Transverse instabilities of deep-water solitary waves

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 462, Issue 2071, 2006, Pages 2039-2061

  Deconinck, Bernard ^a   Pelinovsky, Dmitry E ^b   Carter, John D ^c  

^a UNIVERSITY OF WASHINGTON   (United States)

^b MCMASTER UNIVERSITY   (Canada)

^c SEATTLE UNIVERSITY   (United States)

Author keywords

Evans function; Hill's method; Hyperbolic nonlinear Schr dinger equation; Solitary wave; Spectral stability

Indexed keywords

EID: 33845430110     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2006.1670     Document Type: Article

References (25)

1. Ablowitz, M. J. & Segur, H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92, 691-715.
2. Ablowitz, M. J. & Segur, H. 1981 Solitons and the inverse scattering transform. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
3. Anderson, D., Bondeson, A. & Lisak, M. 1979 Stability analysis of lower-hybrid cones. Phys. Scripta 20, 343-345.
4. Chow, S. N. & Hale, J. K. 1982 Methods of bifurcation theory. New York: Springer.
5. Coddington, E. A. & Levinson, N. 1955 Theory of ordinary differential equations. New York: McGraw-Hill.
6. Cohen, B. I., Watson, K. M. & West, B. J. 1976 Some properties of deep-water solitons. Phys. Fluids 19, 345-354. (doi:10.1063/1.861481)
7. Deconinck, B. & Kutz, J.N. Submitted. Computing spectra of linear operators using Hill's method.
8. Gardner, R. A. & Zumbrun, K. 1998 The gap lemma and geometric criteria for instability of viscous shock profiles. Commun. Pure Appl. Math. 51, 797-855. (doi:10.1002/(SICI)1097-0312(199807)51:7〈797::AID-CPA3〉3.0. CO;2-1)
9. Janssen, P. A. E. M. & Rasmussen, J. J. 1983 Nonlinear evolution of the transverse instability of plane-envelope solitons. Phys. Fluids 26, 1279-1287. (doi:10.1063/1.864288)
10. Kapitula, T. & Sandstede, B. 1998 Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Physica D 124, 58-103. (doi:10.1016/S0167-2789(98)00172-9)
11. Kapitula, T. & Sandstede, B. 2002 Edge bifurcations for near integrable systems via Evans function techniques. SIAM J. Math. Anal. 33, 1117-1143. (doi:10.1137/S0036141000372301)
12. Kapitula, T., Kevrekidis, P. G. & Sandstede, B. 2004 Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems. Physica D 195, 263-282. (doi:10.1016/j.physd. 2004.03.018)
13. Kivshar, Y. S. & Pelinovsky, D. E. 2000 Self-focusing and transverse instabilities of solitary waves. Phys. Rep. 331, 117-195. (doi:10.1016/S0370- 1573(99)00106-4)
14. Kuznetsov, E. A., Rubenchik, A. M. & Zakharov, V. E. 1986 Soliton stability in plasmas and hydrodynamics. Phys. Rep. 142, 103-165. (doi:10.1016/0370-1573(86)90016-5)
15. Martin, D. U., Yuen, H. C. & Saffman, P. G. 1980 Stability of plane wave solutions of the two-space dimensional nonlinear Schrödinger equation. Wave Motion 2, 215-229. (doi:10.1016/0165-2125(80)90003-7)
16. Pelinovsky, D. E. 2001 A mysterious threshold for transverse instability of deep-water solitons. Math. Comput. Simul. 55, 585-594. (doi:10.1016/S0378- 4754(00)00287-1)
17. Pelinovsky, D. E. 2005 Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations. Proc. R. Soc. A 461, 783-812. (doi:10.1098/rspa.2004.1345)
18. Pelinovsky, D. E., Kivshar, Y. S. & Afanasjev, V. V. 1998 Internal modes of envelope solitons. Physica D 116, 121-142. (doi:10.1016/S0167-2789(98) 80010-9)
19. Pereira, N. R., Sen, A. & Bers, A. 1978 Non-linear development of lower hybrid cones. Phys. Fluids 21, 117-120. (doi:10.1063/1.862071)
20. Saffman, P. G. & Yuen, H. C. 1978 Stability of a plane soliton to infinitesimal two-dimensional perturbations. Phys. Fluids 21, 1450-1451. (doi:10.1063/1.862364)
21. Skryabin, D. V. 1999 Role of internal and continuum modes in modulational instability of quadratic solitons. Phys. Rev. E 60, 7511-7517. (doi:10.1103/PhysRevE.60.7511)
22. Sulem, C. & Sulem, P.-L. 1999 The nonlinear Schrödinger equation. New York: Springer.
23. Yajima, N. 1974 Stability of envelope soliton. Prog. Theor. Phys. 52, 1066-1067. (doi:10.1143/PTP.52.1066)
24. Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67-229.
25. Zakharov, V. E. & Rubenchik, A. M. 1974 Instability of waveguides and solitons in nonlinear media. Sov. Phys. JETP 38, 494-500.