American Institute of Mathematical Sciences

Advanced
July  2018, 17(4): 1371-1385. doi: 10.3934/cpaa.2018067

On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion

Wen Feng , Milena Stanislavova and  Atanas Stefanov ,

Department of Mathematics, University of Kansas, 1460 Jayhawk Boulevard, Lawrence, KS 66045, USA

* Corresponding author

Received  January 2017 Revised  June 2017 Published  April 2018

Fund Project: Feng is supported by a graduate fellowship under grant # 1516245. Stanislavova is partially supported by NSF-DMS, Applied Mathematics program, under grant # 1516245. Stefanov is supported by NSF-DMS, Applied Mathematics program, under grant # 1614734.

We consider standing wave solutions of various dispersive models with non-standard form of the dispersion terms. Using index count calculations, together with the information from a variational construction, we develop sharp conditions for spectral stability of these waves.

Keywords: Spectral stability, standing waves, fractional NLS, fractional KleinGordon equation.
Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 35G30.
Citation: Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
References:
[1]

L. Abdelouhab, J. Bona, M. Felland and J. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360-392.   Google Scholar

[2]

J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Comm. PDE, 17 (1992), 1-22.   Google Scholar

[3]

J. Albert and J. Bona, Total positivity and stability of internal waves in stratified fluids of finite depth, IMA J. Appl. Math., 46 (1991), 1-19.   Google Scholar

[4]

J. Albert, J. Bona and D. Henry, Sufficient conditions for instability of solitary wave solutions of model equation for long waves, Phys. D, 24 (1987), 343-366.   Google Scholar

[5]

T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.   Google Scholar

[6]

Y. Cho, G. Hwang, H. Hajaiej and T. Ozawa, On the orbital stability of fractional Schrö dinger equations, Comm. Pure Appl. Anal, 13 (2014), 1267-1282.   Google Scholar

[7]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in R, Acta Math., 210 (2013), 261-318.   Google Scholar

[8]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions of the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.   Google Scholar

[9]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. PDE, 36 (2011), 247-255.   Google Scholar

[10]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal. 14 (2015), no. 6, 2265-2282. Google Scholar

[11]

T. M. Kapitula, P. G. Kevrekidis and B. Sandstede, Counting eigenvalues via Krein signature in infinite-dimensional Hamitonial systems, Physica D, 3-4 (2004), 263-282.   Google Scholar

[12]

T. Kapitula, P. G. Kevrekidis and B. Sandstede, Addendum: "Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems" [Phys. D 195 (2004) no. 3-4,263-282], Phys. D, 201 (2005), 199-201.   Google Scholar

[13]

T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, 185, Applied Mathematical Sciences, 2013. Google Scholar

[14]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.   Google Scholar

[15]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.   Google Scholar

[16]

M. K. Kwong, Uniqueness of positive solutions of ∆u-u+up = 0 in Rn, Arch. Rational Mech. Anal., 105 (1989), 243-266.   Google Scholar

[17]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 3 (2002), 56-108.   Google Scholar

[18]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, American Mathematical Society, 2 edition, 2001. Google Scholar

[19]

F. Natali and A. Pastor, The fourth order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave, SIAM J. Applied Dyn. Sys., 14 (2015), 1326-1347.   Google Scholar

show all references

References:
[1]

L. Abdelouhab, J. Bona, M. Felland and J. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360-392.   Google Scholar

[2]

J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Comm. PDE, 17 (1992), 1-22.   Google Scholar

[3]

J. Albert and J. Bona, Total positivity and stability of internal waves in stratified fluids of finite depth, IMA J. Appl. Math., 46 (1991), 1-19.   Google Scholar

[4]

J. Albert, J. Bona and D. Henry, Sufficient conditions for instability of solitary wave solutions of model equation for long waves, Phys. D, 24 (1987), 343-366.   Google Scholar

[5]

T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.   Google Scholar

[6]

Y. Cho, G. Hwang, H. Hajaiej and T. Ozawa, On the orbital stability of fractional Schrö dinger equations, Comm. Pure Appl. Anal, 13 (2014), 1267-1282.   Google Scholar

[7]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in R, Acta Math., 210 (2013), 261-318.   Google Scholar

[8]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions of the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.   Google Scholar

[9]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. PDE, 36 (2011), 247-255.   Google Scholar

[10]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal. 14 (2015), no. 6, 2265-2282. Google Scholar

[11]

T. M. Kapitula, P. G. Kevrekidis and B. Sandstede, Counting eigenvalues via Krein signature in infinite-dimensional Hamitonial systems, Physica D, 3-4 (2004), 263-282.   Google Scholar

[12]

T. Kapitula, P. G. Kevrekidis and B. Sandstede, Addendum: "Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems" [Phys. D 195 (2004) no. 3-4,263-282], Phys. D, 201 (2005), 199-201.   Google Scholar

[13]

T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, 185, Applied Mathematical Sciences, 2013. Google Scholar

[14]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.   Google Scholar

[15]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.   Google Scholar

[16]

M. K. Kwong, Uniqueness of positive solutions of ∆u-u+up = 0 in Rn, Arch. Rational Mech. Anal., 105 (1989), 243-266.   Google Scholar

[17]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 3 (2002), 56-108.   Google Scholar

[18]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, American Mathematical Society, 2 edition, 2001. Google Scholar

[19]

F. Natali and A. Pastor, The fourth order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave, SIAM J. Applied Dyn. Sys., 14 (2015), 1326-1347.   Google Scholar

[1]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359
[2]

Riccardo Adami, Diego Noja, Cecilia Ortoleva. Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5837-5879. doi: 10.3934/dcds.2016057
[3]

Salvador Cruz-García, Catherine García-Reimbert. On the spectral stability of standing waves of the one-dimensional $M^5$-model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1079-1099. doi: 10.3934/dcdsb.2016.21.1079
[4]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525
[5]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
[6]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111
[7]

François Genoud. Orbitally stable standing waves for the asymptotically linear one-dimensional NLS. Evolution Equations & Control Theory, 2013, 2 (1) : 81-100. doi: 10.3934/eect.2013.2.81
[8]

Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010
[9]

Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121
[10]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[11]

Aslihan Demirkaya, Milena Stanislavova. Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 197-209. doi: 10.3934/dcdsb.2018097
[12]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007
[13]

Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109
[14]

François Genoud, Charles A. Stuart. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 137-186. doi: 10.3934/dcds.2008.21.137
[15]

Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389
[16]

Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic & Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261
[17]

Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193
[18]

Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018
[19]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039
[20]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (95)
  • HTML views (153)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences