Global bifurcation of solitary waves for the Whitham equation

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnstr\"om and Wahl\'en. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.

proposed that the linear dispersion in the KdV equation with Fourier symbol 1 − 1 6 ξ 2 should be replaced by the exact linear dispersion in the Euler equation with Fourier symbol m (ξ ) := tanh(ξ ) ξ .
Note that the dispersion in the KdV equation is the second-order approximation of m at ξ = 0. This leads to the nonlinear nonlocal evolution equation known as the Whitham equation. Here, u(x, t) describes the one-dimensional wave profile and the integral kernel K is given by The function K will be referred to as the Whitham kernel, and the function m as the Whitham symbol. Specializing to traveling waves u = ϕ(x − ct) where c > 0 is the wave speed, integrating and performing a Galilean change of variables, the Whitham equation reduces to the nonlinear integral equation We are interested in functions ϕ : R → R which satisfy (1) pointwise on R, and which we refer to as solutions of (1) with wave speed c. More specifically, the results of this paper will concern solitary solutions, also called solitary-wave solutions. These are solutions ϕ : R → R satisfying lim |x|→∞ ϕ(x) = 0. Despite its simple form, the nonlocal and nonlinear nature of the Whitham equation has made it challenging to study. Recent years have seen a large amount of existence and qualitative results on the solutions of the equation. Traveling small-amplitude periodic solutions were found by Ehrnström and Kalisch [21] using the Crandall-Rabinowitz bifurcation theorem. Then, Ehrnström et al. [19] proved the existence of solitary waves using a variational method for a class of Whitham-type equations. This was followed up by Arnesen [4] where a class covering the Whitham equation with surface tension was considered. By applying a different technique-the implicit function theorem-Stefanov and Wright [31] achieved the same result. Ehrnström and Wahlén [22] showed the existence of a traveling cusped periodic wave ϕ using global bifurcation theory, and proved that c 2 − ϕ(x) ∼ |x| 1/2 near the origin. This wave attains the highest amplitude possible and is referred to as an extreme wave solution. They also conjectured that ϕ is convex and ϕ = c 2 − √ π/8|x| 1/2 + o(x) as x → 0.
Convexity of the extreme wave was shown by Encisco et al. [23] using a computer assisted proof. The goal of this paper is to prove the existence of an extreme solitary-wave solution of (1) and our plan is to use a global bifurcation theorem appearing in [11]; see  1 An extreme solitary-wave solution found by taking a limit of elements along the global bifurcation curve in Theorem 4.8. The wave speed c is supercritical, that is, c > 1. The wave profile ϕ is even and smooth on R \ {0}. It has exponential decay as |x| → ∞ and behaves like c 2 − C|x| 1/2 as |x| → 0. Ehrnström and Wahlén conjecture in [22] that C = √ π/8 also [10,15,16]. The first main step is the construction of a local bifurcation curve, emanating from the point (ϕ, c) = (0, 1), and the second is the application of the global bifurcation theorem. A key to our success is the fact that a lot of qualitative properties have been shown for the Whitham kernel, the Whitham symbol and the solutions of (1), thanks to [9,21,22]. These guide us in choosing a convenient function space to study (1) and have been extremely useful in the application of the global bifurcation theorem. In Sect. 2, we list the relevant properties and prove an integral identity. We also study how sequences of solutions converge and the Fredholm properties of important linear operators.
Another key is the recently developed center manifold theorem for nonlocal equations in [24]. This result states that nonlocal equations with exponentially decaying convolution kernels are essentially local equations near an equilibrium. It also provides a method to derive the local equation, which can then be studied using familiar ODE tools. In our case, the equilibrium is (ϕ, c) = (0, 1). Although the Whitham kernel has the required exponential decay, it fails a local integrability condition. Seeing that this condition is only for proving Fredholm properties of linear operators, we directly prove these properties instead. All necessary changes for the general center manifold theorem are listed in Appendix B. In Sect. 3, we state the center manifold theorem for the Whitham equation and compute the corresponding local equation. More specifically, we prove the following. (ϕ ) 2 + 6(c − 1)ϕ + O(|(ϕ, ϕ )|((c − 1) 2 + |ϕ| 2 + |ϕ | 2 )). (2) The ODE in this lemma is a c-dependent family of perturbed KdV equations. Restricting to c > 1 with c sufficiently close to 1, it features a unique positive even solitary-wave solution ϕ with exponential decay for each fixed c. Using Eq. (2), we show that sup y∈R ϕ( · + y) H 3 ([0,1]) c − 1. So for c sufficiently close to 1, ϕ is also a solution to equation (1). We thus arrive at the first main result of this paper (repeated as Theorem 3.3 in Sect. 3). Theorem 1.2 There exists a unique local bifurcation curve C loc which emanates from (ϕ, c) = (0, 1) and consists of the non-trivial even solitary-wave solutions ϕ to (1) with wave speeds c ∈ V satisfying sup y∈R ϕ( · + y) H 3 ([0,1]) < δ .
While both [19] and [31] contain existence results for supercritical solitary waves, the additional information provided by the center manifold approach concerning uniqueness is crucial in the subsequent analysis. To end Sect. 3, we use the center manifold theorem to prove that the linearization of the left-hand side of (1) along C loc is invertible. This is in preparation for the global bifurcation theorem.
The global bifurcation theory in [11] can now be applied to extend C loc and this extension is referred to as the global bifurcation curve C. The theory dictates several possible behaviors for C and the content of Sect. 4 is the exclusion of unwanted behaviors. We rule out the loss of compactness alternative using qualitative properties of the solutions, how sequences of solutions converge and an integral identity for (1). Then, we show that the blowup alternative happens as the Sobolev norm blows up and that an extreme solitary-wave solution is obtained in the limit. More precisely, we have the following result (repeated as Theorem 4.8 in Sect. 4).

