Periodic H\"older waves in a class of negative-order dispersive equations

We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal $ \alpha $-H\"older continuity in a class of fractional negative-order dispersive equations of the form \begin{equation*} u_t + (| \mathrm{D} |^{- \alpha} u + n(u) )_x = 0 \end{equation*} for every $ \alpha \in (0, 1) $ with homogeneous Fourier multiplier $ | \mathrm{D} |^{ - \alpha} $. We tackle nonlinearities $ n(u) $ of the type $ | u |^p $ or $ u | u |^{p - 1} $ for all real $ p>1 $, and show that when $ n $ is odd, the waves also feature antisymmetry and thus contain inverted cusps. Tools involve detailed pointwise estimates in tandem with analytic global bifurcation, where we resolve the issue with nonsmooth $ n $ by means of regularisation. We believe that both the construction of highest antisymmetric waves and the regularisation of nonsmooth terms to an analytic bifurcation setting are new in this context, with direct applicability also to generalised versions of the Whitham, the Burgers--Poisson, the Burgers--Hilbert, the Degasperis--Procesi, the reduced Ostrovsky, and the bidirectional Whitham equations.


Introduction
1.1 Main result. In this paper, we shall be concerned with singular periodic travelling-wave solutions to a class of nonlinear and dispersive evolution equations of the form u t + (|D| −α u + n(u)) x = 0. ( This family may be viewed as a kind of generalised fractional Korteweg-de Vries (KdV) equations of negative-order, where we refer to [3] for Precisely, we obtain the following result, with corresponding numerical illustrations in Figure 1. increasing on (−π, 0) and exactly α-Hölder continuous at x ∈ 2π , that is, uniformly around 2πℓ for ℓ ∈ .
(for α ∈ (0, 1)) prove that solutions of (1) blow up in finite time for certain initial data. The resulting singularities occur in at least two ways: wave breaking [19], in which the spatial derivative of a bounded solution of (1) blows up, or sharp crests in travelling-wave solutions -reminiscent of the highest Stokes' wave [31] -which is the subject of this paper. We refer also to [13,22,23,25,29] for other results concerning singularities, well-posedness, persistence, and existence time of solutions.
Classical Fourier analysis shows that |D| −α constitutes a singular convolution operator on with kernel | · | α−1 and describes an eigenfunction of when α = 1 2 . Related to this case is the recent work by Ehrnström & Wahlén [12] on the Whitham equation [33], being a shallow-water model of type (1) with inhomogeneous dispersion tanh(D)/D and n(u) = u 2 . Its corresponding symbol behaves as that of the KdV equation for small frequencies and decays like |ξ| − 1 2 as |ξ| → ∞, for which the kernel may be written as |x| − 1 2 plus a regular term. The existence of a highest, cusped steady solution whose behaviour at the crest is like 1 − |x| D N α ∈ (0, 1) α = 1 α > 1
including the Whitham equation. We note that global existence of weak solutions are guaranteed by [4] in the homogeneous case of α = 1 known as the Burgers-Hilbert equation. Finally, the waves are all Lipschitz [5,24] when α > 1, and one naturally conjectures that the Lipschitz angles vanish as α ց 1. See further [16] for uniqueness and instability of the highest wave when α = 2, corresponding to the reduced Ostrovsky equation, and also [2] for the existence of extreme Lipschitz waves for Degasperis-Procesi equation. Table 1, we complete the regularity picture for extreme periodic waves in the given negative-order class of dispersive equations. The regularity analysis emerges from the overall structure of [12] with the following key differences:

Contributions. As promised and illustrated in
i) Whereas [1,5,12] obtain monotonicity properties of the kernels based on a general characterisation of completely monotone functions or sequences, we establish monotonicity of the singular kernel on by computing an explicit integral representation valid for all α > 0, and use the Poisson summation formula to derive its precise singular behaviour ( |x| α−1 as |x| → 0) from the situation on .
ii) Since K α has only algebraic but not exponential decay (unlike the kernels in [1,12,27]), extra care must be applied to the finite-difference estimates for |D| −α u when α is arbitrarily close to 1. In fact, we must exercise order-optimal estimates in order for the integrals to converge.
iii) We treat a class of nonlinearities, including sign-dependent ones (2 sgn ), in the regularity estimates, which amongst others requires the use of suitable properties of composition operators on Hölder spaces.
The study of nonsmooth nonlinearities -with both slow (p 1) and arbitrary (polynomially) fast growth in (2) -also poses new challenges since analytic bifurcation theory cannot be applied directly. We resolve this issue by analytically regularising the nonlinearities and proving that important features related to wave regularity and speed hold uniformly as the regularisation vanishes. The approach is strikingly simple (see (35)).
In the special case of smooth n(x) = x p with 2 p ∈ , we also compute in Theorem 4.4 local bifurcation formulas for all p, which may be of independent interest. We are also able to deduce the overall structure of the bifurcation formulas along the entire local bifurcation curve when p is odd. . .

5/33
As for the case of general sign-dependent nonlinearities (2 sgn ), we establish that the highest waves exhibit antisymmetry and thus also contain an inverted cusp at the troughs, as illustrated in Figure 1.
This construction appears to be completely new in the context of large-amplitude singular waves and sheds light on underlying symmetry principles.
With appropriate modifications, these results are also transferable to other nonlocal dispersive equations. In particular, one may obtain such "doubly-cusped" periodic solutions (with zero mean) in the full scale of equations in Table 1 with generalised nonlinearities of type (2). Specifically, consider the evolution equation where L α is any of the dispersive operators for α ∈ (0, ∞) and n is as in (2). By readily adapting the regularity estimates in [1,5,8,11,12,24,27] with the estimates for general nonlinearities considered here and applying the regularisation procedure in the global bifurcation analysis, we can also deduce the following analogous result of Theorem 1.1. Here C := 1 for the inhomogeneous operators (the value at the origin for their symbol) and  (2 sgn ). The solution is even (about x ∈ 2π ) and strictly increasing on (−π, 0), and satisfies where again µ = (c/p) 1/(p−1) . Moreover, the estimate holds uniformly around the extreme point x = 0.
A similar statement may be formed for extreme Lipschitz waves in a generalised version of the Degasperis-Procesi equation with the nonlinearities (2) by combining the analysis here with that of [2,27].
1.4 Outline of the analysis. For homogeneous dispersion one has a choice as to what class of functions |D| −α should act upon in the interpretation of (1). We restrict our attention to functions 6/33 with zero mean (´ u(·, x) dx = 0), but note that other alternatives such as equivalence classes of functions that differ by a constant are possible; see for instance [26] on homogeneous Sobolev-type spaces.
We set up (1) in steady variables u(t, x) := ϕ(x − ct) with wave speed c > 0, so that, after integration, (1) takes the form where we have introduced N (ϕ; c) := cϕ − n(ϕ), and the mean ffl n(ϕ) := 1 2π´ n(ϕ( y)) d y of n(ϕ) is the constant of integration. One may observe that in which the value being the first positive critical point for N (ϕ), turns out to be the maximum of the highest wave.
As the regularity analysis will reveal, the quadratic nature near ϕ = µ, where N ′′ is strictly negative, causes in partnership with |D| −α the singular behaviour of ϕ at the crest (and through, in case (2 sgn )).
With regards to the precise C α regularity estimates, we consider as in [12] fine details of local regularity and first-and second-order differences of both u, |D| −α u, and n(u) in connection with the Hölder seminorm. We first establish global C β regularity for all β < α, then the exact α-Hölder estimate at 0, and finally global C α regularity with help of an interpolation argument. A key property in this setting is that |D| −α is α-smoothing on the scale of Hölder-Zygmund spaces, and that if |D| −α u is (2α)-Hölder continuous at a point, then u is α-Hölder continuous at that point for α ∈ 0, 1 2 . However, when α > 1 2 (remember that [12] corresponds to α = 1 2 ), |D| −α u passes index 1 on the Hölder-Zygmund scale, and we must partially work with derivatives as in [8].
When it comes to the bifurcation analysis, we first establish small-amplitude waves by the local Crandall-Rabinowitz bifurcation theorem [6,Theorems 8.3.1 and 8.4.1]. It is interesting to note that the regularisation of n lightens the computation of the local bifurcation formulas; case (2 abs ) acts essentially as u 2 and case (2 sgn ) behaves like u 3 . As for the construction of the highest waves, we make use of the analytic global bifurcation theory of Buffoni and Toland [6]. One obtains, after ruling out certain possibilities, a global, locally analytic curve s → ϕ ε (s) of smooth sinusoidal waves, along which max x∈ ϕ ε (s)(x) approaches a particular value µ ε depending on the wave speed and the regularisation parameter ε (see (6) and (36)). Although we are not able to establish unconditional antisymmetry in case (2 sgn ), we enforce this property along the global branch by working in a subspace. Coupled with the a priori regularity estimates for solutions touching µ ε from below, it is then possible to extract a subsequence of (ϕ ε (s)) s converging to a solution ϕ ε with both max ϕ ε = µ ε and the exact, global α-Hölder continuity. Finally, we show that ϕ ε converges to a solution of (1) with the same properties as ε ց 0.
The outline of the paper is as follows. In Section 2 we focus on properties and representations of |D| −α and K α on together with the relevant function spaces. In Section 3 we study a priori properties of solutions-especially, what concerns the α-Hölder continuity when max ϕ = µ, which is the most technical part. Finally, in Section 4 we first consider the local bifurcation analysis, and then study what happens at the end of the global bifurcation curve, supported by the theory in Section 3. 2 Properties of |D| −α and functional-analytic setting 2.1 Representations of the kernel. On the real line it is well known that the inverse Fourier transform of the symbol | · | −α for α ∈ (0, 1) equals in the sense of (tempered) distributions, with γ −1 α := 2Γ (α) sin π 2 (1 − α) using the normalisation Here Γ is the gamma function, and we observe that γ α ց 0 as α ց 0 and γ α ր ∞ as α ր 1. We are interested in the action of |D| −α in the periodic setting, and by the convolution theorem this amounts to understanding the periodic kernel Here ϕ has zero mean so that ϕ(0) = 0, and the normalisation is again A naïve application of the Poisson summation formula yields that which, although it is nonsense due to divergence on both sides, nevertheless suggests that the kernel on mimics the singularity of the kernel on . In fact, [ on (−π, π), where K α,reg is an even, smooth function. In particular, K α ∈ L 1 ( ).
Proof. Let ̺ be an even, smooth cut-off function that vanishes in a neighbourhood of ξ = 0 and equals 1 for |ξ| 1, and define F(ξ) := ̺(ξ)|ξ| −α for ξ ∈ . Then F is the Fourier transform of an integrable function of the form where f reg ∈ C ∞ ( ) and | f (x)| = O(|x| −m ) as |x| → ∞ for all m 1. Indeed, writing it follows directly from (7) that f = −1 F has the given form, where we remember that the inverse Fourier transform of an integrable function of bounded support (in this case (̺ − 1)| · | −α ) is smooth.
Since (x m f (x)) ∼ F (m) is integrable for all m 1, it must be the case that x m f (x) is bounded. In particular, f ∈ L 1 ( ), and the Poisson summation formula then gives .
this proves the result with K α,reg : In Figure 2 we display the integral kernels on both and for various values of α. Whereas the kernels on are all nonnegative, those on become negative away from the positive singularity at 0, because K α has zero mean. In both cases the profiles are monotone on either side of the singularity; this is obvious on , and on we deduce this by means of the following integral representation of K α , which is valid for all α ∈ (0, ∞). Although the formula is known [28,Section 5.4.3], we include a slick computation of it using the gamma distribution. We remark that we shall only need the monotonicity of K α in our work, and for that property one may alternatively use the theory of completely monotone sequences and the discrete analogue of Bernstein's theorem; see [5, Theorem 3.6 b)].
Proof. By recognising k −α in the definition of the gamma distribution with shape α and rate k, whose probability density function is t → k α t α−1 e −kt /Γ (α) on (0, ∞), we find that using the dominated convergence theorem and a trick with geometric series. Leibniz's integral rule next yields that which shows that K α is strictly increasing on (−π, 0).

