doi: 10.3934/dcds.2021203
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

On the one dimensional cubic NLS in a critical space

BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country - Spain

*Corresponding author: Marco Bravin

Received  July 2021 Early access January 2022

Fund Project: Marco Bravin is supported by ERC-2014-ADG project HADE Id. 669689 (European Research Council). Luis Vega is supported by ERC-2014-ADG project HADE Id. 669689 (European Research Council), MINECO grant BERC 2018-2021, PGC2018-094522-B-I00 and SEV-2017-0718 (Spain)

In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.

Citation: Marco Bravin, Luis Vega. On the one dimensional cubic NLS in a critical space. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021203
References:
[1]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc. (JEMS), 14 (2012), 209-253.  doi: 10.4171/JEMS/300.  Google Scholar

[2]

V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament, Arch. Ration. Mech. Anal., 210 (2013), 673-712.  doi: 10.1007/s00205-013-0660-6.  Google Scholar

[3]

V. Banica and L. Vega, Evolution of polygonal lines by the binormal flow, Annals of PDE, 6 (2020). doi: 10.1007/s40818-020-0078-z.  Google Scholar

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[5]

R. Carles and T. Kappeler, Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces, Bull. Soc. Math. France, 145 (2017), 623-642.   Google Scholar

[6]

M. ChristJ. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.  doi: 10.1353/ajm.2003.0040.  Google Scholar

[7]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo, 22 (1906), 117-135.   Google Scholar

[8]

F. De la Hoz and L. Vega, Vortex filament equation for a regular polygon, Nonlinearity, 27 (2014), 3031-3057.  doi: 10.1088/0951-7715/27/12/3031.  Google Scholar

[9]

A. Grünrock, Bi-and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not., 2005 (2005), 2525-2558.  doi: 10.1155/IMRN.2005.2525.  Google Scholar

[10]

A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar

[11]

B. Harrop-Griffiths, R. Killip and M. Visan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, arXiv: 2003.05011. Google Scholar

[12]

H. Hasimoto, A solution on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.  doi: 10.1017/S0022112072002307.  Google Scholar

[13]

R. L. Jerrard and D. Smets, On the motion of a curve by its binormal curvature, J. Eur. Math. Soc., 17 (2015), 1487-1515.  doi: 10.4171/JEMS/536.  Google Scholar

[14]

C. E. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633.  doi: 10.1215/S0012-7094-01-10638-8.  Google Scholar

[15]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.   Google Scholar

[16]

N. Kita, Mode generating property of solutions to the nonlinear Schrödinger equations in one space dimension, GAKUTO Internat. Ser., Math. Sci. Appl., 26 (2006), 111-128.   Google Scholar

[17]

T. Oh, A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.  doi: 10.1619/fesi.60.259.  Google Scholar

[18]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.  doi: 10.1016/j.jde.2019.12.017.  Google Scholar

[19]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

[20]

A. Vargas and L. Vega, Global well-posedness for 1D non-linear Schrödinger equation for data with an infinite $L^2$ norm, J. Math. Pures Appl, 80 (2001), 1029-1044.  doi: 10.1016/S0021-7824(01)01224-7.  Google Scholar

show all references

References:
[1]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc. (JEMS), 14 (2012), 209-253.  doi: 10.4171/JEMS/300.  Google Scholar

[2]

V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament, Arch. Ration. Mech. Anal., 210 (2013), 673-712.  doi: 10.1007/s00205-013-0660-6.  Google Scholar

[3]

V. Banica and L. Vega, Evolution of polygonal lines by the binormal flow, Annals of PDE, 6 (2020). doi: 10.1007/s40818-020-0078-z.  Google Scholar

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[5]

R. Carles and T. Kappeler, Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces, Bull. Soc. Math. France, 145 (2017), 623-642.   Google Scholar

[6]

M. ChristJ. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.  doi: 10.1353/ajm.2003.0040.  Google Scholar

[7]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo, 22 (1906), 117-135.   Google Scholar

[8]

F. De la Hoz and L. Vega, Vortex filament equation for a regular polygon, Nonlinearity, 27 (2014), 3031-3057.  doi: 10.1088/0951-7715/27/12/3031.  Google Scholar

[9]

A. Grünrock, Bi-and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not., 2005 (2005), 2525-2558.  doi: 10.1155/IMRN.2005.2525.  Google Scholar

[10]

A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar

[11]

B. Harrop-Griffiths, R. Killip and M. Visan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, arXiv: 2003.05011. Google Scholar

[12]

H. Hasimoto, A solution on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.  doi: 10.1017/S0022112072002307.  Google Scholar

[13]

R. L. Jerrard and D. Smets, On the motion of a curve by its binormal curvature, J. Eur. Math. Soc., 17 (2015), 1487-1515.  doi: 10.4171/JEMS/536.  Google Scholar

[14]

C. E. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633.  doi: 10.1215/S0012-7094-01-10638-8.  Google Scholar

[15]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.   Google Scholar

[16]

N. Kita, Mode generating property of solutions to the nonlinear Schrödinger equations in one space dimension, GAKUTO Internat. Ser., Math. Sci. Appl., 26 (2006), 111-128.   Google Scholar

[17]

T. Oh, A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.  doi: 10.1619/fesi.60.259.  Google Scholar

[18]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.  doi: 10.1016/j.jde.2019.12.017.  Google Scholar

[19]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

[20]

A. Vargas and L. Vega, Global well-posedness for 1D non-linear Schrödinger equation for data with an infinite $L^2$ norm, J. Math. Pures Appl, 80 (2001), 1029-1044.  doi: 10.1016/S0021-7824(01)01224-7.  Google Scholar

[1]

Liren Lin, Tai-Peng Tsai. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 295-336. doi: 10.3934/dcds.2017013

[2]

Yvan Martel, Tiễn Vinh Nguyến. Construction of 2-solitons with logarithmic distance for the one-dimensional cubic Schrödinger system. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1595-1620. doi: 10.3934/dcds.2020087

[3]

Huifang Jia, Gongbao Li, Xiao Luo. Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2739-2766. doi: 10.3934/dcds.2020148

[4]

Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934/cpaa.2011.10.127

[5]

Alessandro Michelangeli. Strengthened convergence of marginals to the cubic nonlinear Schrödinger equation. Kinetic & Related Models, 2010, 3 (3) : 457-471. doi: 10.3934/krm.2010.3.457

[6]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803

[7]

Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379

[8]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036

[9]

Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565

[10]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[11]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284

[12]

Yuji Sagawa, Hideaki Sunagawa. The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5743-5761. doi: 10.3934/dcds.2016052

[13]

Christopher Chong, Dmitry Pelinovsky. Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1019-1031. doi: 10.3934/dcdss.2011.4.1019

[14]

Nan Lu. Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3533-3567. doi: 10.3934/dcds.2015.35.3533

[15]

Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237

[16]

Jason Murphy, Fabio Pusateri. Almost global existence for cubic nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2077-2102. doi: 10.3934/dcds.2017089

[17]

J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka, T. Tao. Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 665-686. doi: 10.3934/dcds.2008.21.665

[18]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2022, 15 (1) : 79-93. doi: 10.3934/dcdss.2021030

[19]

Benedetta Noris, Hugo Tavares, Gianmaria Verzini. Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6085-6112. doi: 10.3934/dcds.2015.35.6085

[20]

Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (31)
  • HTML views (28)
  • Cited by (0)

Other articles
by authors

[Back to Top]