doi: 10.3934/dcdsb.2022039
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 Cauchy problem for a nonlocal nonlinear Schrödinger equation

1. 

School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China

2. 

Department of Mathematics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan

*Corresponding author: Amin Esfahani

Received  August 2021 Revised  January 2022 Early access March 2022

Fund Project: H. W. was supported by Key Project of Natural Science Foundation of Educational Committee of Henan Province(No. 20A110007). A. E. was supported by the Social Policy Grant (SPG) funded by Nazarbayev University, Kazakhstan

This paper considers the one-dimensional Schrödinger equation with nonlocal nonlinearity that describes the interactions of nonlinear dispersive waves. We obtain some the local well-posedness and ill-posedness result associated with this equation in the Sobolev spaces. Moreover, we prove the existence of standing waves of this equation. As corollary, we derive the conditions under which the solutions are uniformly bounded in the energy space.

Citation: Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022039
References:
[1]

P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. 

[2]

H. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4.

[3]

J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.  doi: 10.1016/S0021-7824(97)89957-6.

[4]

D. CaiA. MajdaD. McLaughlin and E. Tabak, Dispersive wave turbulence in one dimension, Phys. D., 152/153 (2001), 551-572.  doi: 10.1016/S0167-2789(01)00193-2.

[5]

Y. ChoG. HwangS. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.

[6]

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

[7]

A. Erdélyi, Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, 1954.

[8]

A. Esfahani, Anisotropic Gagliardo-Nirenberg inequality with fractional derivatives, Z. Angew. Math. Phys., 66 (2015), 3345-3356.  doi: 10.1007/s00033-015-0586-y.

[9]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math, 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[10]

P. Gérard, Description du defaut de compacite de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107.

[11]

A. Grunrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Notic., 41 (2005), 2525-2558.  doi: 10.1155/imrn.2005.2525.

[12]

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

[13]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equation revisited, Int. Math. Res. Notic., (2005), 2815–2828. doi: 10.1155/imrn.2005.2815.

[14]

C. E. KenigY. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré, 28 (2011), 853-887.  doi: 10.1016/j.anihpc.2011.06.005.

[15]

F. LinaresD. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.

[16]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^nd$ edition, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[17]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincare Anal. Non-linéaire, 1 (1984), 109-145.  doi: 10.1016/s0294-1449(16)30428-0.

[18]

A. MajdaD. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.  doi: 10.1007/BF02679124.

[19]

S. Oh and A. Stefanov, On quadratic Schrödinger equations on $\mathbb{R}^{1+1}$: A normal form approach, J. London Math. Soc., 86 (2012), 499-519.  doi: 10.1112/jlms/jds016.

[20]

H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differ. Equ., 4 (1999), 561-580. 

[21]

H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Equations, (2001), 1–23.

[22]

T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.

[23]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.

[24]

K. TrulsenI. KliakhandlerK. Dysthe and M. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids, 12 (2000), 2432-2437.  doi: 10.1063/1.1287856.

show all references

References:
[1]

P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. 

[2]

H. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4.

[3]

J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.  doi: 10.1016/S0021-7824(97)89957-6.

[4]

D. CaiA. MajdaD. McLaughlin and E. Tabak, Dispersive wave turbulence in one dimension, Phys. D., 152/153 (2001), 551-572.  doi: 10.1016/S0167-2789(01)00193-2.

[5]

Y. ChoG. HwangS. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.

[6]

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

[7]

A. Erdélyi, Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, 1954.

[8]

A. Esfahani, Anisotropic Gagliardo-Nirenberg inequality with fractional derivatives, Z. Angew. Math. Phys., 66 (2015), 3345-3356.  doi: 10.1007/s00033-015-0586-y.

[9]

R. L. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math, 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[10]

P. Gérard, Description du defaut de compacite de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107.

[11]

A. Grunrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Notic., 41 (2005), 2525-2558.  doi: 10.1155/imrn.2005.2525.

[12]

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

[13]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equation revisited, Int. Math. Res. Notic., (2005), 2815–2828. doi: 10.1155/imrn.2005.2815.

[14]

C. E. KenigY. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré, 28 (2011), 853-887.  doi: 10.1016/j.anihpc.2011.06.005.

[15]

F. LinaresD. Pilod and J.-C. Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal., 46 (2014), 1505-1537.  doi: 10.1137/130912001.

[16]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^nd$ edition, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[17]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincare Anal. Non-linéaire, 1 (1984), 109-145.  doi: 10.1016/s0294-1449(16)30428-0.

[18]

A. MajdaD. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.  doi: 10.1007/BF02679124.

[19]

S. Oh and A. Stefanov, On quadratic Schrödinger equations on $\mathbb{R}^{1+1}$: A normal form approach, J. London Math. Soc., 86 (2012), 499-519.  doi: 10.1112/jlms/jds016.

[20]

H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differ. Equ., 4 (1999), 561-580. 

[21]

H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Equations, (2001), 1–23.

[22]

T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.

[23]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.

[24]

K. TrulsenI. KliakhandlerK. Dysthe and M. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids, 12 (2000), 2432-2437.  doi: 10.1063/1.1287856.

Figure 1.  The graphs of $ K $ with negative values of $ \beta $ are shown in the left and in the right figures with both signs
Figure 2.  Plots of ground states of (1) with $ \omega = 1 $ and various values of $ \beta $
[1]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100

[2]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093

[3]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082

[4]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations and Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15

[5]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[6]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[7]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure and Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261

[8]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072

[9]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123

[10]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37

[11]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010

[12]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023

[13]

Chao Yang. Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4631-4642. doi: 10.3934/dcdss.2021136

[14]

Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122

[15]

Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2721-2757. doi: 10.3934/dcdsb.2021156

[16]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205

[17]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905

[18]

Shaoming Guo, Xianfeng Ren, Baoxiang Wang. Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4207-4253. doi: 10.3934/dcds.2021034

[19]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863

[20]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure and Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (234)
  • HTML views (110)
  • Cited by (0)

Other articles
by authors

[Back to Top]