American Institute of Mathematical Sciences

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June  2012, 5(2): 261-281. doi: 10.3934/krm.2012.5.261

A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation

Hua Chen 1, and  Ling-Jun Wang 2,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072
2. 

College of Science, Wuhan University of Science and Technology, Wuhan 430065

Received  March 2011 Revised  December 2011 Published  April 2012

We study the spectral stability of the one-dimensional small amplitude periodic traveling wave solutions of the Zakharov-Kuznetsov equation with respect to two-dimensional perturbations, which are either periodic in the direction of propagation with the same period as the one-dimensional underlying traveling wave, or non-periodic (localized or bounded). Relying upon the perturbation theory for linear operators with periodic coefficients, we show that the small periodic traveling waves are transversely spectrally unstable, with respect to both types of perturbations.
Keywords: ZK equation, small periodic traveling wave., spectral stability.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 37K4.
Citation: Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic and Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261
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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