Modified scattering type asymptotic behavior for a quadratic nonlinear Schr\"odinger system under the mass-resonance condition in two dimensions

Kota Uriya; Satoshi Masaki

arxiv: 2606.02088 · v1 · pith:KH5MOXXKnew · submitted 2026-06-01 · 🧮 math.AP

Modified scattering type asymptotic behavior for a quadratic nonlinear Schr\"odinger system under the mass-resonance condition in two dimensions

Satoshi Masaki , Kota Uriya This is my paper

Pith reviewed 2026-06-28 13:44 UTC · model grok-4.3

classification 🧮 math.AP

keywords nonlinear Schrödinger equationsasymptotic behaviormass-resonance conditionmodified scatteringJacobi elliptic functionsquadratic nonlinearitytwo space dimensions

0 comments

The pith

Small solutions to this 2D quadratic nonlinear Schrödinger system follow modified scattering profiles governed by an integrable ODE solved with Jacobi elliptic functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks a complete description of the long-time behavior of small solutions to the system in two dimensions. It reduces the PDE asymptotics to an associated ODE system and solves that ODE explicitly for arbitrary initial data. The amplitude takes the form of Jacobi elliptic functions while the phase is given by elliptic integrals of the third kind. This yields an explicit modified-scattering profile for every small solution rather than only special cases.

Core claim

The asymptotic profile of solutions is given by a modified-scattering-type behavior whose evolution follows an integrable ODE system; for arbitrary initial data the amplitude is expressed explicitly via Jacobi elliptic functions and the phase via elliptic integrals of the third kind.

What carries the argument

The associated system of ordinary differential equations that governs the asymptotic profile, equipped with an explicit phase-amplitude representation in Jacobi elliptic functions.

If this is right

Long-time behavior takes the form of a modified-scattering profile that continues to evolve according to the integrable ODE structure.
Elliptic-function asymptotics hold for all sufficiently small initial data, not merely special solutions.
The same ODE reduction supplies a full characterization of the asymptotic dynamics in two dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit ODE solutions may supply a template for constructing approximate solutions or testing stability in related quadratic systems.
Similar phase-amplitude reductions could be attempted for mass-resonance systems in other dimensions once an analogous ODE is derived.
The integrability of the profile ODE suggests the possibility of conserved quantities that might constrain scattering maps for the original PDE.

Load-bearing premise

The long-time profile of the PDE solutions is controlled by the associated ODE system.

What would settle it

An explicit small solution or high-precision numerical evolution whose amplitude or phase at large times deviates from the Jacobi-elliptic form predicted by the ODE for the same initial data.

read the original abstract

We study a system of nonlinear Schrodinger equations under the mass-resonance condition and provide a complete description of the asymptotic behavior of small solutions in two space dimensions. Our analysis is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile. We establish a phase-amplitude representation for this ODE with arbitrary initial data, where the amplitude is expressed explicitly in terms of Jacobi elliptic functions and the phase is given by elliptic integrals of the third kind. As a consequence, we obtain a fully explicit characterization of the asymptotic profile for the original PDE. In particular, the long-time behavior is described by a modified-scattering-type profile whose profile evolves according to the above integrable structure. While elliptic-function-type asymptotics were previously constructed for special solutions in final-state problems, the present work provides the first complete characterization, in two space dimensions, of asymptotic dynamics associated with arbitrarily small initial data through elliptic-function-type profiles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit Jacobi-elliptic description of modified scattering for arbitrary small data in this 2D resonant system, but the ODE-to-PDE transfer needs checking.

read the letter

The main takeaway is that the authors solve an associated ODE system explicitly for any initial data, writing the amplitude in Jacobi elliptic functions and the phase via elliptic integrals of the third kind, then claim this yields the full asymptotic profile for the PDE. This extends earlier work that only constructed such profiles for special solutions in final-state problems.

The ODE analysis itself looks like the strongest part. It supplies a complete phase-amplitude representation without extra restrictions on the data, which is a concrete advance for the model equation.

The softer point is the justification that solutions of the original PDE actually follow this ODE flow with integrable errors. The abstract states the analysis rests on the ODE governing the asymptotic profile, but the quantitative control on the difference—typically via Duhamel estimates showing remainders converge—must be verified in detail. If those bounds hold only under additional smallness or decay assumptions, the claim for arbitrary small data would not go through as stated.

