined for all x ∈ (0, ν].We also adopt the function Λ from (2.17), whose utility comes from the identity which holds by linearity of δ 2 x .Seeking to determine the limit of g at zero, we begin with a lemma that asymptotically rephrases (2.4) in terms of g.Lemma 4.1.With ℓ and g as in (4.1) and (4.2), respectively, we have the equation as x → 0.Moreover, the square bracket is nonnegative and satisfies while the final term admits the bound as x → 0.
Proof.Dividing each side of (2.4) by ℓ(x) 2 , we find where, since lim x→0 n(u(x)) = 0 by Assumptions 1 and 3, the left-hand side is indeed like that of (4.4).As for the right-hand side, notice that when ρ is defined through x K(y)u(y) dy.We exploit that u is increasing on (0, ν] to conclude that ρ is bounded on this interval.This is because 3) and Lemma 3.3.In particular, the first term on right-hand side of (4.7) is o(1)g(x), and after using u(y) = ℓ(y)g(y) for the second term, we obtain the right-hand side of (4.4).
Next, the nonnegativity of the expression inside the square bracket in (4.4) is an immediate consequence of the first part of Lemma 3.2.To prove (4.5), we use Lemma 3.3 and the boundedness of ℓ on (0, ν] to conclude that ν 0 δ 2 x R(y)ℓ(y) dy = O(x 2 ).Thus 2 x dτ from (4.3) and the change of variables τ → τ x.By simplifying the integrand, setting z = ν/x, and splitting the integral, this last limit is equal to The first equality follows from an application of L'Hôpital's rule to each of the two terms, while the second follows from the observation that Λ(τ ) = 1/τ 2 + O(1/τ 3 ) as τ → ∞.The latter can be seen directly from its definition in (2.17).Finally, the bound in (4.6) follows from Lemma 3.2 and u being nonnegative and bounded.□ We are ready to prove the first limit of Main Theorem.Proposition 4.2.Under Assumptions 1 to 3, the solution enjoys the limit when K has a logarithmic singularity.
Proof.With g as in (4.2), our strategy is to prove that lim sup which clearly implies the desired limit.We first prove that m ≥ 1/2.The function g defined by is nondecreasing on (0, ν], and we find as x → 0 from (4.4).Here we have used both (4.5), and the nonnegativity of the square bracket and the final term.Choose now a sequence {x k } k∈N ⊂ (0, ν] realising m.Assuming, for the sake of contradiction, that m = 0, we may specifically ensure that g = g along this sequence.This is possible by positivity and continuity of g on (0, ν].Then (4.11) yields as k → ∞, after division by g(x k ).Thus, in fact, m > 0, and we instead arrive at from (4.11).Taking the limit ν → 0, we conclude that m ≥ 1/2.For M , we similarly define which is nonincreasing on (0, ν], and find from (4.4).The last term can no longer be discarded, but can be combined with the first term.This is because of (4.6), and the fact that m > 0 entails that 1/g is bounded on (0, ν].Choosing a realising sequence for M in an analogous way, we find M < ∞ and whence M ≤ 1/2, after taking the limit ν → 0. □ 4.1.The limit for the derivative.We move on to proving the second limit of Main Theorem.Analogously to the function g in (4.2), we introduce the quotient which is well defined on (0, ν]; where u ′ is also continuous and nonnegative.By L'Hôpital's rule, a limit of h at zero would immediately imply a limit for g.Perhaps curiously, we will go the other way; that is, prove the limit of h by exploiting the already established limit for g in Proposition 4.2.
We also introduce -for notational convenience -the function which will serve a similar role to that of Λ from (2.17).It is positive on R + and appears in the relation as x → 0.Moreover, the expression inside the square brackets satisfies and is positive for small x > 0.
Proof.Recalling the antiderivative K from (3.1), the equation takes the form x K(y)u(y) dy after integrating by parts on the right-hand side of (2.4).Subsequently differentiating, we get for x ∈ (0, ν), where ñ(t) := n(t) + 1 2 tn ′ (t).This computation is justifiable because of Assumptions 1 to 3.
