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A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation
1. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072 |
2. | College of Science, Wuhan University of Science and Technology, Wuhan 430065 |
References:
[1] |
J. Angulo Pava, Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Differential Equations, 235 (2007), 1-30. |
[2] |
J. Angulo Pava, Jerry L. Bona and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374. |
[3] |
N. Bottman and B. Deconinck, KdV cnoidal waves are spectrally stable, Discrete Contin. Dyn. Syst., 25 (2009), 1163-1180.
doi: 10.3934/dcds.2009.25.1163. |
[4] |
T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Differential Equations, 234 (2007), 544-581. |
[5] |
M. Hărăguş, Transverse spectral stability of small periodic traveling waves for the KP equation, Stud. Appl. Math., 126 (2011), 157-185.
doi: 10.1111/j.1467-9590.2010.00501.x. |
[6] |
M. Hărăguş and T. Kapitula, On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D, 237 (2008), 2649-2671.
doi: 10.1016/j.physd.2008.03.050. |
[7] |
M. Haragus, E. Lombardi and A. Scheel, Spectral stability of wave trains in the Kawahara equation, J. Math. Fluid Mech., 8 (2006), 482-509.
doi: 10.1007/s00021-005-0185-3. |
[8] |
M. A. Johnson, The transverse instability of periodic waves in Zakharov-Kuznetsov type equations, Stud. Appl. Math., 124 (2010), 323-345.
doi: 10.1111/j.1467-9590.2009.00473.x. |
[9] |
M. A. Johnson and K. Zumbrun, Transverse instability of periodic traveling waves in the generalized Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 42 (2010), 2681-2702.
doi: 10.1137/090770758. |
[10] |
Tosio Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. |
[11] |
A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853.
doi: 10.1007/s002200050230. |
show all references
References:
[1] |
J. Angulo Pava, Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Differential Equations, 235 (2007), 1-30. |
[2] |
J. Angulo Pava, Jerry L. Bona and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374. |
[3] |
N. Bottman and B. Deconinck, KdV cnoidal waves are spectrally stable, Discrete Contin. Dyn. Syst., 25 (2009), 1163-1180.
doi: 10.3934/dcds.2009.25.1163. |
[4] |
T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Differential Equations, 234 (2007), 544-581. |
[5] |
M. Hărăguş, Transverse spectral stability of small periodic traveling waves for the KP equation, Stud. Appl. Math., 126 (2011), 157-185.
doi: 10.1111/j.1467-9590.2010.00501.x. |
[6] |
M. Hărăguş and T. Kapitula, On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D, 237 (2008), 2649-2671.
doi: 10.1016/j.physd.2008.03.050. |
[7] |
M. Haragus, E. Lombardi and A. Scheel, Spectral stability of wave trains in the Kawahara equation, J. Math. Fluid Mech., 8 (2006), 482-509.
doi: 10.1007/s00021-005-0185-3. |
[8] |
M. A. Johnson, The transverse instability of periodic waves in Zakharov-Kuznetsov type equations, Stud. Appl. Math., 124 (2010), 323-345.
doi: 10.1111/j.1467-9590.2009.00473.x. |
[9] |
M. A. Johnson and K. Zumbrun, Transverse instability of periodic traveling waves in the generalized Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 42 (2010), 2681-2702.
doi: 10.1137/090770758. |
[10] |
Tosio Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. |
[11] |
A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853.
doi: 10.1007/s002200050230. |
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