Theorem 1.3
There exists a sequence of elements (ϕ n , c n ) on the global bifurcation curve C such that lim n→∞ ϕ n H 3 = ∞ and (ϕ n , c n ) → (ϕ, c) locally uniformly, where ϕ is a solitary-wave solution of (1) with supercritical wave speed c > 1. The solitary solution ϕ is even, bounded, continuous, exponentially decaying, smooth everywhere except at the origin and The function ϕ in the above theorem is the extreme solitary-wave solution we set out to find and is illustrated in Fig. 1. By demonstrating the use of recent spatial-dynamics tools, this paper serves as an example to studies of other nonlocal nonlinear evolution equations. In particular, these results will likely extend to a larger class of equation, such as in [5] and [20].
Finally, it is interesting to compare our results with the global bifurcation theory for the water wave problem. The existence of an unbounded, connected set of solitary water waves, including a highest wave in a certain limit, was proved by Amick and Toland [2,3] following several earlier small-amplitude results. Around the same time, Amick et al. [1] verified Stokes' conjecture for both periodic and solitary water waves, showing in particular that the limiting solitary wave is Lipschitz continuous at the crest with a corner enclosing a 120 • angle. Thus, the behavior at the crest is different from the extreme Whitham wave, which has no corner due to the C 1/2 cusp in Theorem 1.3. The construction of the global solution continua in [2,3] is also different from ours. While both proofs are based on nonlinear integral equations, the common approach in [2,3] is to first apply global bifurcation theory to a regularized problem and then pass to the limit. On the other hand, we use global bifurcation theory directly on the solitary Whitham problem. A similar approach has in fact recently been used for solitary water waves with vorticity and stratification, but based on a PDE formulation [11,32]. For the water wave problem with vorticity and stratification, the limiting behavior of largeamplitude waves is more complex and there is numerical and some analytical evidence of overhanging waves; see for example [14,17,18,28] and references therein.

Notation
We use the following notations for function spaces.
-The space of pth power integrable functions on an interval I ⊂ R with respect to a measure μ is denoted by and For σ ∈ R, we write L p (I )| := L p (I , dx), L p := L p (R, dx) and L p where dx is the Lebesgue measure and ω σ : R → R is a positive and smooth function, which equals exp(σ |x|) for |x| > 1. In particular, functions in L p η when η > 0 are necessarily exponentially decaying while functions in L p −η can grow exponentially.
-The Sobolev space is denoted by -C k denotes the space of k times continuously differentiable functions f : R → R. BU C k ⊂ C k denotes the space of functions with bounded and uniformly continuous derivatives of order up to and including k. C k,α denotes the Hölder spaces -C k (X , Y) denotes the space of k times Fréchet differentiable mappings between two normed spaces X and Y.
We use the following scaling of the Fourier transform:

The Whitham kernel and the Whitham symbol
The Whitham kernel K is given by Since m(0) = 1, we have R K dx = 1. However, since m / ∈ L 1 , K is singular at the origin. More specifically, where K reg is real analytic on R; see Proposition 2.4 in [22]. In addition, as |x| → ∞, by Corollary 2.26 in [22]. Since m is an even function, so is K . The fact that K is a positive function has been shown in Proposition 2.23 in [22]. Because the Whitham symbol m satisfies is bounded; see for example Proposition 2.78 in [6].

Properties of solutions
When choosing appropriate function spaces for (1), we will rely on the following qualitative properties of solutions.
Item (i) is stated in Lemma 4.1 in [22]. Items (ii) and (iii) are Proposition 3.13 and Theorem 4.4 in [9] respectively. The optimal exponent η = η c depends on c and is given implicitly by √ tan(η c )/η c = c, with η c ∈ (0, π/2); see [5], Theorem 6.2. We remark that the requirement sup x∈R ϕ(x) < c/2 in (iii) is not mentioned in [9] despite its importance in the proof; see the introduction in [30] for a detailed discussion. Items (iv), (v) and (vi) can be found in Theorem 4.9, Theorem 5.1 and Theorem 5.4 in [22]. The upper bound in (viii) comes from [22], Eq. (6.9). The lower bound in (vii) comes from non-negativity and the following proposition. (1) with wave speed c, such that the limits lim x→±∞ ϕ(x) exist. Then

Proposition 2.2 Let ϕ be a bounded and continuously differentiable solution to
In The latter statement is in fact true for bounded and continuous solutions ϕ with finite limits lim x→±∞ ϕ(x).
Proof In general, if ϕ is any bounded and continuously differentiable function, the limits lim x→±∞ ϕ(x) exist, and K is any non-negative even function with R K dx = 1, then A proof of this can be found in [8], pp. 113-114.
Since ϕ solves (1) with wave speed c, and since the Whitham kernel K is a non-negative even function with R K dx = 1, By non-negativity of bounded and continuous solutions with c ≥ 1, ϕ − (c − 1) must be sign-changing, ϕ ≡ 0, or ϕ ≡ c − 1, otherwise the generalized integral cannot converge to zero. This proves the claim for bounded and continuously differentiable functions ϕ.
If ϕ is only a bounded and continuous solution, convolution with a non-zero smooth and compactly supported test function φ ≥ 0 gives The left-hand side equals due to associativity and commutativity of convolution. By Lebesgue's dominated convergence theorem, the function φ * ϕ is bounded, continuously differentiable and the limits as x → ±∞ exist. It follows that Again, we must have ϕ ≡ 0, ϕ ≡ c − 1, or ϕ − (c − 1) is sign-changing.

Convergence of solution sequences
Modes of convergence of solution sequences will be important in ruling out alternatives from the global bifurcation theorem. We start with pointwise convergence, using the Arzelà-Ascoli theorem and the smoothing property of convolution with K .
n=1 be a sequence of continuous and bounded solutions to (1) such that each ϕ n has wave speed c n ∈ [1, 2] and sup x∈R ϕ n (x) ≤ c n /2. Then, there exists a subsequence (ϕ n k ) ∞ k=1 satisfying for every x ∈ R. The convergence is uniform on every bounded interval of R. The limit ϕ is a continuous, bounded and non-negative solution of (1) with wave speed c, and sup x∈R ϕ(x) ≤ c/2.

