AbstractWe study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity $$\begin{aligned} \left\{ \begin{array}{l} i\partial _{t}\left( u-\partial _{x}^{2}u\right) +\partial _{x}^{2}u-a\partial _{x}^{4}u=\lambda \left| u\right| ^{2}u,\text { } t>0,{\ }x\in {\mathbb {R}}\mathbf {,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,{\ }x\in {\mathbb {R}}\mathbf {,} \end{array} \right. \end{aligned}$$
i
∂
t
u
-
∂
x
2
u
+
∂
x
2
u
-
a
∂
x
4
u
=
λ
u
2
u
,
t
>
0
,
x
∈
R
,
u
0
,
x
=
u
0
x
,
x
∈
R
,
where $$a>\frac{1}{5},$$
a
>
1
5
,
$$\lambda \in {\mathbb {R}}$$
λ
∈
R
. 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 $${\mathbf {L}}^{2}$$
L
2
-boundedness of the pseudodifferential operators.
On the Asymptotics of Solutions of the Klein - Gordon - Fock Equation with Meromorphic Coefficients in the Neighborhood of Infinity
