# American Institute of Mathematical Sciences

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

 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.

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. [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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##### 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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