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

Previous Article
Boltzmann equation and hydrodynamics at the Burnett level
KRM Home
This Issue
Next Article
Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension

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

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

Abstract
Related Papers

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 & 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.
[2]	J. Angulo Pava, Jerry L. Bona and M. Scialom, Stability of cnoidal waves,, Adv. Differential Equations, 11 (2006), 1321.
[3]	N. Bottman and B. Deconinck, KdV cnoidal waves are spectrally stable,, Discrete Contin. Dyn. Syst., 25 (2009), 1163. 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.
[5]	M. Hărăguş, Transverse spectral stability of small periodic traveling waves for the KP equation,, Stud. Appl. Math., 126 (2011), 157. 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. 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. 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. 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. doi: 10.1137/090770758.
[10]	Tosio Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition, (1980).
[11]	A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation,, Comm. Math. Phys., 189 (1997), 829. 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.
[2]	J. Angulo Pava, Jerry L. Bona and M. Scialom, Stability of cnoidal waves,, Adv. Differential Equations, 11 (2006), 1321.
[3]	N. Bottman and B. Deconinck, KdV cnoidal waves are spectrally stable,, Discrete Contin. Dyn. Syst., 25 (2009), 1163. 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.
[5]	M. Hărăguş, Transverse spectral stability of small periodic traveling waves for the KP equation,, Stud. Appl. Math., 126 (2011), 157. 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. 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. 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. 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. doi: 10.1137/090770758.
[10]	Tosio Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition, (1980).
[11]	A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation,, Comm. Math. Phys., 189 (1997), 829. doi: 10.1007/s002200050230.

[1]	Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
[2]	Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507
[3]	Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[4]	Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841
[5]	M. M. Cavalcanti, V.N. Domingos Cavalcanti, D. Andrade, T. F. Ma. Homogenization for a nonlinear wave equation in domains with holes of small capacity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 721-743. doi: 10.3934/dcds.2006.16.721
[6]	Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
[7]	Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925
[8]	Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[9]	Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359
[10]	Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265
[11]	Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[12]	Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455
[13]	Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065
[14]	Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084
[15]	Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[16]	Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[17]	Lina Wang, Xueli Bai, Yang Cao. Exponential stability of the traveling fronts for a viscous Fisher-KPP equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 801-815. doi: 10.3934/dcdsb.2014.19.801
[18]	Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823
[19]	Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[20]	Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815

2018 Impact Factor: 1.38

American Institute of Mathematical Sciences