A coupling problem for entire functions and its application to the long-time asymptotics of integrable wave equations

We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has purely discrete spectrum. To this end, we introduce a natural coupling problem for entire functions, which serves as a replacement for the usual Riemann-Hilbert problem, which does not apply in these cases. As a prototypical example, we investigate the long-time asymptotics of the dispersionless Camassa-Holm equation.


Introduction
Integrable wave equations play a key role in understanding numerous phenomena in science. In this connection, understanding the long-time asymptotics of solutions is crucial. Roughly speaking, the typical behavior is that any (sufficiently fast) decaying initial profile splits into a number of solitons plus a decaying dispersive part. This has been first observed numerically for the Korteweg-de Vries equation [30]. Corresponding asymptotic formulas were derived and justified with increasing level of rigor over the last thirty years. To date, the most powerful method for deriving such long-time asymptotics is the nonlinear steepest descent method from Deift and Zhou [11], which was inspired by earlier work of Manakov [24] and Its [19]. More on this method and its history can be found in the survey [12]; an expository introduction to this method for the Korteweg-de Vries equation can be found in [17].
Although this method has found to be applicable to a wide range of integrable wave equations, there are still some exceptions. The most notable one is the Camassa-Holm equation, also known as the dispersive shallow water equation, where u ≡ u(x, t) is the fluid velocity in the x direction, κ ≥ 0 is a constant related to the critical shallow water wave speed, and subscripts denote partial derivatives. It was first introduced by Camassa and Holm in [7] and Camassa et al. [8] as a model for shallow water waves, but it actually already appeared earlier in a list by Fuchssteiner and Fokas [16]. Regarding the hydrodynamical relevance of this converges locally uniformly to an entire function of exponential type zero [3,Lemma 2.10.13], [21,Theorem 5.3]. Furthermore, we introduce the quantities η λ ∈ R∪{∞} for each λ ∈ σ which are referred to as the coupling constants.
Definition 2.1. A solution of the coupling problem with data {η λ } λ∈σ is a pair of entire functions (Φ − , Φ + ) of exponential type zero such that the following three conditions are satisfied: (C) Coupling condition: (G) Growth and positivity condition: In order to be precise, if η λ = ∞ for some λ ∈ σ, then the coupling condition (C) in this definition has to be read as Φ − (λ) = 0. The growth and positivity condition (G) means that the meromorphic function is a so-called Herglotz-Nevanlinna function, which satisfy certain growth restrictions (to be seen from their integral representations; [1,Chapter 6], [28,Chapter 5]). Moreover, let us mention that since the residues of such a function are known to be nonpositive, condition (G) also requires the necessary presumption are finite, where ρ ± denote the sets of all (necessarily simple) zeros of the functions Φ ± . As a consequence, these functions can be written as the canonical products bearing in mind the normalization condition (N). Finally, we mention the bounds upon roughly estimating (2.6) and employing the interlacing condition once more.
The simplest case of a coupling problem prevails when the set σ consists of only one point. In this case, it is indeed possible to write down the solution explicitly in terms of the one single coupling constant.
Proposition 2.2. Suppose that σ = {λ 0 } for some nonzero λ 0 ∈ R. If the coupling constant η λ0 ∈ R ∪ {∞} is not negative, then the coupling problem has a unique solution given by Proof. It is readily verified that the given polynomials are indeed a solution of the coupling problem. Conversely, if (Ψ − , Ψ + ) is another solution, then for some a ± ∈ R with a − a + = 0. Moreover, we infer that 0 ≤ a ± λ 0 ≤ 1 in view of the Herglotz-Nevanlinna property (more precisely, from the interlacing condition of the poles and zeros). Lastly, the coupling condition (C) takes the form Now if η λ0 ≤ 1, then necessarily a − = 0 since otherwise we get the contradiction Consequently, we may express a + in terms of the coupling constant using the coupling condition. In much the same manner, one may obtain the coefficients a ± if η λ0 ≥ 1 and finally end up with in either case, which finishes the proof.
