LONG-TERM STABILITY FOR KDV SOLITONS IN WEIGHTED H s SPACES

. In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the I -method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The ﬁnite time restriction is due to a lack of global control of the unweighted perturbation.

(1) This is a well-known nonlinear dispersive partial differential equation, which models the behavior of water waves in a long, narrow, shallow canal. It is well known that the KdV equation (1) is completely integrable. This means that, among other things, the equation possesses infinitely many conserved quantities, the first two of which are the mass

BRIAN PIGOTT AND SARAH RAYNOR
The equation is also known to support traveling wave solutions-that is, solitons. Indeed, making the ansatz that the solution of (1) is of the form Q c,x0 (x, t) = ψ c (x − ct − x 0 ) for some profile ψ c and some speed c > 0 and horizontal shift x 0 ∈ R, we find that the soliton is given by The stability of these solitons has been an area of intense study for many years and is the main topic of this paper. One might first be interested in the orbital stability of the soliton. In the Sobolev space H s = H s (R), this means that for all > 0 there is a δ > 0 so that if u 0 − ψ c H s < δ, then there is a continuous function x 0 : [0, ∞) → R such that u(t) − ψ c (· − ct − x 0 (t)) H s < for all t ≥ 0. The study of orbital stability in the energy space H 1 began with Benjamin [1] and Bona [2]; see also [3]. This work was made systematic by Weinstein [22], who established the orbital stability of solitons for nonlinear Schrödinger equations and for generalized KdV equations. The orbital stability of solitons in H s with s < 1 is not as well developed. Merle and Vega [13] showed that the solitons are orbitally stable in H 0 = L 2 using the Miura transform, together with the stability theory for kink solutions of the mKdV equation in H 1 . One might expect that the orbital stability results in L 2 and H 1 imply orbital stability in H s with 0 < s < 1. However, the natural interpolation argument fails because H s functions need not be in H 1 . In the case of H s with 0 < s < 1, the I-method has been used to show that any possible orbital instability of the solitons can be at most polynomial in time; see [21] and [19].
A stronger notion of stability is asymptotic stability-one aims to show that there exist c + ∈ (0, ∞) and x + ∈ R so that u(t) − ψ c+ (· − c + t − x + ) X → 0 as t → +∞ (3) in some Banach space X. By perturbing the main soliton ψ c+ by a very small soliton located sufficiently far to the left of the main soliton, we see that this notion of asymptotic stability cannot hold in any translation-invariant space X. Therefore, in order to investigate asymptotic stability of solitons in the Sobolev spaces H s , the translation invariance of the space must be broken in some way. Within the current literature there appear to be three approaches to this problem: 1. Insert a spatial weight into the Sobolev space so that movement to the left registers as decay. 2. Replace strong convergence in (3) with weak convergence. 3. Truncate the Sobolev space in an appropriate time-dependent way.
The first results on asymptotic stability for KdV solitons were established by Pego and Weinstein in [18]. In that paper, the authors considered solutions of KdV in the exponentially weighted Sobolev space H 1 a = {f | e ax f H 1 < ∞}, for an appropriate choice of a. They were able to prove that solitons are asymptotically stable in H 1 a and that the rate of decay in (3) is exponential. Mizumachi [14] has since proved that the solitons are asymptotically stable in a weighted version of H 1 with polynomial type weight; the decay rate in (3) is shown to be polynomial.
Martel and Merle [10] have shown that KdV solitons are asymptotically stable in H 1 if we replace the strong convergence in (3) with weak convergence. Martel and Merle have gone on to show that for any β > 0, KdV solitons are asymptotically stable in the truncated Sobolev space H 1 (x > βt); see [11,12]. Merle and Vega [13] used this approach together with asymptotic stability of kink solutions to the modified KdV equation to show that KdV solitons are asymptotically stable in L 2 loc . Buckmaster and Koch [4] have shown that for any β > 0 KdV solitons are asymptotically stable in H k (x > βt) for k ∈ {−1, 0, 1, . . .}. More recently, Mizumachi and Tzvetkov [15] have used the approach of Pego and Weinstein together with the Miura transform to show that solitons for KdV are asymptotically stable in L 2 (x > βt), thus offering an alternative proof of the result of Merle and Vega.
The principal goal of this paper is to investigate the asymptotic stability of KdV solitons in H s with 0 < s < 1. As in the case of orbital stability, the standard interpolation argument does not enable us to conclude that solitons are asymptotically stable in H s for fractional values of s < 1. Instead we turn to the I-method and implement it in the setting of the exponentially weighted spaces used by Pego and Weinstein. To that end we define I : denotes the Fourier transform of f and the multiplier m is given by with smooth, even patching on the intervening intervals, and with N a parameter that we will choose during the course of our analysis.