Using the definition of h and L from (4.13), the left-hand side of (4.18) can be written as x → 0, where we have applied Proposition 4.2 and the properties of n from Assumption 1.In particular, this motivates dividing (4.18) by xL(x) 2 .We investigate each term on the right-hand side separately: In the first term, we have δ 2x K(ν) ≤ 0 by monotonicity of K on R + , so ≤ 0 for all x ∈ (0, ν).Concerning the tail, we see that convexity of K on R + implies that δ 2x K ′ (y) ≥ 0 for all 0 < x < y, and thus when x ∈ (0, ν).Combined, we therefore have on (0, ν), where the final inequality again is due to the convexity of K. Consequently, as x → 0. Turning to the final, evidently dominant, term on the right-hand side of (4.18), we see that by (4.15) and (4.13).We have also made the change of variables τ → τ x in the first integral.Since by Assumption 2, we arrive at as x → 0. Thus, dividing (4.18) by xL(x) 2 ; followed by inserting (4.19), (4.20), and (4.21); we obtain as x → 0. The expression inside the square brackets is clearly nonnegative for 0 < x ≤ ν ≪ 1, and the auxiliary limit (4.17) follows directly from integrability of Ψ(τ ) and Ψ(τ )L(τ ) on (0, 2).The proof will therefore be complete once the limit lim is established.
To demonstrate this limit, we first argue that, for each fixed δ ∈ (0, ν), we have as x → 0. Indeed, assuming x is small enough for δ/x ≥ 2 to hold, we get that τ for all τ ≥ δ/x.Using this, and the fact that h is bounded on [δ, ν] (by continuity), we obtain (4.23).The second limit, found in (4.24), follows from an argument very similar to the one we used to prove (4.5).
Since u(x) = xL(x)g(x) and u ′ (x) = L(x)h(x) by definition, see (4.2) and (4.13), we may use integration by parts to compute that as x → 0. On the second line, we deal with the first term by using that Ψ(δ/x) = O(x) and for every 0 < δ < ν.Thus (4.22), and hence (4.16), follows by Proposition 4.2.□ We may now prove the desired limit for the derivative in Main Theorem.Proposition 4.4.Under Assumptions 1 to 3, the derivative of the solution enjoys the limit when K has a logarithmic singularity.
Proof.With h as in (4.13), the result follows immediately from (4.16) and (4.17); provided we are able to show that h is bounded near the origin.We know that h is nonnegative on (0, ν], so it is sufficient to prove that h is bounded above on this set.For the sake of contradiction, suppose that this is not the case; which by continuity of h necessitates blow-up at the origin.As a result, the set of points where h is larger than subsequent values must have the origin as an accumulation point.Furthermore, the limit lim must also hold.
Observing that ν ∈ A, we see that the intersection [x, ν] ∩ A is nonempty and closed for any x ∈ (0, ν].In particular, the point x := min([x, ν] ∩ A), exists, and enjoys the property (4.28) for every x ∈ (0, ν].For convenience, we define the accompanying scaling factor (4.29) as x → 0, by the aforementioned fact that A admits zero as an accumulation point.
By differentiating, one sees that that x → xL(x) 2 is increasing on (0, e −2 ), which we may assume entirely contains (0, ν].From this, we obtain 1 2 , which will be exploited in our next calculation: The change of variables τ → τ /τ x yields where we have also used that Ψ is positive, and increasing on (0, 1).This can be seen directly from its definition in (4.14).
As we shall see, (4.30) actually implies that h(x) is comparable to h(x).Taking the difference of (4.16) evaluated at x and x, respectively, we get as x → 0, after using (4.29).On the right-hand side, by (4.30), and positivity of the integrand.Thus as x → 0, in view of (4.29), (4.28), and (4.17).As a consequence of (4.31), we conclude that lim inf holds.This leads to our contradiction: With g as in (4.2), we see through (4.8) and integration by parts that 1 dy → 1 2 as x → 0. For this to be the case, we must necessarily have lim inf x→0 h(x) < ∞; contradicting what we just demonstrated.In conclusion, h is bounded on (0, ν], and the proof is complete.□ 4.2.The bidirectional Whitham equation.By inserting the steady-wave ansatz ϕ(t, x) = φ(x − ct) and (t, x) → v(x − ct) into (1.2) and integrating, the time-independent Whitham-Boussinesq system is obtained.The constants of integration have been set to zero in order to match the setting of [13].By subsequently eliminating φ, we find the steady bidirectional Whitham equation for v.Given a solution to (4.33), the associated φ can easily be recovered through the second equation in (4.32).We see that even if (4.33) arose from a system, it is of the exact same type as (2.2).Repeating the procedure in Section 2, we first discern that the right-hand side of (4.33) increases to the left of a local maximum at v = (1 − 1/ √ 3)c.If v is even, and assumes this value at the origin, then satisfies the equation x (πK B )(y)u(y) dy, which is precisely of the form (2.4).Moreover, Assumption 1 holds trivially, and the formula 3) and [29]*I.7.37 shows that Assumption 2 is satisfied with S = L.
In [13]*Theorem 5.9, the existence of a limiting 2π-periodic solution (v, c) of (4.33) is established.This solution is even, assumes v(0) = (1 − 1/ √ 3)c at the crest, decreases on the half-period [0, π], and is smooth on (0, 2π).In particular, Assumption 3 holds both for this solution, and for similar solutions with a different period.The hypotheses of Propositions 4.2 and 4.4 are therefore satisfied for the rescaled variable in (4.34).From this, we may deduce the asymptotic behaviour of v, and in turn that of φ.
The authors of [13] also pose a natural question about the global regularity of these waves: What is a reasonable function space that can capture the kind of asymptotic behaviour in (4.35) in an optimal way?A sensible candidate is the space of log-Lipschitz functions [11].This space appears, for instance, in critical Sobolev embeddings, and as a simple example of a class of non-Lipschitz right-hand sides for which the Osgood criterion [19] for the Picard-Lindelöf theorem holds.This global regularity is not a direct consequence of the local behaviour in (4.35).Oscillations may, even under additional assumptions of monotonicity and smoothness, cause the estimates to blow up in the limit.We will show that the highest waves indeed are log-Lipschitz by combining (4.36) with fairly straightforward bounds.To get the result, it is advantageous to introduce the concept of a modulus of continuity, commonly used in approximation theory.
We shall say that ω : R + 0 → R + 0 is a modulus of continuity if it is increasing, concave, continuous, and vanishes at the origin.Any function f : I → R is then said to admit ω as a modulus of continuity if for all x, y ∈ I.The following simple lemma is ours, but is very likely known in some form in the literature.It can be viewed as a kind of L'Hôpital's rule for moduli of continuity.Lemma 4.6.Suppose that f is absolutely continuous on an open interval I ∋ 0, and that for a modulus of continuity ω.Then there are M, δ > 0 such that f admits M ω as a modulus of continuity on (−δ, δ).
Since ω is concave on R + 0 , and ω(0) ≥ 0, it is also subadditive.Thus and a similar line of reasoning works for the case x ≤ y ≤ 0. Finally, if by monotonicity of ω.This concludes the proof.□ It is furthermore straightforward to show that if f admits M i ω as a modulus of continuity on an interval I i for i = 1, 2, and )ω as a modulus of continuity on I 1 ∪ I 2 .This follows since for any x ∈ I 1 and y ∈ I 2 , there is some z ∈ I 1 ∩ I 2 between x and y, whence  [5].Small solitary-wave solutions to the Whitham-Boussinesq system (1.2) were constructed in [28], but at present there is no existence result for extreme solutions in the solitary case.

Homogeneous kernel
Like in the previous section, we shall adopt Assumptions 1 to 3, but now take the singular part of the kernel K to be S(x) = H(x) = |x| −1/2 .The same restriction of x to (0, ν] for some 0 < ν ≪ 1 will also be made.We will here prove the final two limits of Main Theorem, and so analogously to (4.2) define for x > 0 in this section.We further remind the reader of Φ from (2.12), which appears in the identity 2) due to (2.8).
Understanding the properties of Φ will clearly be paramount for the calculations in this section, and we therefore start with a lemma listing a few of them.The bounds are certainly not optimal, but sufficient for our purposes.See also Figure 1 in Section 2.
