Boussinesq's equation for water waves: asymptotics in Sector I

In a recent paper, we showed that the large $(x,t)$ behavior of a class of physically relevant solutions of Boussinesq's equation for water waves is described by ten main asymptotic sectors. In the sector adjacent to the positive $x$-axis, referred to as Sector I, we stated without proof an exact expression for the leading asymptotic term together with an error estimate. Here we provide a proof of this asymptotic formula.


Introduction
In a recent paper [2], we developed an inverse scattering approach to the Boussinesq [1] equation (1.1) This approach yields an expression for the solution u(x, t) in terms of the solution n(x, t, k) of a 1 × 3-row-vector Riemann-Hilbert (RH) problem, and by performing a Deift-Zhou steepest descent analysis of this RH problem, it is possible to derive asymptotic formulas for the large (x, t) behavior of u(x, t).In [2], we identified in this way ten main sectors in the half-plane t ≥ 0 describing the asymptotics of u(x, t) for a class of physically relevant initial data (see Figure 1), and in each sector we provided an asymptotic formula for the solution.For conciseness, the formula in the sector adjacent to the positive x-axis, referred to as Sector I in [2], was stated without proof.Here we provide a proof of this formula.Equation (1.1) first appeared in a paper by Boussinesq from 1872 [1], where it was derived as a model for long small-amplitude waves in a channel.About a century later, Hirota discovered that (1.1) supports soliton solutions [8], and soon afterwards Zakharov [10] constructed a Lax pair for (1.1), thus demonstrating its formal integrability.In the early 1980's, Deift, Tomei, and Trubowitz [6] developed an inverse scattering formalism for an equation closely related to the Boussinesq equation (more precisely, for the equation obtained from (1.1) by deleting the u xx -term).However, the problem of developing an inverse scattering transform formalism for (1.1) is substantially more complicated, because (1.1) involves a spectral problem with coefficients that do not decay to zero at infinity.This introduces new jumps on the unit circle in the RH problem, and it turns out that all the main asymptotic contributions to u(x, t) as (x, t) → ∞ originate from these new jumps.In particular, as shown below, the main contributions to the asymptotic behavior of u(x, t) in Sector I stem from a global parametrix which is used to cancel jumps on certain parts of the unit circle, as well as from six saddle points on the unit circle.
E-mail address: christophe.charlier@math.lu.se, jlenells@kth.se.To describe the large (x, t) asymptotics of u(x, t), it is sufficient to consider Sectors I-V because Sectors IV-X are related to Sectors I-V by symmetry.The derivation of the asymptotic formula in Sector I presented here features a number of differences compared to the derivations in Sectors II-V.For example, in Sector I, x plays the role of the large parameter instead of t, because t is not uniformly large as (x, t) → ∞ in Sector I.The analysis needed for Sector I is also complicated by the fact that as (x, t) approaches the x-axis, the six saddle points relevant for the analysis merge with the six points {ie πij/3 } 5 j=0 , which are all intersection points for the jump contour of the RH problem.This makes the local analysis near the saddle points more involved.It also forces us to perform an additional transformation of the RH problem (see Section 5) and for this purpose additional analytic approximations of the reflection coefficients r 1 and r 2 are required on the six rays that start at the origin and pass through {ie πij/3 } 5 j=0 .Since another set of analytic approximations of r 1 and r 2 is needed on the unit circle, this leads to the curious situation that we have two different approximations of the same functions on overlapping domains.Nevertheless, thanks to certain nonlinear relations obeyed by r 1 and r 2 (see (4.4) and (4.5)), the relevant new jumps can still be estimated (see Lemma 7.3).

Main result
Let u 0 (x) and u 1 (x) be real-valued functions in the Schwartz class S(R) and consider the initial value problem for (1.1) on the line with initial conditions u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) for x ∈ R. (2.1) 2.1.Assumptions.Our results will be valid under the same asumptions on u 0 and u 1 as in [2, Theorem 2.14].To formulate these assumptions, we define two scattering matrices s(k) and s A (k) by where the following notation is used: • L = diag (l 1 , l 2 , l 3 ) where the functions {l j (k)} 3 j=1 are defined by with ω := e • U(x, k) is given by where v 0 (x) := x −∞ u 1 (x )dx and • X(x, k) and X A (x, k) are the unique solutions of the Volterra equations Our assumptions on {u 0 , u 1 } can now be formulated as follows: (i) Mass conservation: we suppose that R u 1 (x)dx = 0. (ii) Absence of solitons: we assume that s(k are the open subsets of the complex plane shown in Figure 2, ∂D is the unit circle, and Q = {κ j } 6 j=1 ∪ {0} with κ j = e πi(j−1) 3 , j = 1, . . ., 6, the sixth roots of unity.(iii) Generic behavior of s and s A near k = 1 and k = −1: for k = ±1, we assume that (iv) Existence of a global solution of the initial value problem: we suppose that r 1 (k) = 0 for all k ∈ [0, i], where [0, i] is the vertical segment from 0 to i.