Proof
We can without loss of generality assume that lim n→∞ c n = c ∈ [1,2]. For each n, we have A rearrangement gives We claim that the right-hand side forms an equicontinuous sequence. Indeed, ϕ → K * ϕ is a bounded map from L ∞ ⊂ B 0 ∞,∞ to B 1/2 ∞,∞ = C 1/2 according to (6). Because c n ∈ [1,2], this gives Hence, (K * ϕ n ) ∞ n=1 is an equicontinuous sequence of functions. The square root of a non-negative equicontinuous sequence is an equicontinuous sequence. So, (ϕ n ) ∞ n=1 is equicontinuous. The Arzelà-Ascoli theorem gives a subsequence (ϕ n k ) ∞ k=1 , which converges uniformly to a function ϕ on each bounded interval of R. Also, ϕ is continuous and bounded by c/2.
Finally, since sup x∈R ϕ n (x) ≤ 1 and K L 1 = 1, Lebesgue's dominated convergence theorem gives K * ϕ n (x) → K * ϕ(x) as n → ∞ for all x. It follows that ϕ is a solution to (1) with wave speed c.
Here are several immediate consequences.
n=1 be a sequence of even solutions which are decreasing on [0, ∞). Define τ x n ϕ n := ϕ n ( · + x n ) for a sequence of real numbers x n with lim n→∞ x n = ∞. Then, the sequence of translated solutions τ x n ϕ n is a sequence of solutions to (1). It has a non-increasing locally uniform limitφ.
Proof Item (iii) is straightforward. We only prove items (i) and (ii) .
Each τ n ϕ is a solution of (1) with wave speed c ∈ [1,2]. We have By Proposition 2.3, a is a constant solution to (1) with wave speed c and hence by Proposition 2.2, we have a = 0 or a = c − 1.
The evenness and monotonicity in (ii) are clear, so assume that lim |x|→∞ ϕ(x) = 0 and fix > 0. Then, there exists R > 0 such that Due to lim n→∞ ϕ n (x) = ϕ(x) locally uniformly, there exists an N > 0 such that In particular, we have |ϕ n (R) − ϕ(R)| ≤ , which in turn implies |ϕ n (R)| ≤ 2 by (8). Since ϕ n is non-increasing, we have |ϕ n (x)| ≤ 2 for |x| ≥ R. But then, again by (8), |ϕ n (x) − ϕ(x)| ≤ 3 for |x| ≥ R and n > N , and the claim about uniform convergence is proved. Now, we consider convergence in H j for j > 0. Combining the smoothing property of convolution with K and (1), we use a bootstrap argument to increase the regularity of the solutions, starting with convergence in L 2 = H 0 .

Proposition 2.5
Let ϕ n , c n , ϕ and c be as in Proposition 2.3. If (a) ϕ n → ϕ uniformly and ϕ n → ϕ in L 2 , (b) sup x∈R ϕ n (x) < c n /2 and sup x∈R ϕ(x) < c/2, then ϕ n → ϕ in H j for any j > 0.
Proof Since ϕ n and ϕ solve (1) with wave speed c n and c respectively, we can write Letting ω n = K * ϕ n and ω = K * ϕ, the assumptions imply (a') ω n → ω in H 1/2 and ω n → ω uniformly, A quick calculation gives For each n, g n is smooth and g n (0) = 0. Moreover, the range of ω n belongs to the domain of g n . A standard result in the theory of paradifferential operators, for instance Theorem 2.87 in [6], gives where C > 0 depends on j, sup x∈R |ω n | and sup x∈D gn |g n (x)|. A computation shows that |g n | is uniformly bounded in n for c n , c ∈ [1, 2] and x ∈ D g n . Also, sup n ω n H 1/2 < ∞, as well as sup n,x |ω n (x)| < ∞ by (a'). It follows that g n (ω n ) H j is uniformly bounded in n and To deal with the second term, let c be fixed and define Then, we can write Due to (a') and (b'), we have ω n (s) + τ (ω(s) − ω n (s)) ∈ D h for all s ∈ R. Note that h is smooth and h(0) = 0. Theorem 2.87 in [6] gives an estimate for the integral Another standard result in paradifferential calculus, for example Theorem 8.3.1 in [29], gives for all j ∈ (0, 1/2), which tends to 0 as n → ∞ by (a'). So, the right-hand side of (9) tends to 0 in H j for all j ∈ (0, 1/2). Hence, ϕ n → ϕ in H j . Then, convolution with K increases the regularity of ϕ n and ϕ by 1/2. Choosing j = 1/4 and replacing (a') with the critical case of Theorem 8.3.1 in [29] is no longer relevant and the convergence is in H j for all j ∈ (0, 1/4 + 1/2]. By iterating as many times as needed, the claim of the proposition is proved.

Remark 2.6
Since Theorem 2.87 in [6] and Theorem 8.3.1 in [29] are valid for the Besov spaces B s p,q , we can replace the L 2 space in (a) with B 0 p,q and obtain ϕ n → ϕ in B s p,q for s > 0 using the same proof idea.

Fredholmness of linear operators
Let j > 0 be an integer. As a preparation for future bifurcation results, we study the operators where ϕ * is a solitary-wave solution with wave speed c * > 1, satisfying sup x∈R ϕ * (x) < c * /2. These are linearizations of the left-hand side in (1) at (0, 1) and (ϕ * , c * ), respectively. We show that T and L[ϕ * , c * ] are Fredholm with Fredholm index two and zero respectively, using results from [27]. The central idea is to relate a pseudodifferential operator t(x, D) : H j → H j to a positively homogeneous function A via the symbol t(x, ξ). By studying the boundary value and the winding number of A around the origin, the Fredholm property of t(x, D) can be determined. Appendix A summarizes the relevant theorems from [27].
Up until now, the weight η > 0 has remained somewhat mysterious. Since our interest lies in the Fredholm properties of T , η should be chosen so that T is at least bounded. By (4), K is locally L 1 around the origin. From (5), we deduce that K ∈ L 1 where Young's inequality for the L p norms of convolutions is used in the last step. Noting that d n dx n (K * ϕ) = K * ϕ (n) and applying the above estimates on ϕ (n) , for η in the range (0, π/2).

Fredholmness of T
Multiplication with cosh(η · ) is an invertible linear operator. Its inverse is multiplication with 1/ cosh(η · ), mapping H j −η to H j . Conjugating T with these gives and more explicitlỹ T can be rewritten as

Lemma 2.7
The conjugated pseudodifferential operatorT =t(x, D) : H j → H j is a Fredholm operator.

Proof
The idea is to apply Proposition A.1. We define a positively homogeneous function A by In order to apply Proposition A.1, we need to check that A is smooth in and that A(x 0 , x, ξ 0 , ξ) = 0 on , where is the boundary of S. can be decomposed into the arcs We compute the value of A along each arc i and show that A is nowhere vanishing. From the computations, it will be apparent that A is smooth on S.

Similar computations yield
which is nowhere vanishing by the same argument. Along the other arcs, So, A along is nowhere vanishing and Proposition A.1 gives the desired conclusion.
The next result is about the Fredholm index ofT .
Proof According to Proposition A.1, the total increase of the argument of A as is traversed with the counter-clockwise orientation which is positive for θ ∈ R and η ∈ (0, π/2). This means that m 2 stays in the first and fourth quadrant of C. The sign of the imaginary part equals the sign of which is a strictly increasing function in θ taking the value zero at θ = 0. This means that at θ = 0, m 2 enters the first quadrant from the fourth. By computing the value of m 2 as θ → −∞, at θ = 0 and as θ → ∞, we can conclude that m 2 along 1 makes one counter-clockwise revolution about 1. Taking the square root of m 2 preserves the signs of the real and imaginary part. Then, multiplication with −1 flips the signs and addition with 1 corresponds to horizontal translation to the right by 1; see Fig. 2. Finally, we arrive at the conclusion that the increase of the argument of A along 1 from (0, 1, 0, −1) to (0, 1, 0, 1) is 2π . A similar analysis shows that an additional increase of 2π is gained along 2 . On Conjugation with the invertible linear operator M cosh preserves Fredholmness and the Fredholm index. Hence, We have proved the first part of the main result of this section, which is the following. Proof The statement concerning the Fredholm properties of T is already proved. Note that solving T ϕ = 0 for ϕ ∈ L 2 −η using the Fourier transform is problematic because Fϕ is not necessarily a tempered distribution. Thus, we consider the L 2 -adjoint of T : η corresponds to (1 − m)Fψ = F g on the Fourier side, where Fψ and F g are analytic functions bounded on the strip | Im z| < η. In view of (3), 1 − m(ξ ) vanishes to second order at ξ = 0 and is bounded away from zero if ξ is. As a consequence, the range of T on L 2 where smoothness of ϕ * is from Proposition 2.1(iv). The corresponding positively homogeneous function B is As before, we verify that B does not vanish at any point along = ∪ 1≤i≤4 i ; see the proof of Lemma 2.7. Along 1 and 2 , as ϕ * is smooth and lim |t|→∞ ϕ * (t) = 0. Since m ≤ 1, B cannot attain the value zero. Along 3 and 4 , B is c * − 2ϕ * (x/ √ 1 − x 2 ). Since sup x∈R ϕ * (x) < c * /2 by assumption, B cannot take the value zero. Moreover, the argument of B is constant along as B is real-valued and the claim is proved.

Local bifurcation
We apply an adaptation of the nonlocal center manifold theorem in [24] to (1) in order to construct a small-amplitude solitary-solution curve emanating from (ϕ, c) = (0, 1). For convenience, we work with a different bifurcation parameter ν := c−1 which will be small and positive along the local curve. In the notation of [24], Eq. (1) becomes where T is defined in Sect. 2.4, and Equation (10) will be studied for ν ∈ (0, ∞), in the Sobolev space of even functions H 3 even and the weighted Sobolev spaces H 3 −η . This regularity choice j = 3 is with regard to Proposition 2.1. A solution ϕ ∈ H 3 even with wave speed c > 1, such that sup x∈R ϕ(x) < c/2, is smooth on R. Moreover, ϕ < 0 on (0, ∞) and ϕ has exponential decay. We also prove that the bifurcation curve of non-trivial even solitary solutions is locally unique in H 3 u × (0, ∞), and refer to it as C loc .
The global bifurcation theorem demands L[ϕ * , ν * ] to be invertible in H 3 even where (ϕ * , ν * ) ∈ C loc and ν * = c * − 1. Seeing that the Fredholm index of L[ϕ * , ν * ] is zero, it suffices to show that the nullspace of L[ϕ * , ν * ] is trivial. We consider Eq. (10), together with the linearized equation L[ϕ * , ν * ]φ = 0, and formulate a center manifold theorem for this system. Exploiting the previous reduction for (10), we simplify the reduced equation for the linearized problem and are able to solve it completely in H 3 even .

Center manifold reductions
Two center manifold reductions are presented: one for the nonlinear problem (10) and the other for the linearized problem L[ϕ * , ν * ]φ = 0. For (10), we use an adaptation of the center manifold theorem in [24]. In this reference, it is assumed that the convolution kernel belongs to W 1,1 η , which is not the case for the Whitham kernel K , as K is not locally L 1 according to (4). Seeing that this requirement is only used for proving the Fredholm properties of the linear part T , we replace it with requirements on the Fredholm properties on T ; see Hypothesis B.1(ii) in Appendix B.1. The rest of the proof of the center manifold theorem in [24] remains the same.
We consider (10) together with the modified equation where χ δ (ϕ) is a nonlocal and translation invariant cutoff operator defined in Appendix B. We have We have shown that Ker T has dimension two in H 3 −η and equals span{1, x}. Hence, elements A + Bx ∈ Ker T will often be identified with (A, B) ∈ R 2 . We define a projection on Ker T , which could also be considered as a mapping from H 3 −η to R 2 . Finally, the shift ϕ → ϕ( · + ξ) will be denoted by τ ξ . Theorem 3.1 For equation (10), there exist a neighborhood V of 0 ∈ R, a cutoff radiu s δ, a weight η * ∈ (0, π/2) and a map as its graph. We have −η * is a solution of the modified equation (11) with parameter ν; (iv) (local reduction) any ϕ solving (10) with parameter ν and ϕ H 3 Proof We use Theorem B.5. Proposition 2.9 shows that Hypothesis B.1 is met. Also, the fact that N is a Nemytskii operator verifies Hypothesis B.3. In particular, N ∈ C ∞ and N commutes with the translations τ ξ for all ν ∈ (0, ∞). This means that in Hypothesis B.3, we can choose any regularity k ≥ 2, possibly at the price of a smaller cutoff radius δ and weight η * . Since a quadratic-order Taylor expansion of suffices for our purposes, k = 3 is chosen. Statement (vi) concerning the reflection symmetry R follows directly from K being an even function. Hence, Theorem B.5 applies and gives items (i)-(vi). Equation (14) in (v) is given by Theorem B.5(vii). We use Q defined in (12) to compute the reduced vector field. Let ϕ ∈ M ν 0 . According to Theorem B.5(viii), M ν 0 is invariant under translation symmetries. Hence, τ ξ ϕ is also an element of M ν 0 for all ξ ∈ R. Applying Theorem B.5(vii) on τ ξ ϕ gives We compute ϕ (ξ ) by noting that , and since τ ξ ϕ ∈ M ν 0 , Hence, which is (13). To prove Eq. (14), we compute the Taylor expansion of . In view of (0, 0, 0) = 0 and D (A,B) (0, 0, 0) = 0, the Taylor expansion of : Differentiating twice with respect to x and evaluating at x = 0 shows equation (14).
When solving the linearized problem L[ϕ * , ν * ]φ = 0, we want to take advantage of the assumption that ϕ * ∈ M ν * 0 is a solution of (10) with parameter ν * . Hence, we consider (10) and L[ϕ * , ν * ]φ = 0 simultaneously: where T : 2 is an onto Fredholm operator with Fredholm index four given by The modified system is Since we only cut off in ϕ, the modified linearized equation coincides with the original one, as long as ϕ ∈ M ν 0 is sufficiently small in the H 3 u topology. Hence, all solutions to the linearized equation will be captured. The downside of this scheme is that our previous adaptation of the results in [24] cannot be applied directly. We replace the contraction principle with a fiber contraction principle to achieve the following result; see Appendix B.2. (15), there exist a cutoff radius δ, a neighborhood V of 0 ∈ R, a weight η * ∈ (0, π/2), two mappings 1 and 2 , where  D (A,B) 1 (A, B, ν), so 2 [A, B, ν] is a bounded linear operator from Ker T to Ker Q. Also, 2 is C k−1 in (A, B, ν). Suppose that ϕ * ∈ M ν * 0,1 is sufficiently small in the H 3 u norm, so that ϕ * is a solution of (10) with parameter ν * , uniquely determined by (A * , B * ). Then,

Theorem 3.2 For
Proof Theorem B.6 applies and gives (i)-(iv). The cutoff radii δ given by Theorem 3.1 and Theorem 3.2 are not necessarily the same, but the smallest one can be chosen to have (i). Arguing along the same lines as the proof of Theorem 3.1 gives equation (iv) and differentiating the Taylor expansion of f in (14) gives Eq. (16).

Local bifurcation curve
The center manifold theorem states that a solution ϕ sufficiently small in the H 3 u norm solves the reduced ODE (14), which is local in nature and allows spatial dynamics tools. Let ϕ = P, ϕ = Q and regard the spatial variable x as "time" t. Equation (14) defines the following system of ODEs which is reversible by Theorem 3.1(vi). We aim to rescale (17) into a KdV-equation Hence, we set Differentiating, substituting into (17) and identifying coefficients yield which are satisfied by The resulting rescaled system is For ν = 0, (18) is the KdV-equation with the explicitly known pair of solutions which corresponds to a symmetric and homoclinic orbit. For ν > 0, the symmetric homoclinic orbit persists by the same argument as in [26], p. 955. Undoing the rescaling and switching back to P = ϕ as well as Q = ϕ give for ν > 0. The supercritical solitary-wave solution ϕ is exponentially decaying, so both ϕ and ϕ belong to H 3 ⊂ H 3 u . Also, they depend continuously on the parameter ν. We denote this solution as ϕ * ν * with parameter ν * and define for some ν > 0. The main result of this section is reached.
Proof The function ϕ * ν * belongs to M ν * 0 by the one-to-one correspondence between (17) and M ν * 0 in Theorem 3.1(v). From (19) combined with the fact that ϕ * ν * is exponentially decaying, we have ϕ * ν * H 1 u ν * . Since the reduced vector field f in (13) is superlinear in ϕ * ν * (x) and (ϕ * ν * ) (x) by Theorem 3.1(ii), the bound by ν * in (19) is carried over to (ϕ * ν * ) . Differentiating (13) gives where D 1 f and D 2 f are bounded in view of Theorem 3.1(i). Hence, (ϕ * ν * ) (3) is also bounded by ν * . We obtain the improvement ϕ * ν * H 3 u ν * and then by choosing ν * sufficiently small, ϕ * ν * is indeed a solution of (10) according to Theorem 3.1(iv). The existence of C loc in H 3 even is now established since ϕ * ν * ∈ H 3 is an even function. Our argument for the uniqueness of C loc is similar to the one in [11], Lemma 5.10. Suppose that (ϕ, ν) is a non-trivial even solitary wave solution which is small enough in H 3 u × (0, ∞) that ϕ lies on the center manifold M ν 0 . Then (P, Q) = (ϕ, ϕ ) is a reversible homoclinic solution of the ODE (17), whose phase portrait is qualitatively the same as in Fig. 3. The homoclinic orbit in the right half plane corresponds to the case ϕ = ϕ ν and hence (ϕ, ν) ∈ C loc . Any other solution must therefore approach the origin along the portions of its stable and unstable manifolds lying in the left half plane. But this would force P(t) = ϕ(t) < 0 for sufficiently large |t|, contradicting (i) in Proposition 2.1.