Note that there is no solution of the coupling problem in Proposition 2.2 if the coupling constant is negative, since it would violate the positivity condition (G).

Asymptotic analysis
We shall now derive a general result on the asymptotic behavior of solutions to the coupling problem. Therefore, let ∆ be a first-countable topological space (that is, every point has a countable neighborhood basis) and fix some δ ∞ ∈ ∆. Again, we denote with σ ⊆ R a discrete set such that the sum (2.1) is finite and define the entire function W by (2.2). Moreover, for every δ ∈ ∆ we consider a set of coupling constants η λ (δ) ∈ R ∪ {∞} indexed by λ ∈ σ.
Proof. First consider a sequence δ k ∈ ∆, k ∈ N with δ k → δ ∞ as k → ∞ such that the entire functions Φ ± ( · , δ k ) converge locally uniformly as k → ∞. The respective limits are entire functions of exponential type zero in view of (2.7) and will be denoted by Ψ ± . Due to assumption (3.1) and the coupling condition, we conclude that Ψ ± (λ) = 0 for λ ∈ σ ± (also observe that the quantities Φ ± (λ, δ) are uniformly bounded in δ ∈ ∆). As a consequence, the meromorphic Herglotz- has only one pole and thus at most two zeros, which are necessarily simple. Consequently, we may write (keep in mind that these functions are of exponential type zero and that their zeros have genus zero) where P ± are polynomials such that P − P + has at most one zero, which is simple. Moreover, the pair (P − , P + ) satisfies the coupling condition where the constant η λ0,∞ ∈ R ∪ {∞} is given as the limit Hereby note that the limit is nonnegative because of (2.4). In view of Proposition 2.2 we now may write down the polynomials P ± explicitly and conclude that as k → ∞, locally uniformly in z ∈ C\σ. Finally, from the very definition of the constants η λ0,∞ we may also rewrite this as as k → ∞, locally uniformly in z ∈ C\σ. Finally, if the claim of the theorem was not true, then there would be a compact set K ⊆ C\σ and a subsequence δ k ∈ ∆, k ∈ N with δ k → δ ∞ as k → ∞ such that for all z ∈ K, k ∈ N and some ε > 0. However, a compactness argument (recall (2.7) and apply Montel's theorem) shows that there is a subsequence δ k l such that Φ ± ( · , δ k l ) converges locally uniformly as l → ∞. In view of the first part of the proof, this gives a contradiction to (3.5).
The assumptions in Theorem 3.1 allow one of the coupling constants to be arbitrary. This will turn out to be crucial to obtain long-time asymptotics of the Camassa-Holm equation which are valid uniformly in sectors. However, in the case when all of the coupling constants are supposed to converge to zero or infinity, one obtains the following result.
Proof. Similarly to the first part of the proof of Theorem 3.1, one infers that as k → ∞, locally uniformly for all z ∈ C\σ as long as the functions Φ ± ( · , δ k ) are assumed to converge locally uniformly. In fact, this follows since the function in (3.4) is now known to have no poles at all. Now the claim follows in much the same manner as in the second part of the proof of Theorem 3.1, invoking a compactness argument.

Applications to the Camassa-Holm equation
As anticipated in the introduction, we will now demonstrate that our results provide a powerful tool to derive long-time asymptotics for the dispersionless Camassa-Hom equation. To this end, let u be a solution of with decaying spatial asymptotics. To be precise, we will assume that the quantities are finite signed measure for each time t ≥ 0. These conditions guarantee (see [15,Theorem 3.1]) that for every time t ≥ 0 and z ∈ C, there are unique solutions φ ± (z, · , t) of the differential equation (the prime denotes spatial differentiation) with the spatial asymptotics In view of [15,Theorem 4.1], it is known that these solutions are real entire and of exponential type zero with respect to the spectral parameter. Now the importance of the spectral problems (4.3) lies in the well-known fact that their spectra are invariant under the Camassa-Holm flow, that is, they are the same for all times t ≥ 0 (for example, we refer to [2, Section 2], [7], [9, Section 3], [14,Theorem 5.1]). For this reason, we may simply denote the spectrum of (4.3) with σ, which is known to be real and purely discrete such that the sum is finite, in view of [15,Proposition 3.3]. The Wronskian of our two solutions turns out to be independent of space x ∈ R and time t ≥ 0. Indeed, this function is the characteristic function of the spectral problem (4.3), that is, in view of [15,Corollary 4.2].
