ASYMPTOTICS FOR THE MODIFIED WITHAM EQUATION

. We consider the modiﬁed Witham equation where (cid:112) a 2 − ∂ 2 x means the dispersion relation which correspond to nonlinear Kelvin and continental-shelf waves. We develop the factorization technique to study the large time asymptotics of solutions.

To exclude the translation in the equation we make a change x → x − at. Also we rewrite the equation in the potential form introducing a new dependent variable u = ∂ −1 x v = x −∞ vdx , hence we get the Cauchy problem Our method is related to a series of papers [2,3,4,5], [6]. We survey the results and ideas of these papers.
In [3,4,5], we introduced the factorization formula for the case of the fourthorder Schrödinger equations of the derivative types under the zero mass condition, 1408 NAKAO HAYASHI, PAVEL I. NAUMKIN AND ISAHI SÁNCHEZ-SUÁREZ and the defocusing case under the non zero mass condition. The free evolution group of the fourth order Schrödinger equation is defined by U (t) = e − it 4 ∂ 4 x = F −1 EF, where the multiplication factor E (t, ξ) = e − it 4 ξ 4 . Then we write . If we define the new dependent variable ϕ = FU (−t) u (t), then we obtain the factorization formula We also have the representation for the inverse evolution group FU (−t) where D −1 t φ = |t| √ 2π e itS(x,ξ) φ (x) x 2 dx.
Next applying the operator FU (−t) to equation (4) and using FU (−t) L = i∂ t FU (−t) , L = i∂ t − 1 4 ∂ 4 x and u (t) = D t BM V ϕ, we get the ordinary differential equation for (4). In the case of (2) we have when the data satisfy zero mass condition. For the non zero mass condition we have for (3), where Vφ = |t| 1 4 √ 2π e itS(x,ξ) φ (ξ) dξ, V * φ = 3 |t| 3 4 √ 2π e −itS(x,ξ) φ (x) x 2 dx. By (7) and the non zero mass condition, in [3], we showed that small solutions for (4) decay in time faster than those of the corresponding linear problem. By (9) and the non zero mass condition, in [5], we showed that the small solutions of (3) are stable in the neighborhood of the self-similar solution of (3). In the case of the zero mass condition it was shown in [4], that the asymptotics of solutions requires a suitable phase correction similar to the case of the usual cubic nonlinear Schrödinger equation. In [2], the third-order nonlinear Schrödinger equation in the critical case was studied by the similar method to [3].
In [6], we studied the large time asymptotics of small solutions to the Cauchy problem for the modified Korteweg-de Vries equation with a fifth order dispersive term where a, b > 0 or a, b < 0. We define the free evolution group of (11) as U (t) = e −itΛ(−i∂x) = F −1 EF, where E (t, ξ) = e −itΛ(ξ) and Λ (ξ) = a 3 ξ 3 + b 5 ξ 5 . We have where D t φ = (it) √ 4bx + a 2 − a, such that Λ (η (x)) = x for x > 0. This fact means that the main term of the large time asymptotics of the solutions lies in the positive half-line. We extend η (x) for all x ∈ R by Define the operator (Bφ) (x) = |Λ (η (x))| − 1 2 φ (η (x)) . Since x = η |η| Λ (η) , we get where the multiplication factor M (t, η) = e it(ηΛ (η)−Λ(η))θ(η) , the phase function If u is a real-valued function, we have where the operator Also we have the representation for the inverse evolution group FU (−t) for all ξ ≥ 0 where D −1 t φ = (it) for ξ ≥ 0, is considered as the inverse operator to V. The identities (12) and (13) play the same roles as those given in (5) and (6), respectively.
To obtain of the uniform estimate of ξ ϕ, we need the estimate of J ∂ x u L 2 ,where x . However, J does not work well on the nonlinear terms. Then, instead of the operators J , we apply the modified dilation operator defined by Note that P acts well on the nonlinear terms as the first order differential operator. Also J and P are related via the identity In order to get the estimate of J ∂ x u, we showed the a-priori estimates of Pu, tLu and Iu in [6]. In this point, our method here is more close to that of paper [6] comparing with the previous works [2,3,4,5] concerning the case of a homogeneous symbol.
To state our results precisely we introduce Notation and Function Spaces. We denote the Lebesgue space by We also use the notations H k,s = H k,s 2 , H k = H k,0 shortly, if it does not cause any confusion. Let C(I; B) be the space of continuous functions from an interval I to a Banach space B. Different positive constants might be denoted by the same letter C. Define the stationary points as the roots of equation Λ (ξ) = x, where Λ (ξ) = ξ ( ξ a − a) . These are given by ±µ (x) with The Heaviside function θ (x) is defined by θ (x) = 1 for x > 0 and θ (x) = 0 for x ≤ 0. We now state our main result.
Moreover there exist unique modified final states W j+ ∈ L ∞ such that the asymptotics is valid for t → ∞ uniformly with respect to x ∈ R, where δ > 0 is a small constant.
2. Factorization technique. Define the free evolution group U (t) = F −1 EF, where the multiplication factor E (t, ξ) = e −itΛ(ξ) with the symbol Λ (ξ) = ξ ( ξ a − a) , where ξ a = a 2 + ξ 2 . Then u = U (t) u 0 solves the linear Cauchy problem Lu = 0, u (0) = u 0 . We write there are two stationary points ξ = ±µ (x) which are obtained as the roots of the equation for all x ≥ 0, and they are given in (15). Also for all ξ > 0. We extend the function for all x < 0. We define the cut off function χ (ξ) ∈ C 2 (R) such that χ (ξ) = 0 for ξ ≤ − 1 3 , χ (ξ) = 1 for ξ ≥ 1 3 , and such that χ (ξ) + χ (−ξ) ≡ 1. Thus we have µ |µ| Λ (µ (x)) = x for all x ∈ R and defining the scaling operator (Bφ) (x) = φ (µ (x)) , we write where the multiplication factor the phase function and the operator Also we need the representation for the inverse evolution group FU (−t) for all and the operator applying the operator FU (−t) to equation (1) we get Then since u x = 2ReD t BM ψ 1 , ψ 1 = Viξ ϕ, we get We have the identity for the phase function Then by the definition of the operator V * (t) we obtain Note that Ω 3 = Ω −3 and Ω 1 = Ω −1 = 0. Therefore we find This is our target equation for the study of the large time asymptotic behavior and similar to (14) in the previous work [6]. As it was mentioned above, an important tool for obtaining the time decay estimates of solutions to the nonlinear dispersive equations is implementation of the operator We have the commutator [L, J ] = LJ − J L = 0, with L = ∂ t + iΛ (−i∂ x ) . To avoid the derivative loss we define the operators where ∂ a Λ (ξ) = ξ ξ We organize the rest of our paper as follows. In Section 3, we state main estimates for the decomposition operators V (t) and V * (t) related to the evolution group U (t). We prove a-priori estimates of solutions in Section 4. Section 5 is devoted to the proof of Theorem 1.1.
Then by virtue of formula (18) with g (y) = S (ξ, ξy) , f (y) = Θ (y) |Λ (ξy)| , y 0 = 1, we obtain In particular we have the estimate |Λ (ξ)| Asymptotics for the operator V. In the next lemma we estimate the operator Lemma 3.1. Let j = 0, 1, 2, α ≥ 0. Then the following estimates are valid for all t ≥ 1 Proof. We write for η > 0. For the first summand I 1 we integrate by parts via the identity with Thus we obtain To estimate the second integral I 2 we integrate by parts via the identity with in the domains ξ ≤ 2 3 η and ξ ≥ 3 2 η, we obtain Hence We obtain for 0 < η < 1 changing ξ = ηy In the same manner for 0 < η < 1 changing ξ = ηy In the domain η < 0 we integrate by parts using the identity (20) in the domain ξ > − 1 3 |η| , η < 0. Then as above we obtain 3.3. Asymptotics for the operator V * . We next consider the operator , then by the Riesz interpolation theorem (see [8], p. 52) we have for 2 ≤ p ≤ ∞. In the next lemma we find the asymptotics of V * .

