TY - JOUR

T1 - Modified scattering for higher-order nonlinear Schrödinger equation in one space dimension

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), under exclusive licence to Springer Nature Switzerland AG.

PY - 2021/12

Y1 - 2021/12

N2 - We consider the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Schrödinger equation with critical cubic nonlinearity {i∂tu+12∂x2u-1α|∂x|αu=λ|u|2u,t>0,x∈R,u(0,x)=u0(x),x∈R,where λ∈ R and α= 4 or α≥ 5 , since while estimating pseudodifferential operators below we need that the condition such that the symbol Λ(ξ)=12ξ2+1α|ξ|α∈C5(R). We show that the modified scattering occurs in the uniform norm. We continue to develop the factorization techniques which was started in papers (Ozawa in Commun Math Phys 139(3):479–493, 1991; Hayashi and Ozawa in Ann IHP (Phys Théor) 48:17–37, 1988; Hayashi and Naumkin in Z Angew Math Phys 59(6):1002–1028, 2008; Hayashi and Kaikina in Math Methods Appl Sci 40(5):1573–1597, 2017; Hayashi and Naumkin in J Math Phys 56(9):093502, 2015). The crucial points of our approach presented here are the L2-boundedness of the pseudodifferential operators which are used to obtain estimates of nonlinear terms in weighted Sobolev space.

AB - We consider the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Schrödinger equation with critical cubic nonlinearity {i∂tu+12∂x2u-1α|∂x|αu=λ|u|2u,t>0,x∈R,u(0,x)=u0(x),x∈R,where λ∈ R and α= 4 or α≥ 5 , since while estimating pseudodifferential operators below we need that the condition such that the symbol Λ(ξ)=12ξ2+1α|ξ|α∈C5(R). We show that the modified scattering occurs in the uniform norm. We continue to develop the factorization techniques which was started in papers (Ozawa in Commun Math Phys 139(3):479–493, 1991; Hayashi and Ozawa in Ann IHP (Phys Théor) 48:17–37, 1988; Hayashi and Naumkin in Z Angew Math Phys 59(6):1002–1028, 2008; Hayashi and Kaikina in Math Methods Appl Sci 40(5):1573–1597, 2017; Hayashi and Naumkin in J Math Phys 56(9):093502, 2015). The crucial points of our approach presented here are the L2-boundedness of the pseudodifferential operators which are used to obtain estimates of nonlinear terms in weighted Sobolev space.

KW - Higher order NLS

KW - Large time asymptotics

KW - Nonlinear Schrödinger equation

KW - Time decay estimates

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U2 - 10.1007/s00028-021-00723-0

DO - 10.1007/s00028-021-00723-0

M3 - Article

AN - SCOPUS:85107469844

SN - 1424-3199

VL - 21

SP - 4469

EP - 4490

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 4

ER -