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

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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 & 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+u^p = 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+u^p = 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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