TY - JOUR
T1 - Modified scattering for the higher-order nonlinear Schrödinger equation with the Hartree-type nonlinearity
AU - Hayashi, Nakao
AU - Mendez-Navarro, Jesus A.
AU - Naumkin, Pavel I.
N1 - Funding Information:
We would like to thank the referees for their useful suggestions on the draft. 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:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2023/3
Y1 - 2023/3
N2 - We study the Cauchy problem for the nonlinear nonlocal Schrödinger equation with the Hartree-type nonlinearity in one space dimension {i∂tu+Λu=Ω(uIα|u|2+2α),t>0,x∈R,u(0,x)=u0(x),x∈R,where Λ = F- 1Λ F, Ω = F- 1Ω F are the pseudodifferential operators, defined by their symbols Λ (ξ) , Ω (ξ) , which are real-valued functions, the Riesz potential Iαϕ= | x| -1+α∗ ϕ, α∈ (0 , 1) and ∗ denotes the convolution in R. We generalize the factorization techniques, used in papers (Hayashi and Naumkin in J Evol Equ, 2021. https://doi.org/10.1007/s00028-021-00723-0, in Z Angew Math Phys 73:2, 2022. https://doi.org/10.1007/s00033-021-01635-2). We show that the factorization techniques related to the evolution operator with the psuedodifferential operator works well for the nonlinearity of the Hartree type through the L2-boundedness of the pseudodifferential operators.
AB - We study the Cauchy problem for the nonlinear nonlocal Schrödinger equation with the Hartree-type nonlinearity in one space dimension {i∂tu+Λu=Ω(uIα|u|2+2α),t>0,x∈R,u(0,x)=u0(x),x∈R,where Λ = F- 1Λ F, Ω = F- 1Ω F are the pseudodifferential operators, defined by their symbols Λ (ξ) , Ω (ξ) , which are real-valued functions, the Riesz potential Iαϕ= | x| -1+α∗ ϕ, α∈ (0 , 1) and ∗ denotes the convolution in R. We generalize the factorization techniques, used in papers (Hayashi and Naumkin in J Evol Equ, 2021. https://doi.org/10.1007/s00028-021-00723-0, in Z Angew Math Phys 73:2, 2022. https://doi.org/10.1007/s00033-021-01635-2). We show that the factorization techniques related to the evolution operator with the psuedodifferential operator works well for the nonlinearity of the Hartree type through the L2-boundedness of the pseudodifferential operators.
KW - Asymptotics for large time
KW - Hartree-type nonlinearity
KW - Higher-order nonlinear Schrödinger equation
KW - Modified scattering
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U2 - 10.1007/s00028-022-00852-0
DO - 10.1007/s00028-022-00852-0
M3 - Article
AN - SCOPUS:85143123891
SN - 1424-3199
VL - 23
JO - Journal of Evolution Equations
JF - Journal of Evolution Equations
IS - 1
M1 - 1
ER -