TY - JOUR

T1 - Modified scattering for the nonlinear nonlocal Schrödinger equation in one-dimensional case

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

N1 - Funding Information:
The work of N.H. is partially supported by JSPS KAKENHI Grant Numbers JP20K03680, JP19H05597. The work of P.I.N. is partially supported by CONACYT project 283698 and PAPIIT project IN103221.
Publisher Copyright:
© 2021, The Author(s).

PY - 2022/2

Y1 - 2022/2

N2 - We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity {i∂t(u-∂x2u)+∂x2u-a∂x4u=λ|u|2u,t>0,x∈R,u(0,x)=u0(x),x∈R,where a>15,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 L2-boundedness of the pseudodifferential operators.

AB - We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity {i∂t(u-∂x2u)+∂x2u-a∂x4u=λ|u|2u,t>0,x∈R,u(0,x)=u0(x),x∈R,where a>15,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 L2-boundedness of the pseudodifferential operators.

KW - Cubic nonlinear Schrödinger equation

KW - Decay estimates

KW - Modified scattering

KW - Nonlocal Schrödinger equation

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U2 - 10.1007/s00033-021-01635-2

DO - 10.1007/s00033-021-01635-2

M3 - Article

AN - SCOPUS:85118988821

VL - 73

JO - Zeitschrift fur Angewandte Mathematik und Physik

JF - Zeitschrift fur Angewandte Mathematik und Physik

SN - 0044-2275

IS - 1

M1 - 2

ER -