American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[2]

J. Angulo Pava, Jerry L. Bona and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374.  Google Scholar

[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.  Google Scholar

[4]

T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Differential Equations, 234 (2007), 544-581.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

Tosio Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[11]

A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853. doi: 10.1007/s002200050230.  Google Scholar

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.  Google Scholar

[2]

J. Angulo Pava, Jerry L. Bona and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374.  Google Scholar

[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.  Google Scholar

[4]

T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Differential Equations, 234 (2007), 544-581.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

Tosio Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[11]

A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853. doi: 10.1007/s002200050230.  Google Scholar

[1]

Guo Lin, Shuxia Pan. Periodic traveling wave solutions of periodic integrodifference systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049
[2]

Xiaoxiao Zheng, Hui Wu. Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components. Mathematical Foundations of Computing, 2020, 3 (1) : 11-24. doi: 10.3934/mfc.2020002
[3]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
[4]

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
[5]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129
[6]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[7]

Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841
[8]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
[9]

Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925
[10]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
[11]

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, 2006, 16 (4) : 721-743. doi: 10.3934/dcds.2006.16.721
[12]

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
[13]

Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084
[14]

Lijun Zhang, Peiying Yuan, Jingli Fu, Chaudry Masood Khalique. Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2927-2939. doi: 10.3934/dcdss.2020214
[15]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265
[16]

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
[17]

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
[18]

Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065
[19]

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
[20]

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

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences