Modified scattering for the nonlinear nonlocal Schrödinger equation in one-dimensional case

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity i∂tu-∂x2u+∂x2u-a∂x4u=λu2u,t>0,x∈R,u0,x=u0x,x∈R,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}\left( u-\partial _{x}^{2}u\right) +\partial _{x}^{2}u-a\partial _{x}^{4}u=\lambda \left| u\right| ^{2}u,\text { } t>0,{\ }x\in {\mathbb {R}}\mathbf {,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,{\ }x\in {\mathbb {R}}\mathbf {,} \end{array} \right. \end{aligned}$$\end{document}where a>15,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>\frac{1}{5},$$\end{document}λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in {\mathbb {R}}$$\end{document}. We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {L}}^{2}$$\end{document}-boundedness of the pseudodifferential operators.


Introduction
We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with a critical nonlinearity in one-dimensional case where a > 1 5 , λ ∈ R. Equation (1.1 ) can be considered as a particular form of the higher-order nonlinear Schrödinger equation introduced by [30] to describe the nonlinear propagation of pulses through optical fibers. Also it arises in the context of high-speed soliton transmission in long-haul optical communication system [12]. Equation (1.1) represents the propagation of pulses by taking higher dispersion effects into account than those given by the Schrödinger equation (see [14,22,27,31,34]).
Multiplying Eq. (1.1) by the operator 1 − ∂ 2 x −1 , we rewrite it in the pseudodifferential form where the linear pseudodifferential operator Λ = 1 − ∂ 2 x −1 −∂ 2 x + a∂ 4 x is characterized by its symbol Λ (ξ) = ξ 2 +aξ 4 1+ξ 2 . As far as we know, there are no results on the large time asymptotics of solutions of the Cauchy problem (1.1). The difficulty of the small data scattering problem lies in the slow time decay rate of the L ∞ -norm of solutions to the linear problem. So the problem on the large time asymptotic behavior becomes more difficult for low space dimensions and low order of the nonlinearity. Comparing the time decay rates of the main term and the remainder terms of the nonlinearity in (1.1), we find that this equation represents critical behavior for large time. Indeed, below we will prove the modified scattering of solutions to (1.1 ). On the other hand, the higher-order nonlinear local or homogeneous Schrödinger equations have been widely studied recently. For the local and global well-posedness of the Cauchy problem, we refer to [4,5,29] and references cited therein. The dispersive blow-up was obtained in [1], [2]. The Cauchy problem for the higher-order nonlinear Schrödinger equations was intensively studied by many authors. The existence and uniqueness of solutions to (1.1) were proved in [15,[23][24][25][26]28,33], and the smoothing properties of solutions were studied in [8,10,11,[23][24][25][26].
In the case of a = 1, we find that Λ = −∂ 2 x , and (1.2) is the well-known cubic nonlinear Schrödinger equation studied by many authors, see [6,17] and references cited therein. So we exclude the case a = 1 from our consideration here. Final value problem for (1.2) with a = 1 was studied by Ozawa [32], where the modified wave operator was constructed. Modified scattering of the initial value problem (1.2) with a = 1 was obtained in [16] by using MDFM decomposition of the free Schrödinger evolution group U 1 (t) (a = 1) called as the factorization techniques. More precisely, we have the identity 2π R e −ixy φ (y) dy. The identity U 1 (t) = MDFM was used in [21] to state the relation between the operators x + 2it∂ x , U 1 (t) xU 1 (−t) and Mit∂ x M. In [16], we have used the identity to decompose the cubic nonlinearity in the form of the sum of the main and the remainder terms and it was shown in [16] that R is the remainder term with respect to time decay. However, this method cannot be applied directly to other dispersive equations since we do not know an explicit representation for the free evolution group. Later we have used the decomposition of free evolution group (also called the factorization techniques) in the case of the cubic nonlinear Klein-Gordon equation [18], the inhomogeneous fourth-order Schrödinger equation [19] and the third-order Schrödinger equation [20]. In the present paper, our purpose is to develop this approach (the factorization techniques) for the case of the evolution group U a (t) generated by the nonlocal operator with symbol Λ (ξ) = ξ 2 +aξ 4 1+ξ 2 , in order to find the large time asymptotic behavior of solutions. We will decompose the cubic nonlinear term into the main part and a remainder term where ϕ a = FU a (−t) u. In order to prove desired estimates, we use the L 2 -boundedness of the pseudodifferential operators. This is a crucial point of our approach presented here. Our method can be applied to higher-dimensional cases which will be considered in a separate paper.
To state our results precisely, we introduce Notation and Function Spaces.
We note here that M is different from the one used previously in (1.3). We use the same notation for simplicity.
We are now in a position to state the main result of this paper.
there exists a unique modified scattering state W + ∈ L ∞ such that the asymptotics is valid for t → ∞ uniformly with respect to x ∈ R, where δ > 0.
We note that the main term of the asymptotics differs from the corresponding linear case by the logarithmic oscillation which vanishes in the case of λ = 0.
We organize the rest of our paper as follows. In Sect. 2, we formulate the factorization techniques. We prove the estimates of the defect operators in the uniform metrics and obtain the estimates for derivatives of the defect operator and its adjoint by applying the L 2 -boundedness results for pseudodifferential operators. We estimate the nonlinearity in Sect. 3. Section 4 is devoted to the proof of a priori estimates of solutions to the Cauchy problem (1.1) in the norm where ϕ (t) = FU (−t) u (t) , γ > 0 is small depending on the size of the data. Finally, we prove Theorem 1.1 in Sect. 5.

Factorization techniques
Denote the symbol Λ (ξ) = ξ 2 +aξ 4 1+ξ 2 , then the free evolution group has the form By a direct calculation for any x ≥ 0. By a simple computation, we have 1 5 . Therefore under the condition a > 1 5 , we have Λ (ξ) > 0 for all ξ ∈ R. This guaranties Λ (ξ) is monotone increasing function and the unique stationary point μ (x) , defined as a root of equation for all x ∈ R. Hence, we have , and the defect operator . Also we need the representation for the inverse evolution group FU ) and the adjoint defect operator Indeed we have

Define the new dependent variable
applying the operator FU (−t) to Eq. (1.2), we get where v = Q ϕ. This is our target equation. Our function space is based on the norm defined by (1.5).
For the convenience of the reader, we now state our strategy of the proof of the theorem briefly. By using the stationary phase method and integration by parts, we estimate v = Q ϕ as follows in Lemma 2.1 (estimate of the defect operator). This estimate will be justified if we could be able to obtain the estimate of ∂ ξ ϕ L 2 , which requires us to estimate the norm ∂ ξ Q * |v| 2 v L 2 via Eq. (2.1). In Lemma 2.6 we show that In order to get the estimate of ϕ L ∞ , in Lemma 2.2 we prove the estimate of Q * φ L ∞ (estimate of the adjoint defect operator) with φ = |v| 2 v, such as This can be considered as the adjoint estimate to (2.2). Furthermore we show the estimate for ∂ η v L 2 = ∂ η Q ϕ L 2 in Lemma 2.5, given by Therefore, the crucial estimates are the following In order to get these estimates, we use the L 2 -boundedness of pseudodifferential operators given by Lemma 2.3 and Lemma 2.4.

Estimate for the defect operator in the uniform norm
Consider the kernel To compute the asymptotics of A for large time, we apply the stationary phase method (see [13], p. 110) to find for t → ∞, where the stationary point y 0 is defined by the equation g (y 0 ) = 0. By virtue of formula (2.3), we get In the next lemma, we estimate the defect operator Q in the uniform norm.
Proof. By a simple calculation, we have Integration by parts via identity where Note that Λ (ξ) = O (1) . Also we have Hence the estimate follows Therefore by Hardy and Cauchy-Schwarz inequalities, we obtain Lemma 2.1 is proved.

Estimate for the adjoint defect operator in the uniform norm
We consider the kernel By virtue of formula (2.3), we obtain the large time asymptotics for t → ∞. In the next lemma, we estimate the adjoint defect operator Q * in the uniform norm.
Proof. As above, we integrate by parts via the identity with ) . Then, we find From which via the estimate and Hardy and Cauchy-Schwarz inequalities, we obtain Lemma 2.2 is proved.

Boundedness of pseudodifferential operators
There are many papers devoted to the L 2 estimates of pseudodifferential operators (see, e.g. [3,7,9]). Below we will use the following result on the L 2 boundedness of pseudodifferential operator (see [9]).

Lemma 2.3. Let the symbol a (x, ξ) be such that
Analogously for the conjugate pseudodifferential operator we get the following result.

Estimate for derivative of the defect operator
In the next lemma, we estimate a derivative of the defect operator Q.

Proof. We integrate by parts to get
and Then, we obtain Since and Hence by Lemma 2.3, we get

Thus in view of equalities
and F −1 φ L 2 = φ L 2 , we obtain

Estimate for derivative of the adjoint defect operator
Next we prove the estimates for derivatives of adjoint defect operator Q * .

Asymptotics of the nonlinearity
In the next lemma, we calculate the asymptotic representation for the nonlinearity. Proof. Applying Lemma 2.2 with φ = |v| 2 v, we find In view of Lemma 2.1, we have the asymptotics v = 1 √ iΛ ϕ + O εt γ− 1 4 , and estimate v L ∞ ≤ Cε. Then by Lemma 2.5 and condition of the lemma we find ∂ η v L 2 ≤ Cεt γ . Hence, the result of the lemma follows. Lemma 3.1 is proved.
Proof. By virtue of Lemma 2.6 with φ = |v| 2 v, we get ∂ ξ Q * |v| 2 v L 2 ≤ C v 2 L ∞ v H 1 . Using Lemma 2.1, we find v L ∞ ≤ Cε. Then by Lemma 2.5, we obtain the result of the lemma. Lemma 3.2 is proved.