Embedded solitons: Solitary waves in resonance with the linear spectrum

Physica D: Nonlinear Phenomena

Volumn 152-153, Issue , 2001, Pages 340-354

  Champneys, A R ^a   Malomed, B A ^b   Yang, J ^c   Kaup, D J ^d  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b TEL AVIV UNIVERSITY   (Israel)

^c University of Vermont ^*  (United States)

^d CLARKSON UNIVERSITY   (United States)

Author keywords

Embedded solitons; Linear spectrum; Solitary waves

Indexed keywords

EID: 0035873628     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(01)00178-6     Document Type: Article

References (37)

1. J.P. Boyd, Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics, Kluwer Academic Publishers, Dordrecht, 1998.
2. Champneys A.R., Groves M.D. A global investigation of solitary wave solutions to a two-parameter model for water waves. J. Fluid Mech. 342:1997;199-229.
3. Kichenassamy S., Olver P.J. Existence and non-existence of solitary wave solutions to higher-order model evolution equations. SIAM J. Math. Anal. 23:1996;1141-1166.
4. J. Yang, Dynamics of embedded solitons in the extended Korteweg-de Vries equation, Stud. Appl. Math (2001), in press.
5. Champneys A.R., Malomed B.A., Friedman M.J. Thirring solitons in the presence of dispersion. Phys. Rev. Lett. 80:1998;4168-4171.
6. Yang J., Malomed B.A., Kaup D.J. Embedded solitons in second-harmonic-generating systems. Phys. Rev. Lett. 83:1999;1958-1961.
7. Hunter J.K., Scheurle J. Existence of perturbed solitary wave solutions to a model equation for water-waves. Physica D. 32:1988;253-268.
8. Karpman V.I. Stationary solitary waves of the fifth-order KdV-type equations. Phys. Lett. A. 186:1994;300-308.
9. Craig W., Groves M.D. Hamiltonian long-wave approximations to the water-wave problem. Wave Motion. 19:1994;367-389.
10. Amick C.J., McLeod J.B. A singular perturbation problem in water-waves. Stabil. Appl. Anal. Cont. Media. 1:1991;127-148.
11. Grimshaw R., Joshi N. Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation. SIAM J. Appl. Math. 55:1995;124-135.
12. R. Grimshaw, P. Cook, Solitary waves with oscillatory tails, in: A.T. Chwang, J.H.W. Lee, D.Y.C. Leung, A.A. Balkema (Eds.), Proceedings of Second International Conference on Hydrodynamics, Hong Kong, Hydrodynamics: Theory and Applications, Vol. 1, 1996, pp. 327-336.
13. Fujioka J., Espinosz A. Soliton-like solution of an extended NLS equation existing in resonance with linear dispersive waves. J. Phys. Soc. Jpn. 66:1997;2601-2607.
14. Christodoulides D.N., Joseph R.I. Slow Bragg solitons in nonlinear periodic structures. Phys. Rev. Lett. 62:1989;1746-1749.
15. Aceves A.B., Wabnitz S. Self-induced transparency solitons in nonlinear refractive periodic media. Phys. Lett. A. 141:1989;37-42.
16. Champneys A.R., Malomed B.A. Moving embedded solitons. J. Phys. A. 32:1999;L547-L553.
17. Mak W.C.K., Malomed B.A., Chu P.L. Three-wave gap solitons in waveguides with quadratic nonlinearity. Phys. Rev. E. 58:1998;6708-6722.
18. Champneys A.R., Malomed B.A. Embedded solitons in a three-wave system. Phys. Rev. E. 61:1999;886-890.
19. (3) nonlinearities Opt. Commun. 122:1996;200-211.
20. Bang O., Kivshar Y.S., Buryak A.V. Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities. Opt. Lett. 22:1997;1680-1682.
21. Yu.N. Karamzin, A.P. Sukhorukov, Nonlinear interaction of diffracted light bams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters, JETP Lett. 11 (1974) 339-342 (originally published in Russian as ZhETF Pis. Red. 20 (1974) 734-739).
22. Yu.N. Karamzin, A.P. Sukhorukov, Nonlinear interaction of diffracted light bams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters, JETP Lett. 11 (1974) 339-342 (originally published in Russian as ZhETF Pis. Red. 20 (1974) 734-739).
23. He H., Werner M.J., Drummond P.D. Simultaneous solitary-wave solutions in a nonlinear parametric waveguide. Phys. Rev. E. 54:1996;896-911.
24. Etrich C., Peschel U., Lederer F., Malomed B.A. Collision of solitary waves in media with a second-order nonlinearity. Phys. Rev. A. 52:1995;R3444-R3447.
25. E.J. Doedel, A.R. Champneys, T.R. Fairgrieve, Yu.A. Kuznetsov, B. Sandstede, X.J. Wang, AUTO97 continuation and bifurcation software for ordinary differential equations, 1997 (available by anonymous ftp from FTP.CS.CONCORDIA.CA, directory PUB/DOEDEL/AUTO).
26. L.M. Lerman, Hamiltonian systems with a seperatrix loop of a saddle-center, Russian Acad. Sci. Sb. Math. 10 (1991) 297 (originally published in Russian as Matemat. Sbornik 184, 105-138).
27. L.M. Lerman, Hamiltonian systems with a seperatrix loop of a saddle-center, Russian Acad. Sci. Sb. Math. 10 (1991) 297 (originally published in Russian as Matemat. Sbornik 184, 105-138).
28. Mielke A., Holmes P., O'Reilly O. Cascades of homoclinic orbits to, and chaos near, a hamiltonian saddle-center. J. Dyn. Diff. Eqs. 4:1992;95-126.
29. Buryak A.V. Stationary soliton bound states existing in resonance with linear waves. Phys. Rev. E. 52:1995;1156-1163.
30. Lombardi E. Homoclinic orbits to small periodic orbits for a class of reversible systems. Proc. R. Soc. Edinburgh A. 126:1996;1035-1054.
31. Lombardi E. Non persistence of homoclinic connections for perturbed integrable reversible systems. J. Dyn. Diff. Eqs. 11:1999;124-208.
32. E. Lombardi, Generic non-persistence of homoclinic connections for 4-dimensional reversible vector fields, 1997, Preprint Institute Non Linéaire de Nice.
33. A.R. Champneys, Codimension-one persistence beyond all orders of homoclinic orbits to degenerate saddle centers in reversible systems, Nonlinearity 14 (2001) 87-112.
34. Kolchugina I.A., Mironov V.A., Sergeev A.M. Structure of steady-state solitons in systems with a nonlocal nonlinearity. JETP Lett. 31:1980;304-307.
35. Markov G.L., Mironov V.A., Sergeev A.M., Sokolova I.A. Multibeam self-channeling of plasma waves. Sov. Phys. JETP. 53:1981;1183-1186.
36. M.M. Bogdam, A.M. Kosevich, Radiationless motion of one-dimensional solitons in dispersive medium, in: K.H. Spatschek, F.G. Mertens (Eds.), Nonlinear Coherent Structures in Physics and Biology, Plenum Press, New York, 1994, pp. 373-376.
37. A.R. Champneys, Yu.S. Kivshar, Origin of multikinks in dispersive nonlinear systems, Phys. Rev. E 61 (2000) 2551-2554.