Lemma 5.1.The function Φ is increasing on the interval (0, 1), where it has a unique root )

.4)
Proof.The first integral in (5.3) is a trivial computation, while the second explicit integral is simply a special case of (2.9).That Φ is increasing on (0, 1) follows directly from differentiating (2.12); and, by explicit evaluation, one further sees that Φ( 1 2 ) < 0 < Φ( 23 ).Hence there is a unique root τ 0 on the interval, which necessarily lies in ( 12 , 2 3 ).The positivity on (1, ∞) follows by the same computation as for δ 2 x K in Lemma 3.2, using the strict convexity of H on R + .For (5.4), it is easily verified that for all t ∈ (0, 1).Due to the sign-change of Φ at τ = τ 0 , this integral is maximised there, and lower bounds can be found by evaluation at any other point.In particular, we find by evaluating (5.5) at 2/3.To establish the upper bound, we observe through (5.5) and straightforward algebra that and that this expression is increasing on (0, 1).Exploiting this, we get after using the fact that τ 0 is a root of Φ, and evaluating at 2/3 > τ 0 .□ We next provide an asymptotic rephrasing of (2.4) for g, analogous to the one provided by Lemma 4.1 in the previous section.A fundamental difference from the logarithmic case is that the contribution from the integral x 0 δ 2 x K(y)u(y) dy can no longer be disregarded when passing to the limit.This is unlike (4.4), and makes the subsequent arguments more involved.Lemma 5.2.With g defined as in (5.1), there is a function λ : (0, 1) → (0, 1) so that as x → 0.Moreover, the square bracket is positive and satisfies while the final term admits the bound (5.8) Proof.Dividing each side of (2.4) by x, we find x K(y)u(y) dy, where we observe that the left-hand side is of the same form as in (5.6).On the right-hand side, the third integrals are identical, while in the second we have simply used the definition of g in (5.1) to write u(y) = y 1/2 g(y).The first integral requires more elaboration.
Recall from Lemma 5.1 that the singular part of δ 2 x K(y) changes sign at τ 0 x.As u is increasing and nonnegative on [0, ν], we may still make use of the second mean value theorem for integrals on the first integral.Explicitly, we are able to conclude that, for every x ∈ (0, ν], we have x K(y) dy. for some λ(x) ∈ (0, 1).Here, combining the identity (5.2) with Lemma 3.3, we further have as x → 0. Consequently, we find the first term in (5.6).
The positivity of the expression inside the square brackets in (5.6) for y ∈ (x, ν] is an immediate corollary of Lemma 3.2, while the limit (5.7) follows directly from (5.2) and Lemma 3.3.Finally, by an argument identical to the one used to prove (4.6), we obtain (5.8).□ As we have alluded to, applying the arguments in the proof of Proposition 4.2 to (5.6) will not directly lead us to the desired limit for g.Instead, we will derive a system of two inequalities for the limits inferior and superior of g at zero.As will be demonstrated in Proposition 5.5, these inequalities are in fact sharp enough to ensure the limit for g. ) hold.Here, Φ is as defined in (2.12).
Remark 5.4.Compare (5.9) and (5.10) with the more symmetric (2.15) that was covered in Section 2. Without the refinement of (5.10) over the second inequality in (2.15), the system would be too weak to reach the conclusion of Proposition 5.5.
Proof.We first prove that m > 0. Proceeding as in Proposition 4.2, we deduce from (5.6) that x K(y)y 1/2 x dy (5.11) as x → 0. Here, g is again defined according to (4.10).To bound the integral below, we have used the monotonicity of Φ on (0, 1), along with 1 0 Φ(τ ) dτ < 0; both from Lemma 5.1.Assuming that m = 0, we may pick a realising sequence {x k } k∈N ⊂ (0, ν] for m, in such a way that g = g along the sequence.Then (5.11) reduces to as k → ∞, after having divided by g(x k ).Going to the limit, we obtain where the equality comes from the first integral in (5.3).Meanwhile, the final inequality stems from positivity of the integrand on (1, ∞), which is part of Lemma 5.1.Regardless, this is a contradiction, so m > 0.
Similarly, arguing like for (4.12), one has as x → 0. Assuming that M = ∞, we are again able to choose a realising sequence along which g = g.This results in the contradiction and so we do in fact have M < ∞.
