KDV TYPE ASYMPTOTICS FOR SOLUTIONS TO HIGHER-ORDER NONLINEAR SCHR¨ODINGER EQUATIONS

. We consider the Cauchy problem for the higher-order nonlinear Schr¨odinger equation

Equation (1.1) arises in the context of high-speed soliton transmission in longhaul optical communication system [13].Also it can be considered as a particular form of the higher order nonlinear Schrödinger equation introduced by [42] to describe the nonlinear propagation of pulses through optical fibers.This equation also represents the propagation of pulses by taking higher dispersion effects into account than those given by the Schrödinger equation (see [10,11,25,28,38,32,43,49]).
In this article we are interested in the case of non zero total mass of the initial data R u 0 (x)dx = 0.
Then by (1.1) we obtain the non-zero total mass for the solution R u(t, x)dx = R u 0 (x)dx = 0 for all t > 0. We develop the factorization technique originated in our previous papers [22,21,26,20,44,45,46].The case of zero total mass R u(t, x)dx = R u 0 (x)dx = 0 is easier and the corresponding results can be obtained following the approach in [24].
We denote the Lebesgue space by L p = {φ ∈ S : φ L p < ∞}, with norms x .We also use the notations H m,s = H m,s

2
, H m = H m,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.
In [29], it was proved the existence of the self-similar solutions of the form u = t −1/3 f m (xt −1/3 ) to the reduced equation which is defined by the mean value m = R f m (x)dx = 0. Now we state the main result of this article.We show that the asymptotic behavior of the solutions to (1.1) resembles that of the KdV equation, which was studied intensively (see [15,18,23]).We denote by µ(x) the root of the equation Λ (ξ) = ξ|ξ|(a + b|ξ|) = x for all x ∈ R. Theorem 1.1.Suppose that R u 0 (x)dx = 0, and that the initial data u 0 ∈ H 1,1 have a sufficiently small norm u 0 H 1,1 ≤ ε.Then there exists a unique global solution u ∈ C([0, ∞); H 1,1 ) to the Cauchy problem (1.1).Furthermore there exists W + ∈ L ∞ such that the large time behavior satisfies the uniform norm and the L 2 -estimates of commutators.In Section 5, we prove a priori estimates of the solution u(t) in the norm where ϕ = FU(−t)u(t), U(t) = F −1 e −itΛ(ξ) F, Λ(ξ) = a 3 |ξ| 3 + b 4 ξ 4 , J = x − tΛ (−i∂ x ), I b = ∂ b + it∂ b Λ(−i∂ x ), P b = 3t∂ t + ∂ x x + b∂ b .Section 6 is devoted to the proof of Theorem 1.1.
, we obtain the commutator it[Λ (η), V]φ = −V∂ ξ φ.Also we use the representation for the inverse evolution group FU(−t)φ = V * M B −1 D −1 t , where the inverse dilation operator D −1 t φ = |t| 1/2 φ(xt) and the inverse scaling operator (B −1 φ)(η) = φ(Λ (η)), and the conjugate operator We have since the nonlinearity is gauge invariant.Also we mention that the operator We also use the operators 2.2.Boundedness of pseudodifferential operators.Define the pseudodifferential operator There are many papers devoted to the L 2 -estimates of pseudodifferential operator a(x, D) (see [4,7,9,27]).Below we will use the following results on the L 2boundedness of pseudodifferential operator a(x, D) (see [27]).
Similarly, by considering the conjugate operator.

3.1.
Estimates for commutators.We first obtain the L 2 -estimate for the operator Proof.Integrating by parts we obtain ).
In the next lemma we estimate the commutator [h, V 2 ].
Lemma 3.2.Let the weights Proof.Integrating by parts we obtain where ).Next we change η = µ(x) and the variable of integration ξ = t −1/3 ξ , then we obtain We define the pseudodifferential operators a k (t, x, D)φ ≡ R e ixξ a k (t, x, ξ) φ(ξ)dξ with symbols a k (t, x, ξ) = q k (t, µ(xt −2/3 ), ξt −1/3 ).Then we obtain We will prove the L 2 -boundedness of a k (t, x, D).Note that χ 2 Thus the pseudodifferential operators a k (t, x, D) are L 2 -bounded for k = 1, 2. Then as above we have Now let us prove the L 2 -L ∞ boundedness of the pseudodifferential operator a 3 (t, x, D).We have which yields the result of the lemma.
In the next lemma we estimate the operator V 3 .
Proof.Integrating by parts we obtain η) ).Next we change η = µ(x), and the variable of integration ξ = t −1/3 ξ , then we obtain Define the pseudodifferential operators Then we obtain Let us prove the L 2 -boundedness of the operators a k (t, x, D), k = 1, 2. We have Thus the pseudodifferential operators a k (t, x, D) are L 2 -bounded for k = 1, 2. Then as above we have Now let us prove the L 2 -L ∞ boundedness of the pseudodifferential operator a 3 (t, x, D).We have Thus as above we obtain The proof is complete.
Next we obtain the L 2 -estimate for the operator V 4 .
Lemma 3.4.Let the weight P be such that Assume that the weight Q satisfies the estimate The proof is complete.
In the next lemma we estimate the operator V 5 .
Lemma 3.5.Let the weights P and Proof.Integrating by parts we obtain ).Next we change η = µ(x), and the variable of integration ξ = t −1/3 ξ , then we obtain Define the pseudodifferential operators ).Then we obtain Let us prove the L 2 -boundedness of the operators a k (t, x, D), k = 1, 2. We have Then as above we have Now let us prove the L 2 -L ∞ boundedness of the pseudodifferential operator a 3 (t, x, D).We have for all x, ξ ∈ R, t ≥ 1, j = 0, 1, 2 with some small ν ∈ (0, 1), δ > 1/2.Therefore by Lemma 2.3 we find The proof is complete.
Applying the above lemmas we obtain estimates for the derivatives, We choose in Lemma 3.1 and 3.4

Asymptotic behavior. Define the kernel
With the changing of variables ξ = ηy, we obtain To study the asymptotic behavior of the kernel A j (t, η) for large t we apply the stationary phase method (see [14, p. 110]) for z → +∞, where the stationary point y 0 is defined by the equation g (y 0 ) = 0.By (3.1) with g(y) = −G(y, η), f (y) = χ 2 (y)y j and y 0 = 1, we obtain By the Sobolev embedding theorem and Lemma 3.
for j ≥ 0, Similarly by the Sobolev embedding theorem and Lemma 3.3 we have Then by Corollary 3.7, for j = 0, 1. Similarly by the Sobolev embedding theorem and Lemma 3.5 we have Then by Corollary 3.8, for j = 0, 1, and for j = 0, 1.In the next lemma we estimate the operators V 2 in the uniform norm.

Proof. Integrating by parts and using
where Next we define the pseudodifferential operators a * k (ξ, D)φ = C R e −ixξ a k (x, ξ) φ(x)dx, with symbols a k (x, ξ) = q k (µ(x), ξ), then changing the variable of integration η = µ(x), we obtain We prove the L 2 -boundedness of the pseudodifferential operators a * k (ξ, D) by considering the adjoint operators Note that the symbols Then by Lemma 2.3 we have the estimate a k,2 (x, D)φ L 2 x ≤ C φ L 2 for all t ≥ 1.The symbol a k,3 (x, ξ) satisfies the estimate sup x,ξ∈R |∂ j x ∂ l ξ a k,3 (x, ξ)| ≤ C for j, l = 0, 1, then by Lemma 2.1 we have a k,3 (x, D)φ . By the Cauchy-Schwarz inequality we find that Then we obtain Therefore the pseudodifferential operator The proof is complete.
In the next lemma we estimate the operator V * Kj , where the symbols holds for all t ≥ 1 and j = 3, 4.

Note that |{ξ}
Then by Lemma 2.3 we have the estimate a j,1 (x, D)φ L 2 x ≤ C φ L 2 for all t ≥ 1.The symbol a j,2 (x, ξ) satisfies the estimate sup x,ξ∈R |∂ j x ∂ l ξ a j,2 (x, ξ)| ≤ C for j, l = 0, 1, then by Lemma 2.1 we have a j,2 (x, D)φ L 2 x ≤ C φ L 2 .By the Cauchy-Schwarz inequality we find that | R e ixξ a j,3 (x, ξ) φ(ξ Then we obtain Therefore the pseudodifferential operator a j (x, D) = 3 l=1 a j,l (x, D) is L 2 -bounded.Hence the estimate follows The proof is complete.
We next prove a priori estimate in the L ∞ -norm for ϕ.
Proof.On the contrary we assume that there exists a first time T > 0 such that sup t∈[1,T ] ϕ L ∞ = Cε.We use (2.1) for a new dependent variable ϕ = FU(−t)u(t).
In view of Lemma 5.1, we obtain For the case of |ξ| < t −1/3 , we integrate in time directly For the case of |ξ| ≥ t −1/3 multiplying equation (5.1) by ϕ and taking the real part of the result we obtain Therefore ϕ L ∞ < Cε.The proof is complete.
5.2.L 2 -norms of P a and P b .There are many papers devoted to Kato-Ponce type estimates of the commutators of the form [ i∂ x α , u]v (see, e.g.[17,7,31,35]).Below we will use the following estimates (see [41]).
Next we consider a-priori estimates of solutions in the norm holds for all T > 1, where γ > 0 is small.
Proof.We apply operators P a = 4t∂ t + ∂ x x − a∂ a and P b = 3t∂ t + ∂ x x + b∂ b to equation (1.1).Using the commutators [L, P a ] = 4L and [L, P b ] = 3L, we obtain and In the same manner we obtain for j ≥ 0. Also via Lemma 3.9 we have Integration in time of the above inequality yields The proof is complete.5.3.L 2 -norms of I a and I b .In this subsection we prove L 2 -estimates for I a u(t) and ∂ −1 x I b u(t).We define the norm Lemma 5.6.Let u X T ≤ Cε.Then the estimate holds for any T > 1.
Proof.Arguing by the contradiction we assume that there exists a first time We apply I a and ∂ where Substituting u xxx = v 3 + w 3 , we obtain By Lemma 5.3 we find , by Lemma 3.3, 3.5 we have by Lemmas 3.1 and 3.4 we have and by Lemma 3.2 we find

Next we represent
Next we represent and denoting ρ |1|1 = D t BM V 2 |ξ|(iξ) ϕ, and similarly ω |1|1 , w |1|1 we obtain Thus we obtain In the same manner we obtain and Next we obtain and writing w |1|3 = D t BM V 3 |ξ|(iξ) Next using the factorization formula (2.1), we represent since by Corollary 3.6 and by Lemma 4.2 Thus we obtain In the same manner we transform uu Now we need to transform the terms 2tu 2 w |1|3 and 2tu 2 w 4 .We have Then by a direct calculation we obtain Next we use the relations Then by Lemma 5.4, , and by Lemmas 3.1, 3.2 and 3.4, for l = 1, 2, 4, j = 1, 2, we have ).
and the commutator relations [L, I a ] = [L, I b ] = 0 and [L, P a ] = 4L, [L, P b ] = 3L hold.Using the relation u