Remark 2.3. For α = 1 in Proposition 2.2 one finds the explicit form
as x → 0. Such logarithmic singularities occur for all the kernels in Table 1  2.2 Action of |D| −α on Hölder-Zygmund spaces. As regards the functional-analytic framework, it is desirable to work with spaces which both capture the precise regularity of cusps and interact well with Fourier multipliers. These turn out to be the so-called Hölder-Zygmund spaces, which we explain next.
10/33 -While the standard Hölder norms provide an accurate description of the modulus of continuity of a function (and its derivatives), an alternative, frequency-based characterisation by means of the Littlewood-Paley decomposition is more suitable for Fourier multipliers. To this end, let ∞ j=0 ̺ j (ξ) = 1 be a partition of unity of smooth functions ̺ j on supported on 2 j |ξ| 2 j+1 for j 1 and on |ξ| 2 for j = 0. We then define the Hölder-Zygmund space C s * ( ) for s ∈ [0, ∞) to consist of those functions ϕ for which is the Fourier multiplier with symbol ̺ j , that is, One has that C s in the sense of equivalent norms, whereas there are strict inclusions Since |D| −α is homogeneous, we restrict from now on to the corresponding subspaces C m ( ), C m,β ( ), and C s * ( ) of functions with zero mean in the above spaces, with identical norms. Note that the seminorm | · | C β ( ) is now a norm equivalent to · C 0,β ( ) by the zero-mean restriction and compactness of . We observe in this setting that is α-smoothing for all s ∈ [0, ∞) (and therefore also on the Hölder-space scale for s = 1, 2, . . .), because in terms of Fourier multipliers we have, with j 1, that Finally, the subscript "even" attached to any space on means the subspace of symmetric functions about 0 (mod 2π).
and the first term on the right-hand side is globally C s+α * regular. Moreover, since the integrand in vanishes for y near x and K α is smooth away from 0, it follows that ρ|D| −α (1 − η)ϕ is smooth. Hence, |D| −α ϕ ∈ C s+α * ,loc (U), as claimed.

11/33
Finally, we include a monotonicity property of |D| −α which will be useful in establishing a priori nodal properties of highest waves in Section 3.

Proposition 2.5 ([12, Lemma 3.6]). |D| −α is a parity-preserving operator, and
Proof. Since K α is even, one immediately obtains that |D| −α is parity-preserving from Next Then The latter inequality is a consequence of |x − y| < |x + y| for x, y < 0 and |x − y| < x + y + 2π for x, y > −π. Since K α is strictly decreasing as a function of the distance from the origin to ±π by Proposition 2.2 and is even and periodic, we therefore obtain that for every y ∈ (−π, 0) \ {x}. In particular, |D| −α f (x) > 0 for all x ∈ (−π, 0) as f is strictly positive in an interval around y 0 by continuity.

A priori properties of travelling-wave solutions
In this section we establish many a priori bounds and regularity properties of continuous solutions of (4). Most importantly, we prove exact, global α-Hölder regularity in Theorem 3.8 for solutions touching the highest point µ (see (6)) from below at x = 0. This is obtained with help of a nodal pattern of solutions in Theorem 3.4. We remind the reader of (5): n ′ (ϕ) < c corresponds to solutions which stay away from (±)µ, and n ′ (ϕ) c includes the possibility of also touching (±)µ.
Our first result is a uniform upper bound on both the size of solutions and the wave speed that we will use in Section 4 in compactness arguments.
Lemma 3.1. For all solutions ϕ ∈ C( ) of (4) satisfying n ′ (ϕ) c one has the uniform estimate Proof. In case (2 sgn ) for n ′ (ϕ) c, the bound in L ∞ is immediate since µ ∼ c 1/(p−1) . As regards (2 abs ), we need to control the minimum of a nontrivial ϕ. Let x min and x max be points where ϕ attains its global minimum ϕ min and maximum ϕ max , respectively, where we note that ϕ max > 0 > ϕ min as ϕ has zero mean. From (4) we then find that which leads to In the first situation, n(ϕ min ) n(ϕ max ), so that |ϕ min | ϕ max µ. In the second situation, it suffices to investigate the case ϕ max < |ϕ min | (otherwise we freely get |ϕ min | ϕ max µ). Then For the last part, we use in case (2 abs ) that for a > 0 > b one has Applied to a = ϕ max and b = ϕ min , we reorder (8) and find that One may similarly treat case (2 sgn ).
Next, we want to establish regularity for solutions which stay away from (±)µ. To this end, we need the inverse function theorem for the Hölder scale. We could not find a proof in the literature and therefore provide a short argument.
Proposition 3.2 (Inverse function theorem for C m,β ). Let m 1 be an integer and β ∈ (0, 1], and assume that a function f is C m,β regular and strictly monotone on a compact interval I ⊂ . Then Proof. We only establish β-Hölder regularity of g ′ , where g := f −1 ; the rest follows by the standard inverse function theorem and similar estimates for the higher-order derivatives. To this end, let x, y ∈ f (I) be different and observe that for some z between x and y by the mean value theorem. Hence, where we note that g ′ is bounded on the compact set f (I) due to its continuity by the standard inverse function theorem. In particular, if n is smooth, then so is ϕ on any open set where n ′ (ϕ) < c.
Proof. Note first by translation invariance (|D| −α is a convolution operator) that if ϕ is a solution of (4), then so is ϕ(· + h) for any h ∈ . Accordingly, it suffices to consider open sets U ⊆ (−π, π) where n ′ (ϕ) < c, so that (5) holds uniformly on every compact interval I ⊂ U. By the inverse function theorem (Proposition 3.2), it follows that N −1 exists on ϕ(I) and has the same regularity as n in the Hölder scale. As such, we may then invert (4) to get the pointwise relation for x ∈ I , where G is a nonlinear composition operator, depending implicitly on c via N −1 .
It is clear from the (higher-order) chain rule that an operator loc ( ) by [17, Theorems 2.1, 4.1 and 5.1], and the composition operator is also bounded (maps bounded sets to bounded sets). Therefore, since ϕ ∈ C( ) → C 0 * ,loc (U) and |D| −α is locally C α * smoothing by Lemma 2.4, it follows by bootstrapping of G(·, c) in (9) that ϕ has at least the same C m,β Hölder regularity as N −1 (that is, as n) on I. In particular, ϕ has the given regularity around ∂ (ϕ −1 (0)) by applying this result to a covering {I j } j of compact intervals I j such that (−π, π) ⊃ j I j ⊃ ∂ (ϕ −1 (0)), which proves property ii).
Similarly, when U does not contain ∂ (ϕ −1 (0)), we know that N −1 is smooth on ϕ(I) ∋ 0, and so bootstrapping (9) yields that ϕ is smooth on I. As I ⊂ U was arbitrary, this establishes property i).
We continue by proving a nodal pattern for solutions which stay away from ±µ. The result will be crucial in establishing that the global bifurcation branch of solutions in Section 4 is not periodic. 14/33 -Remark 3.5. We write "in-/decreasing" instead of "nonde-/increasing", so that constant functions are both increasing and decreasing, and add the prefix "strictly" for the nontrivial cases.
If ϕ is also C 2 around 0, we differentiate (4) twice and use that ϕ ′ (0) = 0 to obtain where we have also isolated the singularity of K α and interchanged limits (which is legitimate since and the latter integral vanishes as r ց 0 because K α is integrable and ϕ ′′ is continuous around 0. By Leibniz's rule once more, we also find that where we have utilised that K α is even and ϕ ′ is odd. Observe that K α (r) |r| α−1 and ϕ ′ (r) = O(r) (because ϕ ′′ is continuous around 0) as r ց 0, which means that K α (r)ϕ ′ (r) vanishes in the limit.
Conversely, suppose that n ′ (ϕ) c everywhere. If in fact n ′ (ϕ) < c uniformly, then Lemma 3.3 implies that ϕ ∈ C 1 ( ), which leads to ϕ ′ > 0 on (−π, 0) by the first case of Theorem 3.4. When n ′ (ϕ) touches c, however, we must use a different approach. Note that ϕ is differentiable almost everywhere on (−π, 0) by Lebesgue's theorem for increasing functions, and that we may also use . .

15/33
central differences to compute ϕ ′ . To this end, observe that for x ∈ (−π, 0) and h ∈ (0, π) by periodicity and evenness of K α and ϕ. The second factor in the integrand is nonnegative by assumption, whereas the first factor is strictly positive by Proposition 2.2.
We next start to investigate what happens if solutions touch µ, and begin with a one-sided α-Hölder estimate around 0.
Proof. Let x ∈ I := − 3π 4 , − π 4 and consider first case (2 abs ). Monotonicity of N ′ and ϕ plus (12) show that where we have used that M α := min K α (x − y) − K α (x + y) : x, y ∈ I > 0 by the extreme value theorem and the fact that K α is even and strictly increasing on (−π, 0) by Proposition 2.2. Integrating over I in x then yields after cancelling ϕ − π 4 − ϕ − 3π 4 > 0 on both sides. Suppose to the contrary that there exists a sequence {(ϕ j , c j )} j of such solution pairs for which c j ց 0. Then n ′ (ϕ j (−π)) c j ց 0 as well, contradicting (14). Thus c 1 uniformly and n ′ (ϕ(−π)) does not touch c, so ϕ is smooth around −π by Lemma 3.3.
Finally we come to the main result in this section, which concerns both the global regularity of solutions and the exact α-Hölder regularity at 0 for solutions that touch µ. This is the most technical part of the paper.

17/33
Theorem 3.8 (Regularity). Let ϕ ∈ C even ( ) be a nontrivial solution of (4) that is increasing in (−π, 0) and satisfies n ′ (ϕ) c, with maximum ϕ(0) = µ. Then i) ϕ is strictly increasing on (−π, 0), smooth except at 0 and possibly the point ϕ −1 (0), and features at least the same regularity in the Hölder scale around ϕ −1 (0) as n around 0; ii) ϕ ∈ C α even ( ); and iii) ϕ is exactly α-Hölder continuous at 0, that is, uniformly around 0 we have Proof. Property i) and the lower bound in property iii) follow directly from Lemmas 3.3 and 3.6 and Theorem 3.4. As a consequence, it remains to establish global α-Hölder regularity and the upper bound in property iii). Note that Hölder regularity at a point plus smoothness everywhere except at that point does not in general imply global Hölder regularity-one additionally needs uniform Hölder regularity around the point. In particular, in order to obtain property ii), it suffices to prove C α regularity in a small interval around 0.
We next establish the upper C α estimate at 0 in property iii). In fact, with u(x) := µ − ϕ(x), we shall prove that uniformly over x ∈ (−r, r) and β ∈ [0, α), from which the desired estimate follows by letting β ր α.

19/33
On this route, note from (16) that where y) denotes the second-order central difference operator.
Here we have utilised periodicity of K α in the first transition between the integrals, and averaging, variable change y → − y, and evenness of u (from ϕ) in the last step. Since we have already estab- r). Thus (22) shows after cancelling sup |x|<r u(x)|x| −β once on each side. Now remember that K α = γ α | · | α−1 + K α,reg from Proposition 2.1. In particular, for the regular part we may Taylor expand around y to see that uniformly over y ∈ , because K α,reg is even and K ′′ α,reg and K ′′′ α,reg are bounded on . As such, using that | y| β 1 on independently of β, we obtain that since β < 1. For the singular part, one has with y = xs that The right-hand side is O(1) over x ∈ (−r, r) because of α − β 0 and the following observation: | · | α−1 is locally integrable, and as |s| → ∞, so that for |s| ≫ 1, where α + β − 3 < −1 uniformly over β < α because α ∈ (0, 1) is fixed, thereby guaranteeing integrability at infinity. Hence, sup |x|<r u(x)|x| −β 1 uniformly over β < α, which is (21).
It remains to establish C α continuity around 0.
which comes straight from (4). On this path, we let x ∈ [−r, 0) and h ∈ (0, |x|], and then choose a := ϕ(x + h) and b := ϕ(x − h) in the Taylor expansion with ζ between a and b, to see that uniformly over x ∈ [−r, 0) and h ∈ (0, |x|]. Now note that in light of (13) and the exact C α estimate at Thus (24) for all β < α, where we postpone taking the supremum over x ∈ [−r, 0) until we have estimates for (23). With that in mind, we first consider the regular part in |D| −α ϕ and compute by the mean value theorem and repeated use of parity and periodicity of K α,reg and ϕ. Consequently, .
In case (2 sgn ) we could have assumed that ϕ(−π) = −µ instead of ϕ(0) = µ in Theorem 3.8 and then proved exact α-Hölder continuity at −π. We conjecture that both assumptions imply the other and more generally imply antisymmetry of waves about − π 2 whenever one deals with antisymmetric nonlinearities. This is also the reason why we assumed that ϕ − π 2 = 0 in Lemma 3.7. As a remedy to the lack of proof of the general property, we shall in Section 4 instead construct solutions which are antisymmetric about − π 2 .

Global bifurcation analysis
We first establish nontrivial small-amplitude travelling waves around the line c → (0, c) of trivial solutions by means of local bifurcation theory and then extend the bifurcation curve globally using the analytic theory of Buffoni and Toland [6]. By carefully examining the structure of the global curve in connection with the a priori nodal properties in Section 3, we finally deduce the existence of a limiting sequence along the curve which converges to a highest wave satisfying Theorem 3.8. This establishes Theorem 1.1 when the nonlinearities (2) are smooth, that is, when they equal n(x) = x p for 2 p ∈ , and in Figure 3 we provide a sketch of the analysis.
. Figure 3: Illustrating the global bifurcation diagram in the smooth case n(x) = x p for 2 p ∈ of 2π/kperiodic even solutions obtained in Theorem 4.8 bifurcating from the trivial solution (0, k −α ) and reaching a limiting extreme wave. The dashed vertical lines mark the bounds for the wave speed in Lemma 3.1 and Corollary 4.11, whereas the solid curve displays the possible maximal height from (6) for these waves (plotted for p = 3). Along the dotted bifurcation curve, one may extract a sequence for which possibilities i) and ii) in Theorem 4.8 occur simultaneously, converging to a solution of (4) with the C α properties of Theorem 3.8.
In the general nonsmooth situation, however, one cannot use the analytic bifurcation theory directly. We resolve this issue by regularising n analytically around 0 (where its regularity is only of order p in the Hölder scale) and instead study global bifurcation for the regularised equation of (4) for every 0 < ε ≪ 1. This leads to solutions (ϕ ε , c ε ) at the end of the bifurcation curves, with the optimal α-Hölder continuity of Section 3, that will be shown to converge (up to a subsequence) to a solution of (4) with the same Hölder properties as ε ց 0. Here N ε (ϕ; c) := cϕ − n ε (ϕ) and is a natural analytic regularisation with the same monotonicity properties as n and that converges uniformly to n on compact sets as ε ց 0. In particular, the regularity theory in Section 3 carries over to the new setting by replacing n and N with n ε and N ε , noting that the extreme value corresponding to the first positive critical point for N ε is a continuous function that converges to µ in (6) as ε ց 0 by the implicit function theorem.
In the remainder, we focus on the analysis of the nonsmooth situation, leaving the appropriate modifications ("ε = 0") when n(x) = x p for 2 p ∈ to the reader, but shall nevertheless provide details for the bifurcation formulas in the smooth case as they may be of independent interest.
Observe that F ε is analytic due to the regularisation and that its linearisation around the line of trivial solutions equals ∂ ϕ F ε (0, c) = |D| −α − c id . 24/33 -Hence, for c > 0 the kernel of ∂ ϕ F ε (0, c) is trivial unless c = k −α for some integer k 1, being a simple eigenvalue of |D| −α , in which case is one-dimensional. Furthermore, |D| −α is a compact operator X β → X β since it is α-smoothing and X β ′ is compactly embedded in X β for β ′ > β. Thus ∂ ϕ F ε (0, c) is a compact perturbation of the identity and therefore constitutes a Fredholm operator of index zero. We may therefore apply the defined around s = 0, of nontrivial 2π/k-periodic solutions of (34) that bifurcates from the line of . In a neighbourhood of (0, k −α ) these are all the nontrivial solutions of F ε (ϕ, c) = 0 in X β × + .
Since we have an analytic curve in X β × + , we may compute the associated asymptotic formulas for C ε loc,k (s) as s → 0 by means of direct expansions in the regularised steady equation (34). Alternatively, one could use the more general theory in [21,Section I.6]. in case (2 sgn ): Remark 4.3. It suffices to study the case k = 1 of 2π-periodic solutions in the bifurcation analysis, Thus we focus on C ε loc := C ε loc,1 , ϕ ε := ϕ ε 1 , and c ε := c ε 1 from now on.
Switching to the bifurcation formulas, we analytically expand ϕ ε (s) and c ε (s) into and observe that the coefficients may be found by plugging the expansions into (34) and identifying terms of equal order in s by uniqueness. Note that the Taylor expansion in case (2 abs ); holds in an ε-dependent interval around x = 0, which simplifies the analysis for all sufficiently small s.
We also include asymptotic formulas in the smooth case n(x) = x p for any 2 p ∈ (with no regularisation). Formulas with arbitrary order in p seem to be new, and the result adapts easily to other dispersive operators as well.

27/33
From the last equation it follows that ς 2p−2 = the coefficient of (ϕ 1 =) cos in p ϕ with help of the odd power-reduction formula cos q x = 2 −q q−1 2 j=1 q j cos ((q − 2 j)x) for q = p − 1 and the product-to-sum identity for cosine.
Switching to odd p 5, we similarly obtain that ϕ 1 = cos and ς 0 = 1 and that (39) is true. Moreover, from = 0 for odd p we finally deduce that again by the odd power-reduction formula.
For odd p we can improve upon Theorem 4.4 and obtain the overall structure of the bifurcation formulas near the line of trivial solutions. This shows that ϕ ε=0 (s) is antisymmetric about − π 2 , and agrees with the general conjecture set forth in Section 3.
follows by expanding (ϕ ε=0 (s)) p . Then we obtain By the induction hypothesis we know that ϕ jq+1 ∈ W for all 0 j j. Moreover, each term in Λ is the product of an odd number of (some of) the terms ϕ jq+1 , with repetitions allowed. This establishes the result by noting that W is algebraically closed under an odd number of multiplications, which can be deduced from the identity whenever u, v, and w are odd. General products reduce iteratively to triple products.
Although Proposition 4.5 is promising, it is not clear to us how one can prove antisymmetry everywhere along C ε loc and its upcoming global extension. Thus we instead redefine X β in case (2 sgn ) as the subspace ϕ ∈ C β even ( ) : ϕ is antisymmetric about − π 2 , that is, ϕ(· + π) = −ϕ , for which correspondingly ker ∂ ϕ F ε (0, k −α ) = W and Theorems 4.1 and 4.2 hold for odd k.
We proceed to analyse the global structure of an extension of C ε loc in Theorem 4.1. To this end, let be the set of admissible solution pairs, where is an open set whose boundary contains any solution pair of (34) with the desirable regularity features in Theorem 3.8. We first note the following property of S ε . Lemma 4.7. Bounded, closed subsets of S ε are compact in X β × + .
Proof. It follows from Lemma 3.3 and its proof, that the operator G in (9)-adapted with N ε replacing the nonsmooth N -sends (ϕ, c) to ϕ on S ε and boundedly maps S ε into C m for any m 1. Since X β ′ is compactly embedded in X β for β ′ > β, we find that G maps bounded subsets of S ε into relatively compact subsets of X β . . .