In order to establish (5.10), we note that, since u is increasing on (0, ν], we have x 1/2 = g(x) for every τ ∈ (0, 1) and x ∈ (0, ν].Thus for all such x, τ , and the lower bound as x → 0, is therefore obtained from (5.12).Finally, we are left with (5.10) after first taking the limit along a sequence realising m, and subsequently letting ν → 0. □ While the inequalities (5.9) and (5.10) are more involved than the corresponding inequalities found in the logarithmic case (4.9), it just so happens that the only point (m, M ) ∈ R + × R + that satisfies both (5.9), (5.10), and m ≤ M , is the one given by m = M = π/2.We now prove this, resulting in the third limit of Main Theorem.Proposition 5.5.Under Assumptions 1 to 3, the solution enjoys the limit when K has a homogeneous singularity.
Proof.With M and m as in Lemma 5.3, we first introduce σ := M/m ≥ 1, and rewrite (5.9) and (5.10) purely in terms of σ and m: When σ = 1, both right-hand sides read ∞ 0 Φ(y)y 1/2 dy = π/2 by Lemma 5.1, and so π/2 = m = M/σ = M .We will therefore be done once we are able to show that no m > 0 simultaneously satisfies both (5.13) and (5.14) when σ > 1.To that end, we introduce for σ ≥ 1, where is a positive constant.As ∞ τ 0 Φ(τ )τ 1/2 dτ = π/2 + b, it is not difficult to see that f (σ) is precisely the right-hand side of (5.14) minus that of (5.13).Hence, if we can that f is positive on (1, ∞), then there is no simultaneous solution to (5.13) and (5.14); thereby completing the proof Since Φ is negative on (0, τ 0 ) by Lemma 5.1, we may use the trivial inequality min(1, στ 1/2 ) ≤ στ 1/2 to see that for all σ ≥ 1.Note that the first of the two factors is positive for all σ > 1, while the second factor is a decreasing function of σ.As b < 3/5 by (5.4), we further have and thus conclude that f (σ) > 0 on (1,2].Suppose finally that σ > 2 1/2 .Then σ −2 < 1/2 < τ 0 , by Lemma 5.1, so that which we in turn can use in (5.15) to compute that for all σ > 2 1/2 .Here, 12) and the convexity of H(τ ) = |τ | −1/2 on R + .Therefore, we infer that for all σ > 2 1/2 , by using the bounds on b provided by (5.4).In conclusion, f is increasing on (2 1/2 , ∞), and therefore positive on (1, ∞); seeing as it is positive on (1,2].□ 5.1.The limit for the derivative.We shall now prove the final limit of Main Theorem.Whereas we in Section 4.1 first proved the limit, and then obtained uniform regularity via Lemma 4.6, we will here prove a sharper form of Hölder regularity first.This regularity is then used to establish the limit.These two approaches are complementary.The counterpart to (4.15) this case is still useful, and becomes is also positive on R + .
To illustrate the idea, if one formally differentiates the toy equation in (2.7), then 2 dτ for all x > 0. In principle, it should therefore be the case that lim because of Proposition 5.5.This principal value integral can be shown to, in fact, equal π/2, so we find the "correct" limit.Of course, to rigorously justify this computation, especially for the full equation, we need to work harder.

Lemma 5.6.
There is some ν > 0 so that for all 0 ≤ h ≤ x ≤ ν.As a consequence, u is C 1/2 -Hölder continuous on [0, ν], and Proof.This proof is a variant of the proof of global regularity given for the Whitham equation in [15], but adapted to obtain more information than just Hölder continuity.The aim is to build up regularity by applying a a bootstrap argument to (2.4).We begin by noting that, if we introduce the notation N (t) := (1 + n(t))t 2 for the nonlinearity on the left-hand side of (2.4), and for simplicity extend u to an even function on R, then for all x, h ∈ R.This equation was referred to as a double symmetrisation formula in [15].On the left-hand side, for 0 < h < x as x → 0. This is because N ′ (t) = (2 + o(1))t as t → 0 by Assumption 1, and because u(0) = 0.Moreover, we further have by Proposition 5.5.We next turn to the right-hand side of (5.19), which we split as for < h < x ≤ ν, with H and R as in Assumption 2. Here, the final terms satisfy where the first inequality follows by similar argument as in Lemma 3.2, while the second follows from the bound on R ′ from Lemma 3.3.Furthermore, the first term on the right-hand side of (5.20) satisfies In summary, we have demonstrated that for 0 < h < x ≤ ν, after possibly shrinking ν.Define now the possibly infinite quantity for each α ∈ [0, 1/2].As a function taking extended real values, C is nondecreasing and left-continuous in α, and is at least finite at α = 0. We emphasise that the supremum is not taken over ν.
Let α be such that C(α) is finite.Then (5.21), by Proposition 5.5 and the definition of C(α) in (5.22).We have also used that min(a, b) ≤ √ ab for all a, b ≥ 0. Inserting this into (5.21),we find for all 0 < h < x ≤ ν.The second inequality comes from the fact that z 1/4+α/2 uniformly in z ≥ 1 and α.If we now divide (5.23) by h 1/2+α x 1/4−α/2 , we arrive at for 0 < h < x ≤ ν.In particular, we thus have according to the definition in (5.23).Crucially, the implicit constant does not depend on α.From (5.24), we immediately conclude by induction that since C(0) is finite, we have implies a uniform bound on C(α) for all α ∈ [0, 1/2).Continuity from the left ensures that C(1/2) is finite as well, concluding the proof.
□ Using the regularity furnished by (5.18) in Lemma 5.6, we are now able to fully justify a version of the calculation in (5.17).We purposely avoid having to deal with principal value integrals.
Proposition 5.7.Under Assumptions 1 to 3, the derivative of the solution enjoys the limit when K has a homogeneous singularity.
Proof.We return to (5.19), make the same splitting of the right-hand side as in (5.20), and divide by 2h; so as to get for all 0 < x ≤ ν and 0 < h ≤ x/2.The intention is to obtain the desired limit from this equation, by first letting h → 0, and subsequently x → 0: After doing so to the left-hand side of (5.25), it reads where the final equality follows from N ′ (u(x)) = (2 + o(1))u(x) and Proposition 5.5.Of course, we do not yet know that this limit actually exists.Turning our attention to the right-hand side of (5.25), we next establish the limit of each integral separately.To accomplish this, we shall prove that the integrands are dominated by integrable functions, independently of x and h.Consequently, limits and integrals may be interchanged, by using the dominated convergence theorem.
As for the second term on the right-hand side of (5.27), its integrand may be expressed as (5.32) 1) , establishing its analogue of Lemma 5.6, and then progressing in a similar manner to the proof of Proposition 5.7.We refrain from pursuing this, and will be content with asymptotics for the first derivative.after integration.The integration constant is chosen to be zero, which can be done by Galilean transformation, like in [15].Again, one observes that the right-hand side of (5.36) is increasing to the left of its maximum at φ = c/2.This is the height of a highest wave for this equation.If φ assumes this height at the origin, and is even, then which is of the desired form (2.4).It is immediate that Assumption 1 holds, and it is well-known [15] that Assumption 2 is satisfied.See also the comment on the precise behaviour of K W in Remark 5.9 below.The existence of a limiting 2π-periodic solution (φ, c) of (5.36) was proved in [15].This solution is even, assumes φ(0) = c/2, decreases on (0, π), and is smooth on (0, 2π).In particular, Assumption 3 holds.All assumptions required for Proposition 5.5 and Proposition 5.7 are therefore satisfied, and we may settle a conjecture posed in the aforementioned paper.Our result also applies equally well to the highest solitary waves recently found in [35] and [14].As shown in [7,8], such solitary waves are necessarily even, and smooth and decreasing on R + .for the Whitham kernel.To the best of our knowledge, this expansion of the kernel for standard linear gravity wave dispersion is new.We note, in particular, that the first term is precisely the singular part of the kernel.As written, the series is only conditionally convergent, but this can be remedied by merging the terms corresponding to n = 2k − 1 and n = 2k for k ≥ 1.These all become smooth, even, negative, and increasing on R + .
To prove the series expansion, the key observation is that tanh(ξ) for ξ ̸ = 0, which can be seen by writing the numerator in terms of a binomial series.Here for all n ≥ 1, and in the sense of distributions when n = 0.

Figure 2 .
Figure 2. The inequalities in (2.15) are satisfied by the points below the solid curve, and above the dashed curve, respectively.The refined version of the second inequality corresponds to the dotted curve.

Lemma 4 . 3 .
With h and Ψ as defined in (4.13) and (4.14) respectively, we have the asymptotic equation
.38) by monotonicity of ω.We use this to get the following result.