Spectral stability of wave trains in the Kawahara equation

Journal of Mathematical Fluid Mechanics

Volumn 8, Issue 4, 2006, Pages 482-509

  Haragus, Mariana ^a   Lombardi, Eric ^b   Scheel, Arnd ^c  

^a UNIVERSITÉ DE FRANCHE COMTÉ   (France)

^b INSTITUT FOURIER   (France)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Dispersive equations; Periodic solutions; Stability

Indexed keywords

APPROXIMATION THEORY; MATHEMATICAL MODELS; MECHANICAL WAVES; PARTIAL DIFFERENTIAL EQUATIONS;

DISPERSIVE EQUATIONS; KAWAHARA EQUATION; PERIODIC SOLUTIONS; WEBER NUMBERS;

NONLINEAR EQUATIONS;

APPROXIMATION THEORY; MATHEMATICAL MODELS; MECHANICAL WAVES; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS;

NONLINEAR WAVE; NUMERICAL MODEL; SOLITARY WAVE; WATER WAVE; WAVE EQUATION; WAVE MODELING; WAVE PROPAGATION;

EID: 33845770343     PISSN: 14226928     EISSN: 14226952     Source Type: Journal    
DOI: 10.1007/s00021-005-0185-3     Document Type: Article

References (24)

1. C. J. AMICK and K. KIRCHGÄSSNER, A theory of solitary water-waves in the presence of surface tension, Arch. Rational Mech. Anal. 105 (1989), 1-49.
2. C. J. AMICK and J. B. MCLEOD, A singular perturbation problem in water waves, Stability Appl. Anal. Contin. Media 1 (1991), 127-148.
3. R. A. GARDNER, On the structure of the spectra of periodic travelling waves, J. Math. Pures Appl. 72 (1993), 415-439.
4. M. HARAGUS and A. SCHEEL, Finite-wavelength stability of capillary-gravity solitary waves, Comm. Math. Phys. 225 (2002), 487-521.
5. J. K. HUNTER, and J. SCHEURLE, Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D 32 (1988), 253-268.
6. A. T. IL'ICHEV and A. Y. SEMENOV, Stability of solitary waves in dispersive media described by a fifth-order evolution equation, Theoret. Comput. Fluid Dynamics 3 (1992), 307-326.
7. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980.
8. E. LOMBARDI, Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves, Arch. Rational Mech. Anal. 137 (1997), 227-304.
9. E. LOMBARDI, Non-persistence of homoclinic connections for perturbed integrable reversible systems, J. Dynam. Differential Equations 11 (1999), 129-208.
10. E. LOMBARDI, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible systems, Lecture Notes in Mathematics 1741, Springer-Verlag, Berlin, 2000.
11. T. KAWAHARA, Oscillatory solitary waves in dispersive media, Phys. Soc. Japan 33 (1972), 260-264.
12. F. MERLE and L. VEGA, L2 stability of solitons for KdV equation, Int. Math. Res. Not. 13 (2003), 735-753.
13. A. MIELKE, On the energetic stability of solitary water waves, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 360 (2002), 2337-2358.
14. M. OH and K. ZUMBRUN, Stability of periodic solutions of conservation laws with viscosity: Analysis of the Evans function, Arch. Rational Mech. Anal. 166 (2003), 99-166.
15. M. OH and K. ZUMBRUN, Stability of periodic solutions of conservation laws with viscosity: Pointwise bounds on the Green function, Arch. Rational Msch. Anal. 166 (2003), 167-196.
16. R. L. PEGO and M. I. WEINSTEIN, Asymptotic stability of solitary waves, Comm. Math. Phys. 164 (1994), 305-349.
17. M. REED and B. SIMON, Methods of Modern Mathematical Physics IV, Academic Press, New York, 1978.
18. 2-perturbations of periodic equilibria of reaction diffusion systems, Non-linear Diff. Eqns. Appl. (NoDEA) 3 (1994), 281-311.
19. G. SCHNEIDER, Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys. 178 (1996), 679-702.
20. G. SCHNEIDER and C. E. WAYNE, The rigorous approximation of long-wavelength capillarygravity waves, Arch. Rational Mech. Anal. 162 (2002), 247-285.
21. D. SERRE, The stability of periodic solutions of viscous conservation laws: Large wavelength perturbations, Comm. PDE 30 (2005), 259-282.
22. S. M. SUN and M. C. SHEN, Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave, J. Math. Anal. Appl. 172 (1993), 533-566.
23. S. M. SUN, Non-existence of truly solitary waves in water with small surface tension, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), 2191-2228.
24. G. B. WHITHAM, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974.