29/33
In particular, if {(ϕ j , c j )} j is a sequence in a bounded subset B ⊆ S ε , then a subsequence of {ϕ j } j converges in X β , which together with the Bolzano-Weierstrass theorem imply that a subsequence of {(ϕ j , c j )} j converges in the X β × + -topology. Thus if B also is closed, it is compact.
By means of Lemma 4.7 and the fact that c ε (s) is not identically constant due to Theorem 4.2, we may appeal to [6, Theorem 9.1.1] and obtain a global extension of C ε loc . Note that we do not distinguish between a curve and its image. Theorem 4.8 (Global bifurcation). C ε loc extends to a global continuous curve C ε : + → S ε of solution pairs C ε (s) = (ϕ ε (s), c ε (s)), and either We shall prove that possibility iii) does not happen and that possibilities i) and ii) occur simultaneously, from which it will follow that one finds a highest, α-Hölder continuous wave as a limit along C ε .
In order to eliminate the possibility that C ε is periodic, we make use of a conic refinement of the global bifurcation theorem [6, Theorem 9.1.1]. Specifically, let K := ϕ ∈ X β : ϕ is increasing on (−π, 0) be a closed cone in X β and observe that C ε (s) ∈ K × + for sufficiently small s. Indeed, cosine is strictly increasing on (−π, 0) and strict monotonicity is stable under C 1 -perturbations on a compact set (here, ). Therefore, since ϕ ε (s) = s cos +O(s 2 ) from Theorems 4.1 and 4.2 is smooth on by Proof. According to [6,Theorem 9.2.2], it suffices to show that each (ϕ ε , c ε ) on C ε which also belongs to (K \ {0}) × + lies in the interior of (K \ {0}) × + in the topology of S ε . To this end, observe by Lemma 3.3 and Theorem 3.4 that such ϕ ε with speed c ε is smooth and satisfies (ϕ ε ) ′ > 0 on (−π, 0), with (ϕ ε ) ′′ (0) < 0 and (ϕ ε ) ′′ (−π) > 0. Now let (φ, d) ∈ S ε be another solution (not necessarily on C ε ) lying within δ-distance to (ϕ ε , c ε ) in X β × + . Then φ and d are nonzero, and φ is also smooth (Lemma 3.3). Moreover, (N ε ) −1 is smooth-also as a function of the wave speed. Hence, it follows from [17, Theorems 2.2, 4.2 and 5.2] and iteration of the smoothing effect of |D| −α that G in (9) (with N ε replacing N ) is a continuous map S ε → S ε 1 ∩ X m for any integer m 1, where S ε 1 is the functional component of S ε . As such, Thus for sufficiently small δ, one deduces that φ is strictly increasing on (−π, 0), so that φ ∈ K \ {0}. does not necessarily equal ϕ when d = c and (ϕ, c) ∈ S ε , and it is key to work with open δ-balls around solution pairs (ϕ, c) ∈ S ε and not only around solutions ϕ.
Lemma 3.7 (adapted to (34) with n ε ) and Proposition 4.9 immediately imply the following result.

Conclusion
In this paper, we have established the existence of large-amplitude periodic travelling-wave solutions with exact and optimal α-Hölder regularity in a class of evolution equations with negative-order homogeneous dispersion of order −α for all α ∈ (0, 1). Techniques include elaborate local estimates for nonlocal operators and global bifurcation analysis. A main novelty is the inclusion of generally nonsmooth, power-type nonlinearities in the considered class of equations, which we analyse using a regularisation process. We also obtain that antisymmetric nonlinearities lead to the first existence result of "doubly-cusped" extreme waves with antisymmetry.
These results open up for new investigations. One may, for instance, consider inhomogeneous nonlinearities and also study associated symmetry principles for the existence of large-amplitude waves. Another line of research may seek to establish the convexity of the highest waves and its connection to the order of the dispersive operator and the growth and regularity of the nonlinearity.

Acknowledgements
The authors gratefully acknowledge the in-depth feedback from the anonymous referee. Both authors were partially supported by research grant no. 250070 from The Research Council of Norway.