In order to point out the connection to the coupling problem for entire functions, one observes that the solutions φ + (λ, · , t) and φ − (λ, · , t) are linearly dependent for every eigenvalue λ ∈ σ and time t ≥ 0. Hence there is some nonzero real c λ (t) ∈ R such that we may write (4.8) The time evolution for these quantities is known to be given explicitly by where the dot denotes differentiation with respect to the spectral parameter.
We have now collected all necessary ingredients to prove the announced long-time asymptotics for the solution u of the Camassa-Holm equation. In fact, the proof of this result is almost immediate from the general results on asymptotic analysis for our coupling problem of entire functions derived in the previous section.
Theorem 4.1. Let S ⊆ R×(0, ∞) be a closed sector which contains at most finitely many of the rays r λ , λ ∈ σ given by 2λx = t. Then we have for (x, t) ∈ S as t → ∞, where the phase shifts ξ λ for each λ ∈ σ are given by Proof. For every x ∈ R and t ≥ 0 we introduce the entire functions Φ ± (z, x, t) = e ± x 2 φ ± (z, x, t), z ∈ C, which will turn out to be a solution of a particular coupling problem. In fact, one clearly has the coupling condition (C) (4.13) Moreover, due to [13,Proposition 4.4], the function is a Herglotz-Nevanlinna function, ensuring the growth and positive condition (G). In fact, this can also be verified by a direct calculation, using the differential equation (4.3). Finally, the normalization condition (N) is immediate from the definition.
We will first consider the special case when the sector S contains precisely one ray, say r λ0 for some λ 0 ∈ σ. Upon defining the sets σ ± ⊆ σ by for some ε λ > 0, λ ∈ σ ± . Therefore, the coupling constants in (4.13) satisfy for (x, t) ∈ S as t → ∞. In view of Theorem 3.1 and [15, Lemma 3.4] this yields for (x, t) ∈ S as t → ∞. But this proves the claim in this special case, since for (x, t) ∈ S as t → ∞, as Lebesgue's dominated convergence theorem shows. In order to finish the proof in the general case, not that under our assumptions we may cover the sector S with finitely many sectors of the type considered above.
The typical long-time behavior of a solution u of the Camassa-Holm equation, derived in Theorem 4.1 can be depicted as follows: Hereby, the grey areas represent two sectors in which our long-time asymptotics hold uniformly. Each of the rays r λ , accumulating at the t-axis, corresponds to an eigenvalue λ ∈ σ of the underlying isospectral problem. After long time, one can see that the solution u splits into a train of single peakons, each of which travels along one of the rays, with height and speed determined by the corresponding eigenvalue. Due to the conditions on the sector in Theorem 4.1, we do not obtain long-time asymptotics of solutions, which hold uniformly in sectors around the t-axis (as long as the spectrum is not finite, that is, in the multi-peakon case). However, we are able to derive long-time asymptotics which hold uniformly in strips near the t-axis, that is, as long as x stays bounded (cf. [29,Theorem 1.2]). e −|x− t 2λ +ξ λ | + o(1) (4.14) for |x| ≤ R as t → ∞.
Proof. With the notation from the proof of Theorem 4.1 we see that the coupling constants in (4.13) for every λ ∈ σ satisfy ln e x− t 2λ c λ (0) = x − t 2λ + ln |c λ (0)| → ∞ as t → ∞. Therefore, an application of Corollary 3.2 shows that u(x, t) = o(1) as t → ∞, in view of [15,Lemma 3.4]. To finish the proof note that the right hand side of (4.14) converges to zero as t → ∞, as an application of Lebesgue's dominated convergence theorem shows.