This is a paper for specialists in modified scattering and long-time asymptotics for quadratic dispersive systems. A reader already working on resonant NLS or integrable structures in 2D would get direct value from the explicit ODE solution.

The work shows clear engagement with the relevant literature and a focused technical goal. It deserves a serious referee to examine the error estimates between the PDE and the ODE.

Referee Report

1 major / 0 minor

Summary. The paper studies a quadratic nonlinear Schrödinger system in two dimensions under the mass-resonance condition. It claims to give a complete description of the asymptotic behavior of small solutions by reducing the problem to an associated ODE system for the asymptotic profile, deriving an explicit phase-amplitude representation (amplitude via Jacobi elliptic functions, phase via elliptic integrals of the third kind) valid for arbitrary initial data, and thereby obtaining a fully explicit modified-scattering-type profile for the original PDE.

Significance. If the reduction from the PDE to the exact ODE flow is justified with integrable error terms, the work would supply the first complete, explicit elliptic-function characterization of long-time dynamics for arbitrary small initial data in this 2D setting, extending prior results limited to special solutions in final-state problems. The explicit integrability of the model ODE is a clear technical strength.

major comments (1)

[Abstract / reduction step] The central claim requires that the asymptotic profile extracted from the PDE satisfies the associated ODE up to errors whose Duhamel integrals remain convergent (or o(1)) at infinity. The abstract states that the analysis 'is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile,' but no quantitative control on the difference between the PDE-derived profile and the pure ODE solution is visible in the provided material; without such estimates the elliptic-function description applies rigorously only to the model ODE.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract / reduction step] The central claim requires that the asymptotic profile extracted from the PDE satisfies the associated ODE up to errors whose Duhamel integrals remain convergent (or o(1)) at infinity. The abstract states that the analysis 'is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile,' but no quantitative control on the difference between the PDE-derived profile and the pure ODE solution is visible in the provided material; without such estimates the elliptic-function description applies rigorously only to the model ODE.

Authors: We agree that the reduction step requires explicit quantitative control on the error between the PDE-derived profile and the pure ODE solution to justify applying the elliptic-function description to the original system. While the manuscript derives the ODE and solves it explicitly, the estimates showing that the Duhamel integral of the remainder term remains convergent (using the small-data assumption, the mass-resonance condition, and the decay of the linear propagator in 2D) are only indicated in outline form rather than fully detailed. We will revise the paper by adding a dedicated subsection (likely in Section 3 or 4) that provides these estimates in full, including the necessary a priori bounds and integrability arguments. This will make the passage from the PDE to the integrable ODE rigorous and will be reflected in a clarified statement in the abstract and introduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from PDE to ODE analysis to explicit solution

full rationale

The paper's central step is to associate an ODE system to the PDE asymptotic profile, then solve that ODE explicitly via Jacobi elliptic functions and elliptic integrals for arbitrary data. This is a forward derivation from the model ODE (whose form is standard for modified scattering under mass resonance) to closed-form expressions, followed by the claim that the PDE inherits the same profile. No step reduces a prediction to a fitted input by construction, no self-citation is invoked as a uniqueness theorem to force the result, and the elliptic-function representation is obtained by direct integration of the ODE rather than by renaming or smuggling an ansatz. The provided abstract and description contain no equations that equate the target PDE profile to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mass-resonance condition enabling the ODE reduction and the small-data assumption justifying the asymptotic analysis; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption The mass-resonance condition holds, allowing the asymptotic profile to be governed by the associated ODE system.
Invoked as the foundation for reducing the PDE to the integrable ODE structure.

pith-pipeline@v0.9.1-grok · 5694 in / 1217 out tokens · 26465 ms · 2026-06-28T13:44:36.135294+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 18 canonical work pages · 1 internal anchor

[1]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan,Interactions between light waves in a nonlinear dielectric, Phys. Rev.127(1962), no. 6, 1918–1939, DOI 10.1103/PhysRev.127.1918

work page doi:10.1103/physrev.127.1918 1962
[2]

Byrd and Morris D

Paul F. Byrd and Morris D. Friedman,Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR0277773

1971
[3]

10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003

Thierry Cazenave,Semilinear Schr¨ odinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR2002047

2003
[4]

Chuanwei Gao, Fanfei Meng, Chengbin Xu, and Jiqiang Zheng,Scattering theory for quadratic nonlinear Schr¨ odinger system in dimension six, J. Math. Anal. Appl.541(2025), no. 1, Paper No. 128708, 42, DOI 10.1016/j.jmaa.2024.128708. MR4781063

work page doi:10.1016/j.jmaa.2024.128708 2025
[5]

Ginibre and T

J. Ginibre and T. Ozawa,Long range scattering for nonlinear Schr¨ odinger and Hartree equations in space dimensionn≥2, Comm. Math. Phys.151(1993), no. 3, 619–645. MR1207269 QUADRATIC NLS SYSTEM UNDER MASS-RESONANCE IN 2D 17

1993
[6]

Masaru Hamano,Global dynamics below the ground state for the quadratic Schr¨ odinger system in 5d, 2018, available as arXiv:1805.12245

work page internal anchor Pith review Pith/arXiv arXiv 2018
[7]

Ekvac.64(2021), no

Masaru Hamano, Takahisa Inui, and Kuranosuke Nishimura,Scattering for the quadratic nonlinear Schr¨ odinger system inR 5 without mass-resonance condition, Funkcial. Ekvac.64(2021), no. 3, 261–291, DOI 10.1619/fesi.64.261. MR4360610

work page doi:10.1619/fesi.64.261 2021
[8]

Masaru Hamano and Satoshi Masaki,A sharp scattering threshold level for mass-subcritical nonlinear Schr¨ odinger system, Discrete Contin. Dyn. Syst.41(2021), no. 3, 1415–1447, DOI 10.3934/dcds.2020323. MR4201846

work page doi:10.3934/dcds.2020323 2021
[9]

Naumkin,On a system of nonlinear Schr¨ odinger equations in 2D, Differential Integral Equations24(2011), no

Nakao Hayashi, Chunhua Li, and Pavel I. Naumkin,On a system of nonlinear Schr¨ odinger equations in 2D, Differential Integral Equations24(2011), no. 5-6, 417–434. MR2809614

2011
[10]

Partial Dif- ferential Equations37(2012), no

,Modified wave operator for a system of nonlinear Schr¨ odinger equations in 2d, Comm. Partial Dif- ferential Equations37(2012), no. 6, 947–968, DOI 10.1080/03605302.2012.668256. MR2924463

work page doi:10.1080/03605302.2012.668256 2012
[11]

Differential Equations260 (2016), no

,Nonlinear Schr¨ odinger systems in 2d with nondecaying final data, J. Differential Equations260 (2016), no. 2, 1472–1495, DOI 10.1016/j.jde.2015.09.033. MR3419736

work page doi:10.1016/j.jde.2015.09.033 2016
[12]

Naumkin,Asymptotics for large time of solutions to the nonlinear Schr¨ odinger and Hartree equations, Amer

Nakao Hayashi and Pavel I. Naumkin,Asymptotics for large time of solutions to the nonlinear Schr¨ odinger and Hartree equations, Amer. J. Math.120(1998), no. 2, 369–389. MR1613646

1998
[13]

Differential Equations254(2013), no

,A system of quadratic nonlinear Klein-Gordon equations in 2d, J. Differential Equations254(2013), no. 8, 3615–3646, DOI 10.1016/j.jde.2013.01.035. MR3020890

work page doi:10.1016/j.jde.2013.01.035 2013
[14]

Nakao Hayashi, Tohru Ozawa, and Kazunaga Tanaka,On a system of nonlinear Schr¨ odinger equations with quadratic interaction, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire30(2013), no. 4, 661–690, DOI 10.1016/j.anihpc.2012.10.007. MR3082479

work page doi:10.1016/j.anihpc.2012.10.007 2013
[15]

Takahisa Inui, Nobu Kishimoto, and Kuranosuke Nishimura,Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition, Discrete Contin. Dyn. Syst.39(2019), no. 11, 6299–6353, DOI 10.3934/dcds.2019275. MR4026982

work page doi:10.3934/dcds.2019275 2019
[16]

MR4090442

,Blow-up of the radially symmetric solutions for the quadratic nonlinear Schr¨ odinger system without mass-resonance, Nonlinear Anal.198(2020), 111895, 10, DOI 10.1016/j.na.2020.111895. MR4090442

work page doi:10.1016/j.na.2020.111895 2020
[17]

Tsukasa Iwabuchi, Takayoshi Ogawa, and Kota Uriya,Ill-posedness for a system of quadratic non- linear Schr¨ odinger equations in two dimensions, J. Funct. Anal.271(2016), no. 1, 136–163, DOI 10.1016/j.jfa.2016.04.017. MR3494245

work page doi:10.1016/j.jfa.2016.04.017 2016
[18]

Soichiro Katayama and Daisuke Sakoda,Asymptotic behavior for a class of derivative nonlinear Schr¨ odinger systems, Partial Differ. Equ. Appl.1(2020), no. 3, Paper No. 12, 41, DOI 10.1007/s42985-020-00012-4. MR4336288

work page doi:10.1007/s42985-020-00012-4 2020
[19]

Soichiro Katayama and Satoshi Masaki,Asymptotic analysis without structural condition for systems of critical derivative nonlinear Schr¨ odinger equations, preprint
[20]

Satoshi Masaki,Partial classification of the large-time behavior of solutions to cubic nonlinear Schr¨ odinger systems, Partial Differ. Equ. Appl.6(2025), no. 4, Paper No. 27, 52, DOI 10.1007/s42985-025-00334-1. MR4917200

work page doi:10.1007/s42985-025-00334-1 2025
[21]

,Global existence and large-time behavior of solutions to cubic nonlinear Schr¨ odinger systems without coercive conserved quantity, 2024, available as arXiv:2412.00413

work page arXiv 2024
[22]

Satoshi Masaki, Jun-Ichi Segata, and Kota Uriya,Asymptotic behavior in time of solution to system of cubic nonlinear Schr¨ odinger equations in one space dimension, Mathematical physics and its interactions, Springer Proc. Math. Stat., vol. 451, Springer, Singapore, [2024]©2024, pp. 119–180. MR4775002

2024
[23]

,Non-polynomial conserved quantities for ODE systems and their application to the long-time behavior of solutions to cubic NLS systems, to appear in Trans. Amer. Math. Soc
[24]

Partial Differ

Norman Noguera and Ademir Pastor,On the dynamics of a quadratic Schr¨ odinger system in dimension n= 5, Dyn. Partial Differ. Equ.17(2020), no. 1, 1–17, DOI 10.4310/DPDE.2020.v17.n1.a1. MR4071981

work page doi:10.4310/dpde.2020.v17.n1.a1 2020
[25]

B42, Res

Takayoshi Ogawa and Kota Uriya,Asymptotic behavior of solutions to a quadratic nonlinear Schr¨ odinger system with mass resonance, Harmonic analysis and nonlinear partial differential equations, RIMS Kˆ okyˆ uroku Bessatsu, vol. B42, Res. Inst. Math. Sci. (RIMS), Kyoto, 2013, pp. 153–170. MR3220153

2013
[26]

Differential Equations258(2015), no

,Final state problem for a quadratic nonlinear Schr¨ odinger system in two space dimensions with mass resonance, J. Differential Equations258(2015), no. 2, 483–503, DOI 10.1016/j.jde.2014.09.022. MR3274766

work page doi:10.1016/j.jde.2014.09.022 2015
[27]

Tohru Ozawa,Long range scattering for nonlinear Schr¨ odinger equations in one space dimension, Comm. Math. Phys.139(1991), no. 3, 479–493. MR1121130

1991
[28]

Kota Uriya,Final state problem for a system of nonlinear Schr¨ odinger equations with three wave interaction, J. Evol. Equ.16(2016), no. 1, 173–191, DOI 10.1007/s00028-015-0297-z. MR3466217

work page doi:10.1007/s00028-015-0297-z 2016
[29]

Henri Poincar´ e18(2017), no

,Final state problem for systems of cubic nonlinear Schr¨ odinger equations in one dimension, Ann. Henri Poincar´ e18(2017), no. 7, 2523–2542. MR3665222 18 SATOSHI MASAKI AND KOTA URIYA Department of Mathematics, F aculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan Email address:masaki@math.sci.hokudai.ac...

