Stability of Periodic Travelling Wave Solutions to the Kawahara Equation

We analyze the stability of periodic, traveling-wave solutions to the Kawahara equation and some of its generalizations. We determine the parameter regime for which these solutions can exhibit reso...

[1]  Bernard Deconinck,et al.  Computing spectra of linear operators using the Floquet-Fourier-Hill method , 2006, J. Comput. Phys..

[2]  Todd Kapitula,et al.  On the spectra of periodic waves for infinite-dimensional Hamiltonian systems , 2008 .

[3]  B. Deconinck,et al.  High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs , 2015, 1505.03688.

[4]  J. Wilkening,et al.  The instability of Wilton ripples , 2015, 1512.01562.

[5]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[6]  W. Gao,et al.  Wilton ripples in weakly nonlinear model equations , 2012 .

[7]  Eric Lombardi,et al.  Spectral stability of wave trains in the Kawahara equation , 2006 .

[8]  Takuji Kawahara,et al.  Oscillatory Solitary Waves in Dispersive Media , 1972 .

[9]  D. Henderson,et al.  RESONANT INTERACTIONS AMONG SURFACE WATER WAVES , 1993 .

[10]  Sue Ellen Haupt,et al.  Modeling nonlinear resonance: A modification to the stokes' perturbation expansion , 1988 .

[11]  David P. Nicholls.  Spectral data for travelling water waves: singularities and stability , 2009, Journal of Fluid Mechanics.

[12]  J. A. Zufiria.  Weakly nonlinear non-symmetric gravity waves on water of finite depth , 1987, Journal of Fluid Mechanics.

[13]  David P. Nicholls.  Spectral Stability of Traveling Water Waves: Analytic Dependence of the Spectrum , 2007, J. Nonlinear Sci..

[14]  B. Akers.  Modulational instabilities of periodic traveling waves in deep water , 2015 .

[15]  P. Saffman,et al.  Stability of water waves , 1986, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[16]  Bernard Deconinck,et al.  On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations , 2015 .

[17]  D. Nicholls,et al.  Spectral Stability of Deep Two‐Dimensional Gravity‐Capillary Water Waves , 2013 .

[18]  Guido Schneider,et al.  The Rigorous Approximation of Long-Wavelength Capillary-Gravity Waves , 2002 .

[19]  Thomas J. Bridges,et al.  Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework , 2002 .

[20]  C. S. Gardner,et al.  Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion , 1968 .

[21]  John W. McLean,et al.  Instabilities of finite-amplitude water waves , 1982, Journal of Fluid Mechanics.

[22]  Mathew A. Johnson,et al.  On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions , 2010, 1003.3282.