Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

Hirayama, Hiroyuki; Okamoto, Mamoru

doi:10.3934/cpaa.2016.15.831

Cited by 11 publications

(35 citation statements)

References 19 publications

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“…Asymptotic behavior of the fourth-order NLS and its related equations have been studied by several researchers. See [1,2,[5][6][7][8][9][10][11][12]14,15,19] and references therein. In particular, Ben-Artzi, Koch, and Saut [2] showed the dispersive estimates for the fourth-order Schrödinger equations.…”

Section: Introductionmentioning

confidence: 99%

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Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

Okamoto

Uriya

2021

J. Evol. Equ.

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We consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.

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Section: Introductionmentioning

confidence: 99%

“…Hirayama and the first author [12] showed the small data global well-posedness and the scattering in the scaling critical Sobolev spaceḢ − 1 2 (R). They used the Fourier restriction norm method adapted to the spaces V p of functions of bounded p-variation and their pre-duals U p .…”

Section: Introductionmentioning

confidence: 99%

Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

Okamoto

Uriya

2021

J. Evol. Equ.

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“…The 4NLS equation with Kerr nonlinearity has been introduced by Karpman and Shagalov [20,21] to describe the role of the small fourth order dispersion term in the propagation of intense laser beams in a bulk medium. There are many authors investigating the 4NLS equation with different nonlinearities, see [14,[17][18][19][27][28][29]. Huo and Jia [17][18][19] studied the Cauchy problem of the 4NLS equation with nonlinear component containing the second order derivative nonlinearities on R. In [17], they showed the local well-posedness of the 4NLS equation for the initial data in H s (R) (s ≥ 1 2 ) with nonlinear part not containing the second order derivative term |u| 2 ∂ 2…”

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Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation

Guo

2022

DCDS-B

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<p style='text-indent:20px;'>The main purpose of this paper is to study local regularity properties of the fourth-order nonlinear Schrödinger equations on the half line. We prove the local existence, uniqueness, and continuous dependence on initial data in low regularity Sobolev spaces. We also obtain the nonlinear smoothing property: the nonlinear part of the solution on the half line is smoother than the initial data.</p>

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“…Aoki, Hayashi and Naumkin [2] showed the global existence and scattering of (1.3) with d = 1, 2 and p > 1 + 4/d. We refer also to the series of paper by Hayashi and Naumkin [18,19,21] and work by Hirayama and Okamoto [22] for interesting phenomena on the long time behavior of solutions to (1.3) with a derivative nonlinearity.Recently, a blow-up result is proved by Boulenger and Lenzmann [7] for (1.3) with the mass critical and super critical focusing nonlinearity in the radial case which solves a long standing conjecture suggested by several numerical studies (see [11] for instance). Notice that most of their results hold also when lower dispersive term µ∆u is added.…”

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Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion

Saut

Segata

2019

Discrete &Amp; Continuous Dynamical Systems - A

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Mathematics Subject Classification. Primary: 35Q55; Secondary: 35B40. Key words and phrases. Schrödinger equation with higher order dispersion, scattering problem. 219 220 JEAN-CLAUDE SAUT AND JUN-ICHI SEGATAIn this paper, we study the asymptotic behavior in time of solutions to (1.2).The Cauchy problem for the homogeneous fourth order nonlinear Schrödinger type equationhas been studied by many authors, most of the results holding also when lower dispersive terms are added. By using the Strichartz estimates in [3] one shows that the Cauchy problem is locally well-posed in the energy space H 2 (R d ) for the energy subcritical case (i.e., 1 < p < 1 + 8/(d − 4) when d ≥ 5 and 1 < p < ∞ when d ≤ 4) and in L 2 (R d ) for the mass subcritical case (1 < p < 1 + 8/d). We also refer to Bouchel [6] who studies the Cauchy problem and furthermore gives non-existence, existence and qualitative properties results of solitary wave solutions for (1.1). See also [12] for results on the Cauchy problem for slightly more general situations.There are several results concerning the scattering and blow-up of solutions for (1.3). For the defocusing case λ < 0, the global well-posedness and scattering for (1.3) with the energy-critical nonlinearity (i.e., (1.3) with d 5, and p = 1 + 8/(d − 4)) was studied by Pausader [32] for radially symmetric initial data by combining the concentration-compactness argument by Kenig-Merle [25] and Morawetz-type estimate. Later, Miao, Xu and Zhao [28] proved a similar result for (1.3) in the energy-critical and higher dimensional case d 9 without radial assumption on initial data. In [33], Pausader has shown the global well-posedness and scattering of (1.3) with cubic nonlinearity for the case 5 d 8. Pausader and Xia [34] proved the global well-posedness and scattering for (1.3) with mass supercritical nonlinearity (i.e., (1.3) with p > 1 + 8/d) for low dimensions 1 d 4 by using a virial-type estimate instead of the Morawetz-type estimates.For the focusing case λ > 0, Pausader [31] and Miao, Xu and Zhao [27] independently showed the global well-posedness and scattering for (1.3) with the energycritical nonlinearity for radially symmetric initial data withḢ 2 and energy norms below that of the ground state. When the initial data is sufficiently small, Hayashi, Mendez-Navarro and Naumkin [14] proved the global existence and the scattering for (1.3) with d = 1 and p > 5 by using the factorization technique developed by the authors [16]. In [14] they also shown the small data global existence and the decay estimates for (1.3) with d = 1 and p > 4 under the assumption that the initial data is odd. In the subsequent paper [15], they proved that when d = 1, p = 5 and λ < 0, a solution to (1.3) has dissipative structure and gains additional logarithmic decay. Aoki, Hayashi and Naumkin [2] showed the global existence and scattering of (1.3) with d = 1, 2 and p > 1 + 4/d. We refer also to the series of paper by Hayashi and Naumkin [18,19,21] and work by Hirayama and Okamoto [22] for interesting phenomena on the...

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Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

Cited by 11 publications

References 19 publications

Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation

Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion

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