On the orbital stability of fractional Schrödinger equations

Previous Article
On improvement of summability properties in nonautonomous Kolmogorov equations
CPAA Home
This Issue
Next Article
On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

May 2014, 13(3): 1267-1282. doi: 10.3934/cpaa.2014.13.1267

On the orbital stability of fractional Schrödinger equations

Yonggeun Cho ^1, , Hichem Hajaiej ^2, , Gyeongha Hwang ^3, and Tohru Ozawa ^4,

1.	Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2.	Department of Mathematics, King Saud University, P.O. Box 2455, 11451 Riyadh, Saudi Arabia
3.	Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea
4.	Department of Applied Physics, Waseda University, Tokyo, 169-8555

Received June 2013 Revised November 2013 Published December 2013

Abstract
Related Papers

We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.

Keywords: Fractional Schrödinger equation, finite time blowup., Hartree type nonlinearity, Strichartz estimates.

Mathematics Subject Classification: Primary: 35Q40, 35Q55, 47J3.

Citation: Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1267-1282. doi: 10.3934/cpaa.2014.13.1267

References:

[1]	J. P. Borgna and D. F. Rial, Existence of ground states for a one dimensional relativistic Schrodinger equations,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4726198.
[2]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).
[3]	Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity,, \emph{Funkcialaj Ekvacioj}, 56 (2013), 193.
[4]	Y. Cho, G. Hwang and T. Ozawa, Global well-posedness of critical nonlinear Schrödinger equations below $L^2$,, \emph{DCDS-A}, 33 (2013), 1389. doi: 10.3934/dcds.2013.33.1389.
[5]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, to appear in \emph{Indina Univ. Math. J.}, ().
[6]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry,, \emph{Contem. Math.}, 11 (2009), 355. doi: 10.1142/S0219199709003399.
[7]	Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1121. doi: 10.3934/cpaa.2011.10.1121.
[8]	Daoyuan Fang and Chengbo Wang, Weighted strichartz estimates with angular regularity and their applications,, \emph{Forum Mathematicum}, 23 (2011), 181. doi: 10.1515/FORM.2011.009.
[9]	P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, \emph{P. Roy. Soc. Edinburgh A}, 142 (2012), 1237. doi: 10.1017/S0308210511000746.
[10]	B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4746806.
[11]	H. Hajaiej, Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity,, \emph{J. Math. Anal. Appl.}, 399 (2013), 17. doi: 10.1016/j.jmaa.2012.09.023.
[12]	H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gargliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,, \emph{RIMS Kokyuroku Bessatsu}, B26 (2011), 159.
[13]	A. D. Ionescu and F. Pusateri, Nolinear fractional Schrödinger equations in one dimension,, \emph{To appear in J. Funct. Anal.}, (). doi: 10.1016/j.jfa.2013.08.027.
[14]	P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, II,, \emph{Ann. Inst. H. Poincare' Anal. Non Line'aire}, 1 (1984), 109.
[15]	T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations,, \emph{Cal. Var. PDE.}, 25 (2006), 403. doi: 10.1007/s00526-005-0349-2.

show all references

References:

[1]	J. P. Borgna and D. F. Rial, Existence of ground states for a one dimensional relativistic Schrodinger equations,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4726198.
[2]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).
[3]	Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity,, \emph{Funkcialaj Ekvacioj}, 56 (2013), 193.
[4]	Y. Cho, G. Hwang and T. Ozawa, Global well-posedness of critical nonlinear Schrödinger equations below $L^2$,, \emph{DCDS-A}, 33 (2013), 1389. doi: 10.3934/dcds.2013.33.1389.
[5]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, to appear in \emph{Indina Univ. Math. J.}, ().
[6]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry,, \emph{Contem. Math.}, 11 (2009), 355. doi: 10.1142/S0219199709003399.
[7]	Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1121. doi: 10.3934/cpaa.2011.10.1121.
[8]	Daoyuan Fang and Chengbo Wang, Weighted strichartz estimates with angular regularity and their applications,, \emph{Forum Mathematicum}, 23 (2011), 181. doi: 10.1515/FORM.2011.009.
[9]	P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, \emph{P. Roy. Soc. Edinburgh A}, 142 (2012), 1237. doi: 10.1017/S0308210511000746.
[10]	B. Guo and D. Huang, Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4746806.
[11]	H. Hajaiej, Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity,, \emph{J. Math. Anal. Appl.}, 399 (2013), 17. doi: 10.1016/j.jmaa.2012.09.023.
[12]	H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gargliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,, \emph{RIMS Kokyuroku Bessatsu}, B26 (2011), 159.
[13]	A. D. Ionescu and F. Pusateri, Nolinear fractional Schrödinger equations in one dimension,, \emph{To appear in J. Funct. Anal.}, (). doi: 10.1016/j.jfa.2013.08.027.
[14]	P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, II,, \emph{Ann. Inst. H. Poincare' Anal. Non Line'aire}, 1 (1984), 109.
[15]	T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations,, \emph{Cal. Var. PDE.}, 25 (2006), 403. doi: 10.1007/s00526-005-0349-2.

[1]	Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[2]	Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050
[3]	Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905
[4]	Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[5]	Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210
[6]	Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233
[7]	Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109
[8]	Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[9]	Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323
[10]	Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
[11]	Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475
[12]	Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177
[13]	Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
[14]	Pengyu Chen, Xuping Zhang, Yongxiang Li. A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1975-1992. doi: 10.3934/cpaa.2018094
[15]	Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085
[16]	Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639
[17]	Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[18]	Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102
[19]	Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149
[20]	Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences