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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 and 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. 

[2]

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

[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. 

[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. 

[5]

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

[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. 

[7]

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

[8]

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

[9]

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

[10]

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

[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. 

[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. 

[13]

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

[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. 

[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. 

[16]

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

[17]

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

[18]

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

[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. 

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. 

[2]

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

[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. 

[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. 

[5]

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

[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. 

[7]

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

[8]

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

[9]

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

[10]

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

[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. 

[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. 

[13]

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

[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. 

[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. 

[16]

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

[17]

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

[18]

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

[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. 

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