Invertibility of L[' * , * ]
Let ϕ * := ϕ * ν * ∈ C loc and the corresponding parameter ν * be fixed. The linear operator L[ϕ * , ν * ] is the linearization of the left-hand side of (10) with respect to the ϕ-component. Note that L[ϕ * , ν * ] : H 3 even → H 3 even is Fredholm with Fredholm The invertibility of L[ϕ * , ν * ] has already been shown in [31]. In this section, we showcase an alternative approach exploiting Theorem 3.2 and are able to make quantitative statements for elements in the nullspace of L[ϕ * , ν * ] in H 3 −η * , where η * is as in Theorem 3.2. This approach is inspired by Lemmas 4.14 and 4.15 in [32]. Proof We use Theorem 3.2(iv), that is, elements φ ∈ Ker L ⊂ H 3 −η * have a one-to-one correspondence to the solutions of (16). Letting φ = U , φ = V and regarding x as a time variable t, we cast (16) into a system

Proposition 3.5 The nullspace Ker
This can be considered as a perturbation problem of the form where u : R 2 → R 2 , M ∈ R 2×2 is a matrix with constant coefficients and R : R → R 2 is an integrable remainder term. In this case, so the eigenvalues for M are √ 6ν * and − √ 6ν * . Moreover, the exponential decay of ϕ * and (ϕ * ) guarantees the integrability condition; see the discussion after (19).
Applying for example Problem 29, Chapter 3 in [13], as well as switching back to φ and φ , the statements concerning Ker L are immediate. In particular, Ker L is spanned by a function φ 1 behaving as exp(t √ 6ν * ) and φ 2 behaving as exp(−t √ 6ν * ) as t → ∞. It is a straightforward calculation to show that one exponentially decaying function in Ker L is (ϕ * ) , which is an odd function. Since even functions in H 3 cannot be written as linear combinations of an odd exponentially decaying function and an exponentially growing function, Ker L is trivial in H 3 even .

Global bifurcation
We use a global bifurcation theorem from [11] in a slightly modified form because the open set in our case is not a product set; see Appendix D. For (1), we take Since H 3 ⊂ BU C 2 , the supremum norm is controlled by the H 3 norm and U is thus an open set in H 3 even × R. We aim to use Theorem D.1. Proposition 2.10 verifies Hypothesis (A) in this theorem, while Sect. 3.2 and Proposition 3.5 together verify Hypothesis (B). Here, the local curve C loc bifurcates from (0, 0) ∈ ∂U. We have thus the following global bifurcation theorem for (1) in H 3 even and U.

Theorem 4.1 The local bifurcation curve C loc in Section 3.2 is contained in a curve of solutions C, which is parametrized as
for some continuous map (0, ∞) s → (ϕ s , ν s ). We have (a) One of the following alternatives holds: (ii) (loss of compactness) there exists a sequence s n → ∞ as n → ∞ such that sup n M(s n ) < ∞ but (ϕ s n ) n has no subsequence convergent in X .
In this section, we use the integral identity in Proposition 2.2 to exclude the loss of compactness scenario. An alternative route is to employ the Hamiltonian structures for nonlocal problems in [7]. Even though the Whitham kernel does not fit into this framework, a direct differentiation confirms that equation 43 in [7] indeed gives a Hamiltonian for the Whitham equation. We also study how M(s) blows up as s → ∞.

Preservation of nodal structure
We begin by showing that the nodal structure is preserved along the global bifurcation curve. (ϕ, ν) ∈ C ⊂ U, then ϕ is smooth on R and strictly decreasing on the interval (0, ∞).

Proof
The property sup x∈R ϕ(x) < (1 + ν)/2 for (ϕ, ν) ∈ U implies smoothness on R by Proposition 2.1(iv). Because in addition ϕ ∈ H 3 even is a solitary-wave solution with parameter ν > 0, we have that ϕ is non-increasing by Proposition 2.1(iii). In order to apply Proposition 2.1(v) to conclude that ϕ is strictly decreasing on (0, ∞) we only need to establish that ϕ is non-constant.
The only constant solutions are 0 and ν > 0 and the latter is excluded by the fact that ϕ ∈ H 3 is a solitary-wave solution. To show that ϕ(x) ≡ 0, note that the linearization even is Fredholm of index zero for all ν ∈ I by Proposition 2.10 and Ker L[0, ν] is trivial. So, L[0, ν] is invertible. The implicit function theorem applies and prevents C from intersecting the trivial solution line. Hence, this alternative cannot occur.

Compactness of the global curve C
The following result rules out alternative (ii) in Theorem 4.1(a). where ν n = c n −1. Also, according to Corollary 2.4, ϕ inherits non-negativity, continuity, evenness, boundedness and monotonicity from ϕ n . More precisely, we have shown in Theorem 4.2 that ϕ n is strictly decreasing on (0, ∞). So, ϕ is at least non-increasing on (0, ∞).
First, we verify that ϕ n → ϕ uniformly. Because the sequence of functions ϕ n is uniformly bounded in H 3 , it has a weak limit which coincides with the locally uniform limit ϕ. So ϕ ∈ H 3 . Since ϕ is in addition monotone on the real half-lines, we have lim |x|→∞ ϕ(x) = 0. Corollary 2.4(ii) now confirms the desired uniform convergence of ϕ n to ϕ.
Next, for the L 2 convergence, we use the integral identity in Proposition 2.2. For each n, ϕ n ∈ BU C 2 and lim |x|→∞ ϕ n (x) = 0. Hence, Also, since ν n > 0, the solitary-wave solution ϕ n has exponential decay according to Proposition 2.1(ii) and we are allowed to write Since sup n ϕ n H 3 < ∞, the L 2 integral on the left-hand side is uniformly bounded in n. Because inf n ν n > 0, the L 1 integral on the right-hand side is uniformly bounded as well. Taking into account that lim |x|→∞ ϕ n (x) = 0 uniformly in n, we obtain As n → ∞, we have |x|<R ϕ 2 n dx → |x|<R ϕ 2 dx. Letting → 0 confirms that ϕ n → ϕ in L 2 .
All prerequisites of Proposition 2.5 are now checked and we have ϕ n → ϕ in H 3 .

Analysis of the blowup
Having excluded the loss of compactness alternative, we examine the blowup alternative where (ϕ s , ν s ) ∈ C. In this case, for any sequence s n → ∞ we can extract a subsequence (also denoted {s n }) for which at least one of the following four possibilities holds: where (P3) and (P4) belong to the case when dist((ϕ s n , ν s n ), ∂U) → 0.

Theorem 4.4 The alternatives (P2) and (P3) cannot occur.
Proof Alternative (P2) cannot occur since the definition of U and Proposition 2.1(viii) imply that ν s n ≤ 1.
To exclude alternative (P3), we assume ν s n → 0 as n → ∞. Any locally uniform limit (ϕ, 0) solves (1). Moreover, ϕ is bounded, continuous, and monotone; see Because ϕ is non-negative, we must have ϕ ≡ 0. In particular, lim |x|→∞ ϕ(x) = 0. In virtue of Corollary 2.4(ii), ϕ s n → 0 uniformly and now according to Remark 2.6, ϕ s n → 0 in C k for any k, which implies that ϕ s n → ϕ in H 3 −η for all η > 0 and that (ϕ s n , ν s n ) reenters any small neighborhood of (0, 0) in H 3 u × (0, ∞). This cannot happen in light of Theorem 4.1(c) and the uniqueness of C loc given by Theorem 3.3.
Next, we show a useful characterization for when the H 3 norm stays bounded. Proof The existence of such a subsequence is given by Proposition 2.3. Since Theorem 4.4 has excluded (P3), we must have ν s n → ν > 0, which also implies that inf n ν s n > 0. We focus on proving the last statement. The proof of Theorem 4.3 already gives that sup n ϕ s n H 3 < ∞ implies ϕ ∈ H 3 , thus sup x∈R ϕ(x) < (1 + ν)/2 by Proposition 2.1(v) and lim |x|→∞ ϕ(x) = 0. Conversely, assume on the contrary that there is a subsequence of functions ϕ s n such that ϕ s n H 3 → ∞ as n → ∞, yet its locally uniform limit ϕ satisfies sup x∈R ϕ(x) < (1 + ν)/2 and lim |x|→∞ ϕ(x) = 0. Then, ϕ is smooth by (iv) in Proposition 2.1. Also, Corollary 2.4(ii) gives ϕ s n → ϕ uniformly, so ϕ s n (x) → 0 as |x| → ∞ uniformly in n. Similar to the proof of Theorem 4.3, we get where and R are independent of n. Rearranging gives where sup x∈R ϕ 2 s n (x) < 1 because ν s n ∈ (0, 1]. Recall that inf n ν s n > 0. Choosing = inf n ν s n /2, this shows sup n ϕ s n L 1 < ∞. It follows that the sequence of functions ϕ s n is uniformly bounded in L 2 and arguing as in the proof of Proposition 2.5 gives the uniform boundedness in H 3 , which is a contradiction to the assumption. We can now establish the following equivalence. Proof Let (ϕ s n , ν s n ) ∞ n=1 be a sequence satisfying (P4). By possibly taking a subsequence, Proposition 2.3 gives that ϕ s n → ϕ locally uniformly and ν s n → ν, where ν > 0 as we have excluded (P3). Since each ϕ s n is even and strictly decreasing, (P4) is the same as which is equivalent to By Lemma 4.5, this implies (P1). For the other implication, let (ϕ s n , ν s n ) ∞ n=1 be a sequence satisfying (P1). We also have that ϕ s n → ϕ locally uniformly and ν s n → ν > 0. Once again by Proposition 2.3 and Corollary 2.4, ϕ solves (1) with parameter ν > 0 and is continuous, bounded, even, and ϕ is non-increasing on (0, ∞). Then, the limit lim |x|→∞ ϕ(x) exists. According to Corollary 2.4(i), this can take the value In addition, Proposition 2.2 says The combinations (aC) and (bB) are quickly excluded. If ϕ ≡ 0, then sup x∈R ϕ(x) < (1+ν)/2 and Lemma 4.5 gives that sup n ϕ s n H 3 < ∞, which contradicts the blowup alternative. This rules out (aB). The fact that ϕ is non-increasing on (0, ∞) rules out (bA). Assume (bC), which is just (C). Then, we arrive at a contradiction as follows. Consider the sequence of translated solutions τ x n ϕ s n = ϕ s n ( · + x n ), where each x n is chosen in such a way that Such a numberν exists because ν s n > 0 cannot limit to 0. Moreover, lim n→∞ x n = ∞, or we cannot have ϕ ≡ ν > 0 for every x while each ϕ n has exponential decay. Corollary 2.4(iii) applied to (τ x n ϕ s n ) n gives a bounded, continuous and non-increasing locally uniform limitφ. The functionφ is a solution to (10) with parameter ν. Its limits ϕ(x) as x → ±∞ are guaranteed to exist and these can take the value zero or ν > 0. By construction,φ On the other hand, this also shows thatφ is not a constant function, implying which is a contradiction. We conclude that (C) cannot occur. Hence, we must have where the first condition is the same as (P4). Applying Lemma 4.5 gives the desired implication. Finally, since (P1) and (P4) are equivalent, (P2) and (P3) cannot happen and the blowup alternative must take place, we must have a sequence (ϕ s n , ν s n ) ∞ n=1 ⊂ C such that lim n→∞ ϕ s n H 3 = ∞ and lim n→∞ ν s n = ν > 0. By taking the limit of a subsequence, an extreme solitary-wave solution ϕ attaining the highest possible amplitude (1 + ν)/2 is found; see Fig. 1. The sequence of solutions ϕ s n has a locally uniform limit ϕ. We have (i) ϕ is continuous, bounded, even and non-increasing on the positive real half-line; (ii) ϕ is a non-trivial solitary-wave solution to (10) with parameter ν > 0; (iii) ϕ(0) = (1 + ν)/2 and more precisely near the origin and for some constants 0 < C 1 < C 2 ; (iv) ϕ is smooth everywhere except at x = 0; (v) ϕ has exponential decay. Let A be the class of functions A(x * , ξ * ) ∈ C ∞ (X * × E * ) such that A is positively homogeneous of degree 0 in x * and ξ * , that is, Clearly, each A ∈ A is uniquely determined by its values on S. Conversely, each functionÃ ∈ C ∞ (S) can be uniquely homogeneously extended to X * × E * . So, A ∼ = C ∞ (S). By S 0 A , we denote the set of symbols p A (x, ξ) which are given by for some A ∈ A. For p A ∈ S 0 A , we have the following result, which combines Theorems 4.1 and 4.2 in [27].
where argA(x * ,¸ * )| 0 is the increase in the argument of A(x * , ξ * ) around as is traversed with the counter-clockwise orientation.
We comment that a version of Proposition A.1 is available for matrix-valued symbols p A (x, ξ); see [27,Sect. 4].

B.1 A nonlocal version of the center manifold theorem
Nonlocal nonlinear parameter-dependent problems of the form where are considered in the weighted Sobolev spaces H j −η (R n ) for some η > 0, positive integers n and j = 1. The following presentation focuses on dimension n = 1 and arbitrary regularity j ≥ 1. T will be referred to as the linear part and N as the nonlinear part of (22).
To obtain an appropriate modification of Eq. (22), we introduce the space of uniformly local H j functions, Note that the embeddings H j ⊂ H j u ⊂ H j −η are continuous for all η > 0. Next, let χ : R → R be a smooth cutoff function satisfying In addition, let θ : R → R be a smooth and even function with We define and then The cutoff operator χ : H j −η → H j u is well-defined, Lipschitz continuous and [25] proves these claims for j = 1 and an asymmetric function θ with supp θ ⊂ [−1/4, 5/4]. Generalizing to higher regularity j ≥ 1 and verifying the results in [25] with our choice of θ are straightforward. The scaled cutoff operator χ δ : H j −η → H j u naturally inherits these properties, in particular where where ind T is the Fredholm index of T . In other words, T is onto. [24] is only used to guarantee item (ii). Since the Whitham kernel K / ∈ W 1,1 η , we instead require (ii) directly. Hypothesis B.3 (The nonlinear part N ) There exist k ≥ 2, a neighborhood U of 0 ∈ H j −η and V of 0 ∈ R, such that for all sufficiently small δ > 0, we have

Hypothesis B.4 (Symmetries)
There exists a symmetry group S ⊂ O(1) × (R × O(1)) which contains all translations on the real line and which commutes with the linear part T as well as the nonlinear part N , that is, More precisely, we study where the linear part is and the nonlinear part is We consider the modified system with where N δ (v, μ) is defined in the previous section. Observe that we only cut off in v, which allows capturing all solutions of the linearized Eq. (24). This requires yet another adaptation of the center manifold theorem, where the usual contraction principle is replaced with a fiber contraction principle.  (25), there exist a cutoff radius δ > 0, a weight η * ∈ (0, η 0 ), a neighborhood V of 0 ∈ R and two mappings with the center manifold as its graph, and at each fixed The following statements hold. We have g(v 0 , w 0 , μ) = D v 0 f (v 0 , μ)w 0 .

μ] consists precisely of the solutions to the modified linearized equation
We will use the following fiber contraction theorem in the proof; see Sect. 1.11.8 in [12].

Proposition B.7 Let X and Y be complete metric spaces. Consider a continuous map
: where λ 1 : X → X and λ 2 : X × Y → Y. If λ 1 is a contraction in X , and y → λ 2 (x, y) is a contraction in Y for every fixed x ∈ X , then has a unique fixed point (x 0 , y 0 ) ∈ X × Y.
Proof of Theorem B. 6 We present the necessary changes in the proof of the center manifold theorem in [24] without the parameter μ. The transition to the parameterdependent version is the same as in [24]. In view of Hypothesis B.1(ii), the operator T = (T , T ) is Fredholm and its Fredholm index is twice the index of T . We define the Fredholm-bordered operator By Lemma 3.2 in [24],T is invertible for any η ∈ (0, η 0 ) with T −1 = (T −1 ,T −1 ), and T −1 where C is a continuous function of η. We define the bordered nonlinearitỹ ApplyingT −1 on both sides, then moving the nonlinear term to the right-hand side, we obtain a fixed point equation Choosing δ sufficiently small, the second component is a contraction mapping on H Then, reasoning as in [24] validates statements (iv)-(vi), except for the last claim g(v 0 , w 0 ) = D v 0 f (v 0 )w 0 . This is shown by plugging τ x w = (Id +D v 0 1 (τ x v 0 ))τ x w 0 into the reduced vector field g in (vi), and then identifying the result with

C Computation of the coefficients 9 ijk
We wish to solve subjected to the condition Q( i jk ) = 0 for all i + j +k ≥ 1. This condition is imposed for unique solvability of i jk ; see the proof of Theorem 3.1.
Using the fact that multiplication by x n corresponds to n-times differentiation on the Fourier side, we have R K (x)x n dx = 0 i f n is odd (−1) n/2 m (n) (0) if n is even. Now, we can compute the convolution of K and monomials x n , where n ∈ N. For instance, K * 1 = 1 and R K (y)(x − y) dy = x R K (y) dy − R y K (y) dy = x · (K * 1) − 0 = x, where R y K (y) dy = 0 because the integrand is odd. Utilizing the binomial theorem to expand (x − y) n together with symmetries of the integrands, we arrive at To solve (27), we are motivated by (30) and make the Ansatz 200 = αx 2 −Q(αx 2 ), where subtraction by Q(αx 2 ) is to make sure that Q( 200 ) = 0. Since Q(αx 2 ) = 0 for all α ∈ R, it can be removed. Plugging the Ansatz into (27)

D A global bifurcation theorem
Let X , Y be Banach spaces and U ⊂ X ×R an open set. Consider the abstract operator equation where F : U → Y is an analytic mapping. The following is a version of Theorem 6.1 in [11] which has been slightly modified to better fit the situation in the present paper. The proof remains the same.