2017

[1] [1]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan,Interactions between light waves in a nonlinear dielectric, Phys. Rev.127(1962), no. 6, 1918–1939, DOI 10.1103/PhysRev.127.1918

work page doi:10.1103/physrev.127.1918 1962

[2] [2]

Byrd and Morris D

Paul F. Byrd and Morris D. Friedman,Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR0277773

1971

[3] [3]

10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003

Thierry Cazenave,Semilinear Schr¨ odinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR2002047

2003

[4] [4]

Chuanwei Gao, Fanfei Meng, Chengbin Xu, and Jiqiang Zheng,Scattering theory for quadratic nonlinear Schr¨ odinger system in dimension six, J. Math. Anal. Appl.541(2025), no. 1, Paper No. 128708, 42, DOI 10.1016/j.jmaa.2024.128708. MR4781063

work page doi:10.1016/j.jmaa.2024.128708 2025

[5] [5]

Ginibre and T

J. Ginibre and T. Ozawa,Long range scattering for nonlinear Schr¨ odinger and Hartree equations in space dimensionn≥2, Comm. Math. Phys.151(1993), no. 3, 619–645. MR1207269 QUADRATIC NLS SYSTEM UNDER MASS-RESONANCE IN 2D 17

1993

[6] [6]

Masaru Hamano,Global dynamics below the ground state for the quadratic Schr¨ odinger system in 5d, 2018, available as arXiv:1805.12245

work page internal anchor Pith review Pith/arXiv arXiv 2018

[7] [7]

Ekvac.64(2021), no

Masaru Hamano, Takahisa Inui, and Kuranosuke Nishimura,Scattering for the quadratic nonlinear Schr¨ odinger system inR 5 without mass-resonance condition, Funkcial. Ekvac.64(2021), no. 3, 261–291, DOI 10.1619/fesi.64.261. MR4360610

work page doi:10.1619/fesi.64.261 2021

[8] [8]

Masaru Hamano and Satoshi Masaki,A sharp scattering threshold level for mass-subcritical nonlinear Schr¨ odinger system, Discrete Contin. Dyn. Syst.41(2021), no. 3, 1415–1447, DOI 10.3934/dcds.2020323. MR4201846

work page doi:10.3934/dcds.2020323 2021

[9] [9]

Naumkin,On a system of nonlinear Schr¨ odinger equations in 2D, Differential Integral Equations24(2011), no

Nakao Hayashi, Chunhua Li, and Pavel I. Naumkin,On a system of nonlinear Schr¨ odinger equations in 2D, Differential Integral Equations24(2011), no. 5-6, 417–434. MR2809614

2011

[10] [10]

Partial Dif- ferential Equations37(2012), no

,Modified wave operator for a system of nonlinear Schr¨ odinger equations in 2d, Comm. Partial Dif- ferential Equations37(2012), no. 6, 947–968, DOI 10.1080/03605302.2012.668256. MR2924463

work page doi:10.1080/03605302.2012.668256 2012

[11] [11]

Differential Equations260 (2016), no

,Nonlinear Schr¨ odinger systems in 2d with nondecaying final data, J. Differential Equations260 (2016), no. 2, 1472–1495, DOI 10.1016/j.jde.2015.09.033. MR3419736

work page doi:10.1016/j.jde.2015.09.033 2016

[12] [12]

Naumkin,Asymptotics for large time of solutions to the nonlinear Schr¨ odinger and Hartree equations, Amer

Nakao Hayashi and Pavel I. Naumkin,Asymptotics for large time of solutions to the nonlinear Schr¨ odinger and Hartree equations, Amer. J. Math.120(1998), no. 2, 369–389. MR1613646

1998

[13] [13]

Differential Equations254(2013), no

,A system of quadratic nonlinear Klein-Gordon equations in 2d, J. Differential Equations254(2013), no. 8, 3615–3646, DOI 10.1016/j.jde.2013.01.035. MR3020890

work page doi:10.1016/j.jde.2013.01.035 2013

[14] [14]

Nakao Hayashi, Tohru Ozawa, and Kazunaga Tanaka,On a system of nonlinear Schr¨ odinger equations with quadratic interaction, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire30(2013), no. 4, 661–690, DOI 10.1016/j.anihpc.2012.10.007. MR3082479

work page doi:10.1016/j.anihpc.2012.10.007 2013

[15] [15]

Takahisa Inui, Nobu Kishimoto, and Kuranosuke Nishimura,Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition, Discrete Contin. Dyn. Syst.39(2019), no. 11, 6299–6353, DOI 10.3934/dcds.2019275. MR4026982

work page doi:10.3934/dcds.2019275 2019

[16] [16]