Lemma 3.2. The following estimate is valid
for the case of ξ > 0. In the integral I 3 we use the identity with , we integrate by parts Then applying the estimates Λ .
Then we find In particular, taking , ψ 1 (ξ) = 1, we obtain Next we obtain a more general result.
Then the estimate Integrating two times by parts via the identity Then by the Young inequality we obtain In the next lemma, we estimate the derivative ∂ η V.
In the next lemma we estimate the derivative ∂ t Vφ.

Thus we find
Next by Lemma 3.1 with j = 1, α = 0, β = 0 we have In the same manner we find for the second summand Note that D −1 A * (−t) = 0 and D −3 A * (−3t) = 0 for ξ > 0. Hence as above we get for the third and fourth terms for all t ≥ 1, ξ > 0. Lemma 4.1 is proved.
Next we prove the a priori estimate of ξ 2 ϕ in the L ∞ -norm.
is true for all T > 1.

5.
A priori estimates in L 2 .
In the next lemma we estimate the derivative ∂ t V * φ.
is true for all t ≥ 1. .
Next we consider the commutator By (24) and Lemma 5.2 with p = 2 we get the estimate In the next lemma we estimate the commutator [ ξ , V * ] φ.

A priori estimates.
To get the desired results, we prove the a priori estimates of solutions uniformly in time. Define the following norm where γ > 0 is small.
holds. Then there exists an ε such that the estimate is true for all T > 1.