Remark 1.
Due to the absence of good commutator estimates between the exponential weight and the I-operator, we have chosen to work in the space with norm e ay I 1 f H 1 .

Remark 2.
Note that this result is not a full asymptotic-stability result, because the length of time for which we have control is finite and dependent on the size of the initial error in the unweighted space. However, the control on the weighted perturbation is a uniform exponential decay whose rate does not depend on the finite time chosen. Therefore the finite time restriction is believed to be a technical issue.
The key difficulty in the proof of Theorem 1.1 is to accommodate both the exponential weight which occurs as a weight on the spatial variable and the I-operator which occurs as a weight on the frequency variable. We proceed by first establishing local well-posedness for the exponentially weighted soliton perturbation in a space X s,1/2,1 which embeds into the Bourgain space X s,1/2+ , partially following the local well-posedness work of Molinet and Ribaud [16,17], and Guo and Wang [8] on dispersive-dissipative equations. In so doing we establish multilinear estimates that accommodate the presence of the exponential weight. For technical reasons, this requires that s > 7/8. We then use the I-method to map our solutions into an exponentially-weighted version of H 1 . Finally, we run an iteration scheme inspired by an analogous argument in [20] to establish long-term control of the perturbation in H s and the exponentially weighted space H s a , concluding that the soliton is exponentially stable for long times in H s a for s > 7/8. The paper is organized as follows: In section 2, we will set up our notation and establish basic results. In section 3, we will establish some necessary estimates to establish local well-posedness in section 4. In section 5, we will run the iteration scheme and establish the main result of the paper.
2. Notation and basic results. We will define the Fourier multiplier operator I N by I N f (ξ) = m N (ξ)f (ξ), with m N a smooth, even, decreasing function of |ξ| which satisfies m N (ξ) = 1 for |ξ| < N and m N (ξ) = |ξ| s−1 N s−1 for |ξ| > 10N . In this paper, N will be a function of our time-step n, and, in particular where v 0 H s is taken to be sufficiently small. We make the ansatz that u( Here c(t) and γ(t) are parameters that will be chosen later. We also define w(y, t) := e ay v(y, t) where a ∈ R will be described later on. The perturbation v satisfies the difference equation Moreover, the perturbation w satisfies the difference equation The derivation of these equations is given in [18]. We definẽ v n (t) = I N (n) v(y, t) andw n (t) = e ay I N (n) v(y, t), where y = x − t 0 c(s)ds − γ(t), and c(t), γ(t) are chosen so that, at each time t, for appropriate value of n, w n (t) L 2 is minimized. In order to do so, we first need to consider the difference equations satisfied byṽ andw, and consider their linearizations about the soliton.
Lemma 2.1. The perturbationṽ satisfies the difference equation Moreover, the perturbationw n (t) satisfies the difference equation Proof. The result here comes from applying I to (4) and (5).
For fixed c > 0, define the operator A a = e ay ∂ y (−∂ 2 y + c − 2ψ c )e −ay . We have the following from [18,20]: , the spectrum of A a in H 1 consists of the following: 1. An eigenvalue of algebraic multiplicity 2 at λ = 0. A generator of the kernel of A a is ζ 1 = e ay ∂ y ψ c , and the second generator of the generalized kernel of . For any element λ of this continuous spectrum, the real part of λ is at most The spectrum contains no other elements.