MR4090442

,Blow-up of the radially symmetric solutions for the quadratic nonlinear Schr¨ odinger system without mass-resonance, Nonlinear Anal.198(2020), 111895, 10, DOI 10.1016/j.na.2020.111895. MR4090442

work page doi:10.1016/j.na.2020.111895 2020

[17] [17]

Tsukasa Iwabuchi, Takayoshi Ogawa, and Kota Uriya,Ill-posedness for a system of quadratic non- linear Schr¨ odinger equations in two dimensions, J. Funct. Anal.271(2016), no. 1, 136–163, DOI 10.1016/j.jfa.2016.04.017. MR3494245

work page doi:10.1016/j.jfa.2016.04.017 2016

[18] [18]

Soichiro Katayama and Daisuke Sakoda,Asymptotic behavior for a class of derivative nonlinear Schr¨ odinger systems, Partial Differ. Equ. Appl.1(2020), no. 3, Paper No. 12, 41, DOI 10.1007/s42985-020-00012-4. MR4336288

work page doi:10.1007/s42985-020-00012-4 2020

[19] [19]

Soichiro Katayama and Satoshi Masaki,Asymptotic analysis without structural condition for systems of critical derivative nonlinear Schr¨ odinger equations, preprint

[20] [20]

Satoshi Masaki,Partial classification of the large-time behavior of solutions to cubic nonlinear Schr¨ odinger systems, Partial Differ. Equ. Appl.6(2025), no. 4, Paper No. 27, 52, DOI 10.1007/s42985-025-00334-1. MR4917200

work page doi:10.1007/s42985-025-00334-1 2025

[21] [21]

,Global existence and large-time behavior of solutions to cubic nonlinear Schr¨ odinger systems without coercive conserved quantity, 2024, available as arXiv:2412.00413

work page arXiv 2024

[22] [22]

Satoshi Masaki, Jun-Ichi Segata, and Kota Uriya,Asymptotic behavior in time of solution to system of cubic nonlinear Schr¨ odinger equations in one space dimension, Mathematical physics and its interactions, Springer Proc. Math. Stat., vol. 451, Springer, Singapore, [2024]©2024, pp. 119–180. MR4775002

2024

[23] [23]

,Non-polynomial conserved quantities for ODE systems and their application to the long-time behavior of solutions to cubic NLS systems, to appear in Trans. Amer. Math. Soc

[24] [24]

Partial Differ

Norman Noguera and Ademir Pastor,On the dynamics of a quadratic Schr¨ odinger system in dimension n= 5, Dyn. Partial Differ. Equ.17(2020), no. 1, 1–17, DOI 10.4310/DPDE.2020.v17.n1.a1. MR4071981

work page doi:10.4310/dpde.2020.v17.n1.a1 2020

[25] [25]

B42, Res

Takayoshi Ogawa and Kota Uriya,Asymptotic behavior of solutions to a quadratic nonlinear Schr¨ odinger system with mass resonance, Harmonic analysis and nonlinear partial differential equations, RIMS Kˆ okyˆ uroku Bessatsu, vol. B42, Res. Inst. Math. Sci. (RIMS), Kyoto, 2013, pp. 153–170. MR3220153

2013

[26] [26]

Differential Equations258(2015), no

,Final state problem for a quadratic nonlinear Schr¨ odinger system in two space dimensions with mass resonance, J. Differential Equations258(2015), no. 2, 483–503, DOI 10.1016/j.jde.2014.09.022. MR3274766

work page doi:10.1016/j.jde.2014.09.022 2015

[27] [27]

Tohru Ozawa,Long range scattering for nonlinear Schr¨ odinger equations in one space dimension, Comm. Math. Phys.139(1991), no. 3, 479–493. MR1121130

1991

[28] [28]

Kota Uriya,Final state problem for a system of nonlinear Schr¨ odinger equations with three wave interaction, J. Evol. Equ.16(2016), no. 1, 173–191, DOI 10.1007/s00028-015-0297-z. MR3466217

work page doi:10.1007/s00028-015-0297-z 2016

[29] [29]

Henri Poincar´ e18(2017), no

,Final state problem for systems of cubic nonlinear Schr¨ odinger equations in one dimension, Ann. Henri Poincar´ e18(2017), no. 7, 2523–2542. MR3665222 18 SATOSHI MASAKI AND KOTA URIYA Department of Mathematics, F aculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan Email address:masaki@math.sci.hokudai.ac...

2017