Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one

Daniele Garrisi; Vladimir Georgiev

Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one

School of Mathematical Sciences

Research output: Chapter in Book/Conference proceeding › Conference contribution › peer-review

Abstract

We show that symmetric and positive profiles of ground-state standing-wave of the non-linear Schr¨odinger equation are non-degenerate and unique up to a translation of the argument and multiplication by complex numbers in the unit sphere. The non-linear term is a combination of two or three pure-powers. The class of non-linearities satisfying the mentioned properties can be extended beyond two or three power combinations. Specifically, it is sufficient that an Euler differential inequality is satisfied and that a certain auxiliary function is such that the first local maximum is also an absolute maximum.

Original language	English
Title of host publication	Nonlinear dispersive waves and fluids
Subtitle of host publication	AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows, January 5-6, 2017, Atlanta, Georgia
Editors	Shijun Zheng, Marius Beceanu, J. L. Bona, Geng Chen, Tuoc Phan, Avy Soffer
Publisher	American Mathematical Society
Pages	137-148
Number of pages	12
ISBN (Print)	9781470441098
Publication status	Published - 2019
Event	AMS Special Sessions on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows - Atlanta, Georgia, United States Duration: 5 Jan 2017 → 7 Jan 2017

Publication series

Name	Contemporary mathematics
Volume	725

Conference

Conference	AMS Special Sessions on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows
Country/Territory	United States
City	Atlanta, Georgia
Period	5/01/17 → 7/01/17

Keywords

Wave equation
Nonlinear waves
Nonlinear wave equations

Cite this

Garrisi, D., & Georgiev, V. (2019). Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one. In S. Zheng, M. Beceanu, J. L. Bona, G. Chen, T. Phan, & A. Soffer (Eds.), Nonlinear dispersive waves and fluids : AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows, January 5-6, 2017, Atlanta, Georgia (pp. 137-148). (Contemporary mathematics; Vol. 725). American Mathematical Society.

Garrisi, Daniele ; Georgiev, Vladimir. / Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one. Nonlinear dispersive waves and fluids : AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows, January 5-6, 2017, Atlanta, Georgia. editor / Shijun Zheng ; Marius Beceanu ; J. L. Bona ; Geng Chen ; Tuoc Phan ; Avy Soffer. American Mathematical Society, 2019. pp. 137-148 (Contemporary mathematics).

@inproceedings{b6df5d19591945d7bc6a1981c497de4c,

title = "Uniqueness of standing-waves for a non-linear Schr{\"o}dinger equation with three pure-power combinations in dimension one",

abstract = "We show that symmetric and positive profiles of ground-state standing-wave of the non-linear Schr¨odinger equation are non-degenerate and unique up to a translation of the argument and multiplication by complex numbers in the unit sphere. The non-linear term is a combination of two or three pure-powers. The class of non-linearities satisfying the mentioned properties can be extended beyond two or three power combinations. Specifically, it is sufficient that an Euler differential inequality is satisfied and that a certain auxiliary function is such that the first local maximum is also an absolute maximum.",

keywords = "Wave equation, Nonlinear waves, Nonlinear wave equations",

author = "Daniele Garrisi and Vladimir Georgiev",

year = "2019",

language = "English",

isbn = "9781470441098",

series = "Contemporary mathematics",

publisher = "American Mathematical Society",

pages = "137--148",

editor = "Shijun Zheng and Marius Beceanu and Bona, {J. L.} and Geng Chen and Tuoc Phan and Avy Soffer",

booktitle = "Nonlinear dispersive waves and fluids",

address = "United States",

note = "AMS Special Sessions on Spectral Calculus<br/>and Quasilinear Partial Differential Equations,<br/>and PDE Analysis on Fluid Flows ; Conference date: 05-01-2017 Through 07-01-2017",

}

Garrisi, D & Georgiev, V 2019, Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one. in S Zheng, M Beceanu, JL Bona, G Chen, T Phan & A Soffer (eds), Nonlinear dispersive waves and fluids : AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows, January 5-6, 2017, Atlanta, Georgia. Contemporary mathematics, vol. 725, American Mathematical Society, pp. 137-148, AMS Special Sessions on Spectral Calculus
and Quasilinear Partial Differential Equations,
and PDE Analysis on Fluid Flows, Atlanta, Georgia, United States, 5/01/17.

Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one. / Garrisi, Daniele; Georgiev, Vladimir.

Nonlinear dispersive waves and fluids : AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows, January 5-6, 2017, Atlanta, Georgia. ed. / Shijun Zheng; Marius Beceanu; J. L. Bona; Geng Chen; Tuoc Phan; Avy Soffer. American Mathematical Society, 2019. p. 137-148 (Contemporary mathematics; Vol. 725).

Research output: Chapter in Book/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one

AU - Garrisi, Daniele

AU - Georgiev, Vladimir

PY - 2019

Y1 - 2019

N2 - We show that symmetric and positive profiles of ground-state standing-wave of the non-linear Schr¨odinger equation are non-degenerate and unique up to a translation of the argument and multiplication by complex numbers in the unit sphere. The non-linear term is a combination of two or three pure-powers. The class of non-linearities satisfying the mentioned properties can be extended beyond two or three power combinations. Specifically, it is sufficient that an Euler differential inequality is satisfied and that a certain auxiliary function is such that the first local maximum is also an absolute maximum.

AB - We show that symmetric and positive profiles of ground-state standing-wave of the non-linear Schr¨odinger equation are non-degenerate and unique up to a translation of the argument and multiplication by complex numbers in the unit sphere. The non-linear term is a combination of two or three pure-powers. The class of non-linearities satisfying the mentioned properties can be extended beyond two or three power combinations. Specifically, it is sufficient that an Euler differential inequality is satisfied and that a certain auxiliary function is such that the first local maximum is also an absolute maximum.

KW - Wave equation

KW - Nonlinear waves

KW - Nonlinear wave equations

UR - https://www.ams.org/books/conm/725/conm725-endmatter.pdf

M3 - Conference contribution

SN - 9781470441098

T3 - Contemporary mathematics

SP - 137

EP - 148

BT - Nonlinear dispersive waves and fluids

A2 - Zheng, Shijun

A2 - Beceanu, Marius

A2 - Bona, J. L.

A2 - Chen, Geng

A2 - Phan, Tuoc

A2 - Soffer, Avy

PB - American Mathematical Society

T2 - AMS Special Sessions on Spectral Calculus<br/>and Quasilinear Partial Differential Equations,<br/>and PDE Analysis on Fluid Flows

Y2 - 5 January 2017 through 7 January 2017

ER -

Garrisi D, Georgiev V. Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one. In Zheng S, Beceanu M, Bona JL, Chen G, Phan T, Soffer A, editors, Nonlinear dispersive waves and fluids : AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations, and PDE Analysis on Fluid Flows, January 5-6, 2017, Atlanta, Georgia. American Mathematical Society. 2019. p. 137-148. (Contemporary mathematics).

Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one

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