We also need to consider the elements of the spectrum to A * a , which are dt and θ 1 , θ 2 and θ 3 are appropriate constants to obtain the biorthogonality relationship ζ j , η k = δ jk . We will define the L 2 spectral projections P w = 2 i=1 w, η i ζ i and Qw = w − P w onto the discrete and continuous spectrums of A a respectively, with respect to the fixed initial value of c, c 0 .
Returning to the difference equation (7), for each fixed t we selectċ n (t) andγ n (t) so that Pw n = 0, and Qw n =w n . Defining we have that where A is the matrix 3. Linear and multilinear estimates. In this section we will review the construction of the space X s,1/2,1 and mention the linear estimates which were developed in [20]. At the end of this section we prove a new bilinear estimate which is then used to establish a multilinear estimate that is necessary for the proof of Theorem 1.1. First, we provide a version of the product rule that holds with the multiplier operator I in place of a derivative: Proof. Define ω R (y) = χ {y≤R} e ay , and consider ω R I N ∂ y (f 1 f 2 ) L 2 . Taking the Fourier transform and using duality, we find that this equals In the first case, note that m(ξ 2 + ξ 3 )(ξ 2 + ξ 3 ) ≤ 2m(ξ 2 )ξ 2 by the properties of m, so we have, with ξ 5 = ξ 2 + ξ 3 and ξ 6 = ξ 1 + ξ 5 , By the symmetry between the two cases, we obtain in total that Now, letting R → ∞, since χ {y<R} |e ay I N ∂ y (f 1 f 2 )(y)| 2 is a pointwise-increasing function in R, by the Lebesgue monotone convergence theorem we see that We next recall the definition of the space X s,1/2,1 . We define the sets A j and B k by For s, b ∈ R, the space X s,b,1 is defined to be the completion of the Schwartz class functions in the norm In taking b = 1/2 we have the following embeddings: We will work primarily in the spaces X s,1/2,1 and X s,−1/2,1 , so we adopt the notation X s := X s,1/2,1 and Y s := X s,−1/2,1 .
The spaces X s , Y s were used in the case when s = 1 to prove local well-posedness for the perturbations v and w = e ay v in H 1 (R), see [20]. We review some of the features of these spaces that were used in the aforementioned local well-posedness arguments. Let W 1 (t) denote the standard Airy evolution, Let W 2 (t) be the linear evolution defined for t ≥ 0 by where p a (ξ) = 3aξ 2 + a(c 2 0 − a). We extend this to all of t ∈ R in defining While the Airy evolution W 1 (t) is the linear evolution associated with the unweighted perturbation v, the evolution W 2 (t) is the linear evolution associated with the weighted perturbation w. A key feature of the space X s is that it accommodates both of the semigroups W 1 (t) and W 2 (t), as illustrated in the following linear estimates which are valid for all s ∈ R: and if 0 < a ≤ min(1, c 0 ), then Here ρ : R → R is a cutoff function such that and χ R+ is the indicator function for the set R + := {t ∈ R | t ≥ 0}. The estimates (9), (10) are proved in [9] while the proofs of (11), (12) are given in [20]. Also crucial for the result proved in [20] was the following bilinear estimate, valid for all s ≥ 0 (see Proposition 3 in [20]): We require the following fundamental lemmas, the proofs of which are given in [9].
If Ω ⊆ R 2 satisfies In the case when s = 1 we have the following generalization of the estimate (14).
Proof. Since we work primarily in frequency space, we define X s,b,1 to be the completion of the Schwartz class functions in the norm Here f = f (τ, ξ) is a function of the frequency variables τ and ξ. Adopting the notation X 1 = X 1,1/2,1 and Y 1 = X 1,−1/2,1 , the estimate (19) reads Following the proof of the standard bilinear estimate (14) we decompose f and g on dyadic blocks as follows: Define f j1,k1 := χ Aj 1 χ B k 1 f and g j2,k2 := χ Aj 2 χ B k 2 g.
The proof is divided into the following cases: 1. At least two of j, j 1 , j 2 are less than 20.