2.2.
Statement of the main result.It is shown in [2] that the solution u(x, t) to the initial value problem for (1.1) with initial data u 0 , u 1 satisfying the assumptions (i)-(iv) can be expressed in terms of the solution n of a row-vector RH problem with jump contour Γ = ∪ 9 j=1 Γ j shown and oriented as in Figure 2. In particular, the assumptions (i)-(iv) imply that the solution u(x, t) exists globally.Our main theorem provides the asymptotics of u(x, t) in Sector I given by τ := t/x ∈ [0, τ max ], where τ max ∈ (0, 1) is a constant.The asymptotic formula is given in terms of two spectral functions r 1 (k) and r 2 (k) defined by where Γj = Γ j ∪ ∂D denotes the union of Γ j and the unit circle.We define Φij (τ, k) for 1 where l j (k) are as in (2.2) and the functions z j (k), j = 1, 2, 3, are defined similarly: Finally, we introduce the saddle points {k j = k j (τ )} 4 j=1 of Φ21 which are given by ) and satisfy ).The following is our main result.Theorem 2.1 (Asymptotics in Sector I).Let u 0 , u 1 ∈ S(R) be real-valued functions such that Assumptions (i)-(iv) are fulfilled.For any integer N ≥ 1, there exists a constant τ max ∈ (0, 1) such that the global solution u(x, t) of the initial value problem for (1.1) with initial data u 0 , u 1 satisfies uniformly for τ = t/x ∈ [0, τ max ], where C N (τ ) ≥ 0 is a smooth function of τ that vanishes to all orders at τ = 0, and A(τ ) and α(x, τ ) are defined by Here Γ(k) is the Gamma function, z is given by where the branch of the square root is such that −ik 1 z > 0, and where the path of the integral starts at i, follows the unit circle in the counterclockwise direction, and ends at k 1 .
Remark 2.2.Since k 1 tends to i as τ → 0 and r 1 (k) vanishes to all orders at k = i, it follows that ν, and hence also A(τ ), vanishes to all orders as τ → 0. Consequently, (2.7) implies that In particular, for any fixed t ≥ 0, the formula (2.7) reduces to u(x, t) = O(x −N ) as x → +∞, in consistency with the fact that u(x, t) is a Schwartz class solution.
Remark 2.3.By expressing the asymptotic formula in Theorem 2.1 in terms of ζ = x/t instead of τ = t/x, we see that it is equivalent to the formula stated in [2] for Sector I (note that z here should be identified with z / √ ζ in [2]).We will use the variable τ in this paper, because it allows us to evaluate certain expressions at t = 0 and τ = 0 without taking limits.
Remark 2.4.By combining the result of Theorem 2.1 with the asymptotic formula for u(x, t) in Sector II (see [2]), it can be seen that the conclusion of Theorem 2.1 in fact holds for any choice of τ max ∈ (0, 1).
The proof of Theorem 2.1 is given in Sections 4-12.It is based on a steepest descent analysis of a RH problem derived in [2] for a 1 × 3-row-vector valued solution n, which we next recall.

The RH problem for n
For j = 1, . . ., 6, let Γ j = Γ j \ D be the part of Γ j outside the open unit disk D = {k ∈ C | |k| < 1} and let Γ j := Γ j \ Γ j be the part inside D, where Γ j is as in Figure 2. Let θ ij (x, t, k) = x Φij (τ, k) and denote by Γ = {iκ j } 6 j=1 ∪ {0} the set of intersection points of Γ. Define the function f on the unit circle by and define the jump matrix v(x, t, k) for k ∈ Γ by where v j , v j , v j are the restrictions of v to Γ j , Γ j , and Γ j , respectively.The jump matrix v obeys the symmetries where The following RH problem was derived in [2].
RH problem 3.1 (RH problem for n).Find a 1 × 3-row-vector valued function n(x, t, k) with the following properties: The limits of n(x, t, k) as k approaches Γ\Γ from the left and right exist, are continuous on Γ \ Γ , and are denoted by n + and n − , respectively.Moreover,

Brief overview of the proof
Since u 0 and u 1 satisfy Assumptions (i)-(iv), [2, Theorems 2.6 and 2.12] imply that RH problem 3.1 has a unique solution n(x, t, k) for each (x, is well-defined and smooth for (x, t) ∈ R × [0, ∞), and that is a real-valued Schwartz class solution of (1.1) on R×[0, ∞) with initial data u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x).We can therefore obtain the asymptotics of u(x, t) by analyzing the RH problem 3.1 with the help of Deift-Zhou [7] steepest descent techniques.In this analysis the saddle points of the three phase functions Φ21 , Φ31 , and Φ32 play a central role.The saddle points of Φ21 , i.e., the solutions of ∂ k Φ21 (τ, k) = 0, were denoted by {k j } 4 j=1 in Section 2. Since Φ31 (τ, k) = − Φ21 (τ, ω 2 k) and Φ32 (τ, k) = Φ21 (τ, ωk), it follows that {ωk j } 4 j=1 are the saddle points of Φ31 and that {ω 2 k j } 4 j=1 are the saddle points of Φ32 .For Sector I only the six saddle points {ω j k 1 , ω j k 2 } 2 j=0 play a role in the asymptotic analysis.The signature tables for Φ21 , Φ31 , and Φ32 are shown in Figure 3.The steepest descent analysis proceeds via several transformations n → n (1) → n (2) → n (3) → n (4) → n, such that the RH problems satisfied by n (1) , n (2) , n (3) , n (4) , and n are equivalent to RH problem 3.1 satisfied by n.We will let Γ (j) , Γ and v (j) , v denote the contours and jump matrices of the RH problems for n (j) , n, respectively.The symmetries (3.3) and (3.6) will be preserved by each transformation, so that, for j = 1, 2, 3, 4, and similarly for n and v.We will see that six local parametrices near the saddle points {k j , ωk j , ω 2 k j } 2 j=1 as well as a global parametrix ∆ −1 contribute to the asymptotics of u(x, t).However, due to the symmetries (4.2)-(4.3),we will only need to construct the local parametrix near k 1 explicitly, and we will only need to define the transformations n (j) → n (j+1) and n (4) → n in the sector S := {k ∈ C| arg k ∈ [ π 3 , 2π 3 ]}.In the end, we will find that n satisfies a small-norm RH problem whose solution can be expressed as an integral over the contour Γ.This leads to an asymptotic formula for n(1) (x, t) := lim k→∞ k(n(x, t, k) − (1, 1, 1)), and hence, by inverting the above transformations, an asymptotic formula also for n (1) 3 (x, t).By substituting this formula into (4.1),we obtain the formula of Theorem 2.1.
The following properties of the functions r 1 and r 2 established in [2, Theorem 2.3] will be used repeatedly throughout the proof: has simple zeros at k = ±ω and simple poles at k = ±ω 2 , and r 1 , r 2 have rapid decay as |k| → ∞.Moreover, ) 5. The n → n (1) transformation Let τ = t/x and note that x Φij (τ, k) = θ ij .We will use the notation I := [0, τ max ], where τ max ∈ (0, 1) is a constant, so that Sector I corresponds to τ ∈ I.The jump matrices v 1 and v 4 involve the off-diagonal entries r 1 respectively.As long as τ > 0 stays away from 0, the exponentials e x Φ21 (τ,k) and e −x Φ21 (τ,k) are exponentially small as x → ∞ for k ∈ (0, i) and k ∈ (i, i∞), respectively.However, 2Im k is pure imaginary for all k ∈ iR, meaning that the above exponentials lose their decay as τ → 0. The goal of the first transformation n → n (1) is to deform the jumps v 1 and v 4 into regions where the exponentials retain their decay as x → ∞ uniformly for small τ .For this purpose, we need to introduce analytic approximations of the functions r 1 and r 2 .Let V 1 ⊂ C be the open set (see Figure 4) Lemma 5.1 (Decomposition lemma I).For every integer N ≥ 1, there exists a decomposition such that the functions r1,a , r1,r have the following properties: (a) For each x ≥ 1, r1,a (x, k) is defined and continuous for k ∈ V1 and analytic for k ) where k = −i and the constant C is independent of x, τ, k.
Remark 5.2.We will later need to introduce yet another decomposition of r 1 , see Lemma 6.1.We use tildes on the functions r1,a and r1,r in Lemma 5.1 to distinguish them from the functions r 1,a and r 1,r in Lemma 6.1.The functions r1,a and r 1,a have partially overlapping domains of definition, but are in general not equal on the intersection of these domains.Since the derivations of Lemma 5.1 and Lemma 6.1 use different phase functions (−i Φ(0, k) and −i Φ(τ, k), respectively), it seems difficult to construct r1,a in such a way that it coincides with r 1,a on this intersection.
Proof of Lemma 5.1.The proof follows the idea of [7] but is nonstandard because the real part of Φ21 (τ, k) is nonzero for k ∈ (−i∞, −i) whenever τ > 0. We therefore present a proof, whose main idea is to use −i Φ21 (0, k) instead of −i Φ21 (τ, k) as the phase function.
Let M ≥ N + 1 be a large integer.There exists a rational function f 0 (k) with no poles in V1 such that r 1 and f 0 coincide to order 4M at k = −i and such that f 0 , so we may define ) so by choosing M large enough, we can achieve that F belongs to the Sobolev space H N +1 (R).We conclude that and, by the Plancherel theorem, where ).On the other hand, f a (x, •) is clearly continuous on V1 and analytic in V 1 .Since Re Φ21 (τ, k) ≥ Re Φ21 (0, k) ≥ 0 for k ∈ V1 and τ ∈ I (cf. Figure 3), it also holds that . The contour Γ (1) (solid), the boundary of S (dashed), and the saddle points k 1 (blue) and ω 2 k 2 (green).
yield a decomposition of r 1 with the desired properties.
Lemma 5.1 establishes a decomposition of r 1 .The symmetry (4.5) then yields an analogous decomposition r 2 = r2,a + r2,r of r2 as follows: We are now in a position to define the first transformation n → n (1) .As explained in Section 4, we will focus our attention on the sector S. Let Γ (1) be as in Figure 5.Note that Γ (1) 2 := {e iθ | θ ∈ ( π 3 , π 2 )}.The matrices v 1 and v 4 admit the factorizations 4r , where (5.10) The function n (1) is defined by where the function G (1) is analytic in C \ Γ (1) .It is given for k ∈ S by 1a , k on the − side of Γ 1r , v (5.12) and extended to C \ Γ (1) by means of the symmetry (5.13) The next lemma follows from the signature tables of Figure 3 and Lemma 5.1.
The L 1 and L ∞ norms of v (1) − I on Γ 1r ∪ Γ 4r are uniformly of order O(x −N ) in Sector I as a consequence of Lemma 5.1.The estimate (5.3) of r1,a ensures that n (1) has the same behavior as n as k → ∞ up to terms of O(k −N ).

Û1
Figure 6.The open subsets U 1 and Û1 of the complex k-plane.
(b) For x ≥ 1 and τ ∈ I, the functions r 1,a and r1,a obey where and C > 0 is independent of x, τ, k.In fact, for k = k 1 , the following stronger estimates hold: where C N (τ ) ≥ 0 is a smooth function of τ which is independent of x, k and which vanishes to all orders at τ = 0.
Proof.On each arc of ∂U 1 ∩ ∂D and ∂ Û1 ∩ ∂D, the function θ → −i Φ21 (τ, e iθ ) = (1 − τ cos θ) sin θ is real-valued and monotone.The statement therefore follows using the method introduced in [7]; see also the proof of Lemma 5.1.Lemma 6.1 provides decompositions of r 1 and r1 ; using the symmetry (4.5), we now deduce decompositions of r 2 and r2 .Let U 2 := {k| k−1 ∈ U 1 } and Û2 := {k| k−1 ∈ Û1 }.We define decompositions r 2 = r 2,a + r 2,r and r2 = r2,a + r2,r by where we have omitted the (x, t)-dependence for conciseness.On Γ 2 , we factorize v 2 as follows: and On Γ (2) 5 , we factorize v 5 as follows: where we have used (4.4).Finally, on Γ 8 , we factorize v 8 as follows: (2) We do not write down the long expression for v 5,r , but note that Lemma 6.1 yields Let Γ (2) be the contour shown in Figure 7.The function n (2) is defined by where G (2) is analytic in C \ Γ (2) .It is defined for k ∈ S by and extended to C \ Γ (2) by means of the symmetries The next lemma follows from Lemma 6.1 and the signature tables of Figure 3.
7. The n (2) → n (3) transformation For the third transformation, we will deform contours up and down (see (7.3) below) using the following factorizations: 7 =v 9 =v (3) Let Γ (3) be the contour shown in Figure 8.We define n (3) by where G (3) is given for k ∈ S by and, similarly, Lemma 6.1 implies that 1a .(7.5) Hence, applying the triangle inequality and the identity Re Φ21 (τ, On the other hand, the function r2 (k) vanishes to all orders at k = i and there exists a c > 0 such that Re Φ21 (τ, k) ≤ −c|k − i| 2 uniformly for k ∈ Γ 1a and τ ∈ I. Consequently, using (7.6) and the estimate for r2,a (k) that follows from Lemma 6.1, we arrive at 1a and τ ∈ I, which gives the desired estimate of the (21)-entry of v 1a .We have (v (τ,k) , where ). Recalling (7.6) and estimating r 1,a ( 1 ω 2 k ) by means of Lemma 6.1, we get the bound , the above can be rewritten as On the other hand, considering the relation (4.4) with k replaced by 1 ω 2 k , we infer that the function h(k To estimate |h a (k) − h(k)|, we note that Lemma 6.1 yields uniformly for τ ∈ I.The identities 1a , and hence, using also (7.5) and the relation Φ21 (τ, ωk) = Φ32 (τ, k), for k ∈ Γ 1a and τ ∈ I. Combining the inequalities (7.7), (7.8), (7.9), and (7.10), it transpires that which proves the desired estimate for the (23)-entry of v (3) 1a −I; the (13)-entry can be handled similarly since r 1 ( 1 kω 2 ) + r 2 (kω 2 ) vanishes to all orders at k = i (this follows from (4.4) with k replaced by ωk and the assumption that r 1 (k) vanishes to all orders at k = i).
The next lemma is proved in the same way as [4,Lemma 6.3].
Lemma 7.4.It is possible to choose the analytic approximations of Lemma 6.1 so that the Using Lemmas 7.2, 7.3, and 7.4, we conclude that v (3)

Global parametrix
For each τ ∈ I, let δ(τ, •) : C \ Γ 5 → C be a function obeying the jump condition 5 , and the normalization condition δ(τ, k) = 1 + O(k −1 ) as k → ∞.As mentioned earlier (see the text below (6.1)), 1 + r 1 (k)r 2 (k) > 1 for all k ∈ Γ (3) 5 .Therefore, by taking the logarithm and using Plemelj's formula, we obtain where the path of integration follows the unit circle in the counterclockwise direction, and the principal branch is used for the logarithm.
Lemma 8.1.The function δ(τ, k) has the following properties: (a) δ(τ, k) admits the representation where the path of integration follows ∂D in the counterclockwise direction and ν ≤ 0 is defined in (2.9).
5 .Furthermore, sup where 2 ) and arg s ∈ (−π, π).In particular, for k ∈ ∂D \ Γ 5 , |δ(τ, k)| is constant and given by where where Proof.To prove part (a), it suffices to perform an integration by parts in (8.1).Part (b) is then a consequence of (8.2) and the fact that Im ν = 0.By (8.3) we have, for k ∈ ∂D \ Γ and (8.4) then follows from the assumption that r 1 = 0 on [0, i] and the definition (2.9) of ν.By (8.2), We obtain (8.5) after substituting (8.4) and arg into the above equation.This finishes the proof of part (c).We obtain part (d) using (8.3) and standard estimates and part (e) is immediate from (8.1).
The matrix v (4) − I is not uniformly small on X .The goal of the next section is to build a local parametrix that approximates n (4) in a neighborhood of k 1 .The local parametrices near {k 2 , ωk 1 , ωk 2 , ω 2 k 1 , ω 2 k 2 } will then be obtained using the A-and B-symmetries.

Local parametrix near k 1
As k → k 1 , we have .
In the statement of the next lemma, and in what follows, C N (τ ) denotes a generic nonnegative smooth function of τ that vanishes to all orders at τ = 0. Lemma 10.2.For each x ≥ 2 and τ ∈ I, m k 1 (x, t, k) defined in (10.4) uniformly for τ ∈ I, where m X 1 (q) is given by (A.3) with q = −r(k 1 ) − 1 2 r 2 (k 1 ).

Figure 1 .
Figure 1.The ten asymptotic sectors for (1.1) in the xt-plane.