Finite-wavelength stability of capillary-gravity solitary waves

Communications in Mathematical Physics

Volumn 225, Issue 3, 2002, Pages 487-521

  Haragus, Mariana ^a   Scheel, Arnd ^b  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036110969     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s002200100590     Document Type: Article

References (40)

1. [AGJ90] Alexander, J., Gardner, R. and Jones, C.K.R.T.: A topological invariant arising in the stability analysis of traveling waves. J. Reine Angew. Math. 410, 167-212 (1990)
2. [AK89] Amick, C.J, and Kirchgässner, K.: A theory of solitary water-waves in the presence of surface tension. Arch. Rational Mech. Anal. 105, 1-49 (1989)
3. [Be67] Benjamin, T.B.: Instability of periodic wavetrains in nonlinear dispersive systems. Proc. Roy. Soc. Lond. A 299, 59-75 (1967)
4. [Be72] Benjamin, T.B.: The stability of solitary waves. Proc. R. Soc. Lon. A 328, 153-183 (1972)
5. [BF67] Benjamin, T.B. and Feir, J.E.: The disintegration of wave trains on deep water, Part 1. J. Fluid Mech. 27, 417-430 (1967)
6. [BO80] Benjamin, T.B. and Olver, P.: Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137-185 (1982)
7. [BSS87] Bona, J.L., Souganidis, P.E. and Strauss, W.A.: Stability and instability of solitary waves of Korteweg-de Vries type. Proc. R. Soc. Lon. A 411, 395-412 (1987)
8. [Bou] Boussinesq, M.J.: Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants à l'Académie des Sciences Inst. France (séries 2) 23, 1-680 (1877)
9. [BM95] Bridges, T.J. and Mielke, A.: A proof of the Benjamin-Feir instability, Arch. Rational Mech. Anal. 133, 145-198 (1995)
10. [Co78] Coppel, W.A.: Dichotomies in stability theory. Lect. Notes Math. 629. Berlin: Springer, 1978
11. [Cr85] Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Diff. Eq. 10, 787-1003 (1985)
12. [Ev72] Evans, J.: Nerve axon equations (iii): Stability of the nerve impulses. Indiana Univ. Math. J. 22, 577-594 (1972)
13. [GZ98] Gardner, R. and Zumbrun, K. : The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51, 797-855 (1998)
14. [Ha96] Haragus, M.: Model equations for water waves in the presence of surface tension. Eur. J. Mech. B/Fluids 15, 471-492 (1996)
15. [HS01] Haragus, M. and Scheel, A.: Linear stability and instability of ion-acoustic plasma solitary waves. Preprint.
16. [IS92] Il'ichev, A.T. and Semenov, A.Y.: Stability of solitary waves in dispersive media described by a fifth-order evolution equation. Theoret. Comput. Fluid Dynamics 3, 307-326 (1992)
17. [KN79] Kano, T. and Nishida, T.: Sur les ondes de surface de l'eau avec une justification mathématique des équations des ondes en eau peu profonde. J. Math. Kyoto Univ. 19, 335-370 (1979)
18. [KN86] Kano, T. and Nishida, T.: A mathematical justification for Korteweg-de Vries and Boussinesq equation of water surface waves. Osaka J. Math. 23, 389-413 (1986)
19. [KS98] Kapitula, T. and Sandstede B.: Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Physica D124, 58-103 (1998)
20. [Ka72] T. Kawahara, Oscillatory solitary waves in dispersive media, Phys. Soc. Japan 33 (1972), 260-264.
21. [Ki88] Kirchgässner, K.: Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation. Adv. Appl. Mech. 26, 135-181 (1988)
22. [KdV] Korteweg, D.J. and de Vries, G.: On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves Phil. Mag. 5, 422-443 (1895)
23. [LH84] Longuet-Higgins, M.S.: On the stability of steep gravity waves. Proc. R. Soc. Lon. A 396, 269-280 (1984)
24. [LHT97] Longuet-Higgins, M.S. and Tanaka, M.: On the crest intabilities of steep surface waves. J. Fluid Mech. 336, 51-68 (1997)
25. [MS86] MacKay, R.S. and Saffman, P.G.: Stability of water waves. Proc. R. Soc. Lon. A 406, 115-125 (1986)
26. [Mc82] McLean, J.W.: Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315-330 (1982)
27. [Mi88] Mielke, A.: Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Sci. 10, 51-66 (1988)
28. [Na74] Nalimov,V.I.: The Cauchy-Poisson Problem (in Russian). Dynamik Splosh. Sredy 18, 104-210 (1974)
29. [PW92] Pego, R.L. and Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. Lond. A 340, 47-94 (1992)
30. [PW96] Pego, R.L. and Weinstein, M.I. : Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305-349 (1996)
31. [PW97] Pego, R.L. and Weinstein, M.I.: Convective linear stability of solitary waves for Boussinesq equations. Stud. Appl. Math. 99, 311-375 (1997)
32. [RS95] Robbin, J. and Salamon, D.: The spectral flow and the Maslov index. Bull. London Math. Soc. 27, 1-33 (1995)
33. [Sa91] Sachs, R.L.: On the existence of small amplitude solitary waves with strong surface tension. J. Diff. Equ. 90, 31-51 (1991)
34. [Sa85] Saffman, P.O.: The superharmonic instability of finite amplitude water waves. J. Fluid Mech. 159, 169-174 (1985)
35. [SW00] Schneider, G. and Wayne, C.E.: The long wave limit for the water wave problem I. The case of zero surface tension, Comm. Pure Appl. Math. 53, 1475-1535 (2000)
36. [Ta86] Tanaka, M.: The stability of solitary waves, Phys. Fluids 29, 650-655 (1986)
37. [Wh67] Whitham, G.B.: Nonlinear dispersion of water-waves, J. Fluid Mech. 27, 399-412 (1967)
38. [Wu97] Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39-72 (1997)
39. [Yo82] Yosihara, H.: Gravity waves on the free surface of an incompressible perfect fluid of finite depth. RIMS Kyoto 18, 49-96 (1982)
40. [Za68] Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190-194 (1968)