Modulational instability of two pairs of counter-propagating waves and energy exchange in a two-component system

Physica D: Nonlinear Phenomena

Volumn 214, Issue 1, 2006, Pages 1-24

  Griffiths, S D ^a   Grimshaw, R H J ^b   Khusnutdinova, K R ^b  

^a UNIVERSITY OF WASHINGTON   (United States)

^b LOUGHBOROUGH UNIVERSITY   (United Kingdom)

Author keywords

Amplitude evolution equations; Energy exchange in two component systems; Modulational instability; Nonlinear Schr dinger equations; Sine Gordon equations

Indexed keywords

ASYMPTOTIC STABILITY; COMPUTER SIMULATION; ENERGY MANAGEMENT; INTEGRATION; MODULATION; NONLINEAR EQUATIONS; SET THEORY;

AMPLITUDE EVOLUTION EQUATIONS; ENERGY EXCHANGE IN TWO-COMPONENT SYSTEMS; MODULATIONAL INSTABILITY; NONLINEAR SCHRÖDINGER EQUATIONS; SINE-GORDAN EQUATIONS;

WAVE EFFECTS;

EID: 31144449779     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2005.11.011     Document Type: Article

References (40)

1. C. Martel, E. Knobloch, and J.M. Vega Dynamics of counter-propagating waves in parametrically forced systems Physica D 137 2000 94 123
2. C. Martel, J.M. Vega, and E. Knobloch Dynamics of counter-propagating waves in parametrically driven systems: dispersion vs advection Physica D 174 2003 198 217
3. F.J. Mancebo, and J.M. Vega Standing wave description of nearly conservative, parametrically driven waves in extended systems Physica D 197 2004 346 363
4. T.J. Bridges, and F.E. Laine-Pearson Nonlinear counterpropagating waves, multisymplectic geometry, and the instability of standing waves SIAM J. Appl. Math. 64 2004 2096 2120
5. M. Onorato, D. Ambrosi, A.R. Osborne, and M. Serio Interaction of two quasi-monochromatic waves in shallow water Phys. Fluids 15 2003 3871 3874
6. S.D. Griffiths, R.H.J. Grimshaw, and K.R. Khusnutdinova The influence of modulational instability on energy exchange in coupled Sine-Gordon equations Theoret. and Math. Phys. 137 2003 1448 1458
7. A.C. Newell Solitons in Mathematics and Physics 1985 SIAM
8. V.E. Zakharov Stability of periodic waves of finite amplitude on the surface of a deep fluid J. Appl. Mech. Tech. Phys. 2 1968 190 194
9. T.B. Benjamin, and J.E. Feir The disintegration of wave trains on deep water J. Fluid Mech. 27 1967 417 430
10. E. Knobloch, and J. De Luca Amplitude equations for travelling wave convection Nonlinearity 3 1990 975 980
11. E. Knobloch, and J.D. Gibbon Coupled NLS Equations for counter propagating waves in systems with reflection symmetry Phys. Lett. A 154 1991 353 356
12. R.D. Pierce, and C.E. Wayne On the validity of mean-field amplitude equations for counter-propagating wavetrains Nonlinearity 8 1995 769 779
13. M.G. Forest, D.W. McLaughlin, D.J. Muraki, and O.C. Wright Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDEs J. Nonlinear Sci. 10 2000 291 331
14. M.G. Forest, and O.C. Wright An integrable model for stable:unstable wave coupling phenomena Physica D 178 2003 173 189
15. I.Sh. Akhatov, V.A. Baikov, and K.R. Khusnutdinova Non-linear dynamics of coupled chains of particles J. Appl. Math. Mech. 59 1995 353 361
16. K.R. Khusnutdinova Nonlinear waves in a bi-layer and coupled Klein-Gordon equations A.B. Movchan IUTAM Symposium on Asymptotics, Singularities and Homogenization in Problems of Mechanics, Solid Mechanics and Its Applications vol. 113 2003 Kluwer Academic Publishers 537 546
17. K.R. Khusnutdinova, and V.V. Silberschmidt Lattice modelling of nonlinear waves in a bi-layer with delamination Proc. Estonian Acad. Sci. Phys. Math. 52 2003 63 75
18. S. Yomosa Soliton excitations in deoxyribonucleic acid (DNA) double helices Phys. Rev. A 27 1983 2120 2125
19. L.V. Yakushevich Nonlinear Physics of DNA 1998 Wiley Chichester
20. T.A. Kontorova, and Ya.I. Frenkel On the theory of plastic deformation and twinning I, II Zh. Eksp. Teor. Fiz. 8 1938 89 95 1340-1368
21. O.M. Braun, and Yu.S. Kivshar Nonlinear dynamics of the Frenkel-Kontorova model Phys. Rep. 306 1998 1 108
22. K.R. Khusnutdinova, and D.E. Pelinovsky On the exchange of energy in coupled Klein-Gordon equations Wave Motion 38 2003 1 10
23. L.I. Mandelshtam Lectures on the Theory of Oscillations 1972 Nauka Moscow
24. V.I. Arnold Mathematical Methods of Classical Mechanics 1989 Springer New York
25. D.J. Kaup, and A.C. Newell Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory Proc. Roy. Soc. A 361 1978 413 446
26. D. McLaughlin, and A.C. Scott Perturbation analysis of fluxon dynamics Phys. Rev. A. 18 1978 1652 1680
27. Yu.S. Kivshar, and B.A. Malomed Dynamics of solitons in nearly integrable systems Rev. Modern Phys. 61 1989 763 915
28. G. Kopidakis, S. Aubry, and G.P. Tsironis Targeted energy transfer through discrete breathers in nonlinear systems Phys. Rev. Lett. 87 2001 165501
29. P. Maniadis, G. Kopidakis, and S. Aubry Classical and quantum targeted energy transfer between nonlinear oscillators Physica D 188 2004 153 177
30. P.J. Olver Applications of Lie Groups to Differential Equations 1986 Springer-Verlag New York
31. D.J. Benney, and A.C. Newell The propagation of nonlinear wave envelopes Stud. Appl. Math. 46 1967 133 139
32. M.J. Ablowitz, and H. Segur Solitons and the Inverse Scattering Transform 1981 SIAM Philadelphia
33. M. Abramowitz, and I.A. Stegun Handbook of Mathematical Functions 1979 Nauka Moscow
34. S. Fauve Pattern forming instabilities C. Godrèche P. Manneville Hydrodynamics and Nonlinear Instabilities 1998 Cambridge University Press Cambridge
35. O. Thual, S. Douady, and S. Fauve Parametric instabilities E. Tirapegui D. Villarroel Instabilities and Nonequilibrium Structures II 1989 Kluwer Academic Dordrecht
36. G.B. Whitham Linear and Nonlinear Waves 1974 Wiley New York
37. G.M. Muslu, and H.A. Erbay A split-step Fourier method for the Complex Modified Korteweg-de Vries Equation Comput. Math. Appl. 45 2002 503 514
38. E. Fermi, J. Pasta, S. Ulam, Studies on nonlinear problems, I, Los Alamos Scientific Laboratory Report No. LA-1940 (1955).
39. Reprinted in A.C. Newell (Ed.), Nonlinear Wave Motion, in: AMS Lectures in Applied Math., vol. 15, 1974, pp. 143-156
40. B. Tan, and J.P. Boyd Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations Chaos Solitons Fractals 12 2001 721 734