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Small amplitude solitary waves in the Dirac-Maxwell system
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 |
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.
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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