It follows that
Finally we consider the case when k 2 = k max . Since the expression to be estimated is symmetric in (j 1 , k 1 ) and (j 2 , k 2 ), we can argue as in the case where k 1 = k max to obtain the desired estimate.
We thus obtain Next we estimate We thus find that (20) In the proof of the modified local well-posedness result we will require the following estimate.
Proposition 3. Let s > 7/8. Suppose that u and v are spacetime functions such that u, v ∈ X s and e ay Iu, e ay Iv ∈ X 1 . Then Remark 3. Since s > 7/8 we see that (22) implies Proof of Proposition 3. For a function u(t, x) of spacetime we let u Nj denote the function whose Fourier transform is given by u Nj = η Aj (ξ) u(ξ), where η Aj is a smooth cutoff function adapted to the set A j := {ξ ∈ R | |ξ| ∼ N j } with N j dyadic. We truncate the exponential weight using a spatial cutoff function. Specifically, for R > 1 we let ϑ R : R → R by ϑ R (y) = 1, y < R 0, y > R, and define ω a,R (y) := ϑ R (y)e ay . Observe that ω a,R ∈ H s (R) for all s ∈ R; in particular, it makes sense to speak of the Fourier transform of ω a,R . Furthermore, we have the following approximation result.
Proof. Arguing as in the proof of Lemma 3.1, we find that Observe that e ay f 2 H 1 = e ay f 2 L 2 + e ay (af + f y ) 2 L 2 . One also checks that ω a,R f 2 In light of this calculation and (23), we obtain the conclusion of the lemma.
To prove (22) it suffices to show that where g := ω a,R . Note that by symmetry we may assume that N 2 ≥ N 3 . We adopt the notation N 12 for |ξ 1 + ξ 2 | ∼ N 12 when |ξ 1 | ∼ N 1 and |ξ 2 | ∼ N 2 . We adopt similar definitions for N 13 and N 23 .
Case (1). N 2 N . In this case we see that m(ξ 2 + ξ 3 ) − m(ξ 2 )m(ξ 3 ) = 0, so the expression to be estimated vanishes. Case (2). N 2 N N 3 . We use the mean value theorem to see that It follows that and N 3 N 2 N 13 1/4− N 2 1/4− N 3/4+ 3 Here we split the expression to be estimated into two terms which are then estimated separately: We estimate Term II as in Case (2) to see that which is sufficient. Turning to Term I, we have where in the final inequality we have used that f X s If X 1 . Observe that since N 2 ≥ N 3 and s > 3/4 we have To estimate the other multiplier expression we first note that if N 23 which is acceptable. If N 23 N 3 , then we must have N 2 ∼ N 3 (with the relevant factors being supported at frequencies of opposite sign), in which case may estimate N 3 m(N 2 ) N s−3/4− . The estimate is then completed as above in (25).
From Proposition 3 we have the following result. e ay Iv, e ay ∂ y I(uv) − IuIv H 1 dt N 3/4−s+ e ay Iv X 1 ( e ay Iu X 1 Iv X 1 + Iu X 1 e ay Iv X 1 ) .

4.
Modified local well-posedness. This section is devoted to the proof of local well-posedness for the v-equation and the w-equation. We make the change of variables y → y +γ(t)+ t 0 c(s)ds and find that the initial value problem for v = I N v is given by The equation for w = e ay I N v is given by the modulation equation Upon expanding the operator A a , we find that the initial value problem for w is Before we proceed with our local well-posedness argument, we define the timelocalized space X s δ to be the space with the norm u X s δ := inf{ w X s | w ≡ u on [0, δ]}. The main goal of this section is to prove the following modified local wellposedness result: Proposition 4. Let 0 < a < c 0 /3, s > 7/8, and N > 1. There is an r > 0 such that the following statement holds: If v 0 ∈ H s (R) satisfies v 0 H 1 < r and w H 1 < r where v 0 = I N v 0 and w 0 = e ay I N v, then there is a δ > 0 so that the initial value problems (26) and (27) admit solutions v(t, y), w(t, y), respectively, on [0, δ]. Moreover these solutions satisfy v X 1 Proof. Let ρ : R → R be a smooth cutoff function, as in (13), and let ρ δ (·) = ρ(·/δ). We begin by rewriting the equation for v(t, y), (26), using Duhamel's formula: We will show that the map Φ given by is a contraction on a small ball in X 1 δ . We estimate Φ v in X 1 δ using (9) and (10): : v 0 H 1 + Term I + Term II + Term III + Term IV.
To estimate Term I we first note that In light of Lemma 12.1 from [5] we may conclude that To estimate Term II we use the bilinear estimate (14) to see that Recall that for δ, > 0 sufficiently small we have ψ c X 1 δ δ .

Thus
Term II δ v X 1 δ . Turning to Term III we argue as for Terms I and II to find that Finally, for Term IV we recall that from the modulation equations we have For the w equation we expand the spectral projection Qf = f − 2 j=1 f, η j ζ j and make the change of variables y → y − ((3a 2 − c 0 )t + γ(t) − t 0 c(s)ds), so that the equation for w reads ∂ t w + ∂ 3 y w − 3a∂ 2 y w + a(c 0 − a 2 − c + c 0 ) w − aγ w − e ay I N ∂ y (v 2 ) − e ay (γ∂ y +ċ∂ c )I N ψ c − e ay ∂ y (I N (ψ c v) − ψ c I N v) Rewriting this equation using Duhamel's formula leads us to define the following operator which we hope to show is a contraction on a ball in X 1 δ . We estimate Ψ w in X 1 δ using (11) and (12), which yields To estimate Term I we use e ay ∂ y e −ay = ∂ y −a, v = e −ay w, and the bilinear estimate (14) to see that In estimating Term II we use that γ L ∞ t w X 1 δ , which gives Term II w 2 In order to estimate Term III we note that x .
Since we are restricted to the interval [0, δ], Hölder's inequality gives To estimate Term IV we use (22) and (14) to see that The estimate for Term V is similar to the one we used for the analogous term in the v equation (term (IV )), yielding Term V δ w X 1 δ . Term VI is estimated using (22), (14), and the fact that I N ψ c − ψ c X 1 δ N −C with C as large as need be: Turning to Terms VII and VIII we recall from Lemma 3.5 in [20] that It follows that 1 and let B = v, w ∈ X 1 δ | v X 1 δ ≤ 2cr, w X 1 δ ≤ 2cr . Using the estimates that we have established, it transpires that Φ, Ψ : B → B are contractions following the arguments from Proposition 4 of [20]. The desired result follows.

5.
Iteration. In this section, we prove the main result of the paper, namely the exponential decay of the weighted perturbation given in Theorem 1.1. We will prove the result by induction. Defineċ n andγ n by (8), and let the variable y be defined accordingly as y = x − t 0 c(s)ds − γ(t). Let T > 0 be given. Let . Now, let 1 and c 2 be sufficiently small so that, whenever e ay I N (n) w(t n ) H 1 < 2 1 and I N (n) v(t n ) H 1 < c 2 , it follows that v(t) exists on [t 0 , t 0 + δ], and where C 0 is the implicit constant in the conclusion of Proposition 4. Additionally, assume that c 2 < b 10 . Let n 0 = T δ . Finally, choose 2 sufficiently small that Cr n 0 2 2 < c 2 , with r to be expressed later. We must recall the known control on v. In [19] it is proven that, with H(f ) = |∂ x f | 2 − 2 3 f 3 , ṽ n (n) 2 H 1 ∼ H(ψ +ṽ n (n)) − ψ +ṽ n (n) L 2 ψ L 2 10 3