Instabilities and oscillations of one- And two-dimensional Kadomtsev-Petviashvili waves and solitons

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

Volumn 455, Issue 1992, 1999, Pages 4363-4381

  Infeld, E ^a,b   Rowlands, G ^a,b   Senatorski, A ^a,b  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b SOLTAN INSTITUTE FOR NUCLEAR STUDIES   (Poland)

Author keywords

Benjamin feir instability; Cnoidal waves; Kadomtsev petviashvili equation; One , two And three dimensional solitons; Wavelength doubling instability

Indexed keywords

EID: 0012473625     PISSN: 13645021     EISSN: None     Source Type: Journal    
DOI: 10.1098/rspa.1999.0505     Document Type: Article

References (53)

1. Ablowitz, M. & Scgur, A. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92, 691-715.
2. Alexander, J. C., Pego, R. L. & Sachs, R. L. 1997 On the transverse instability of solitary waves in the KP equation. Phys. Lett. A 226, 187-192.
3. Arkadiev, A., Pogrebkov, A. & Palivanov, M. 1989 Inverse scattering transform method and soliton solution for Davey-Stewartson II equation. Physica D 36, 189-197.
4. Belashov, V. Y. 1994 Nonlinear effects for FUP waves propagating in magnetized plasma. Plasma Phys. Control. Fusion 36, 1CG1-1CG9.
5. Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. Lond. A 328, 153-183.
6. Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wavetrains in deep water. Part I. Theory. J. Fluid Mech. 27, 417-430.
7. Bridges, T. J. & Rowlands, G. 1994 Instability of spatially quasiperiodic states of the LandauGinzburg equation. Proc. R. Soc. Land. A 444, 347-362.
8. Casali, F., Leadke, E. W. & Spatschek, K. H. 1998 Finite size effects at nonlincarly bending solitary pulses. Phys. Lett. A 245, 407-412.
9. Crow, S. C. 1970 Stability theory of a pair of trailing vortoces. AIAA J. 8, 2172-2179.
10. Fokas, A. S. & Ablowitz, M. J. 1984 On the inverse scattering transform of multidimensional nonlinear equation related to first-order systems in the plane. J. Math. Phys. 25, 2494-2495.
11. Fokas, A. S. & Sung, L. A. 1992 On the solvability of the AT-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations. Inverse Problems 8, 673-708.
12. Fokas, A. S. & Zakharov, V. 1992 The dressing method and nonlocal Riemann-Hilbert problems. J. Nonlin. Sei. 2, 109-134.
13. Frycz, P. & Infeld, E. 1989 Self focusing of nonlinear ion acoustic waves and solitons in magnetized plasmas. Part 3. Arbitrary angle perturbations, period doubling of waves. J. Plasma Phys. 4, 441-446.
14. Galasiewicz, Z. M. (ed.) 1971 Applications of quantum mechanics to liquid helium. In Helium 4, pp. 268-313. New York: Pergamon.
15. Hayes, W. D. 1973 Group velocity and nonlinear dispersive wave propagation. Proc. R. Soc. Land. A 332, 199-221.
16. Hunter, J. K. & Vanden-Broeck, J. M. 1983 Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech. 134, 205-219.
17. Infeld, E. 1980 On three dimensional generalizations of the Boussincsq and Korteweg-de Vries equations. Q. Appl. Math. 38, 277-287.
18. Infeld, E. 1981 Three dimensional stability of Korteweg-de Vries waves and solitons. III. Lagrangian methods, KdV with positive dispersion. Acta Phys. Polon. A GO, 623-643.
19. Infeld, E. 1999 Shape preserving two-dimensional solitons in an Abrikosov vortex lattice. Phys. Rev. E60, 9302-9304.
20. Infeld, E. & Rowlands, G. 1979a Three dimensional stability of Korteweg-de Vries waves and solitons. II. Acta Phys. Polon. A 56, 329-332.
21. Infeld, E. & Rowlands, G. 19796 Stability of nonlinear ion sound waves and solitons in plasmas. Proc. R. Soc. Land. A 36G, 537-554.
22. Infeld, E. & Rowlands, G. 1992 Nonlinear waves, solitons and chaos. Cambridge University Press.
23. Infeld, E., Rowlands, G. & Hen, M. 1978 Three dimensional stability of Korteweg-de Vries waves and solitons. Acta Phys. Polon. A 54, 131-139.
24. Infeld, E., Ziemkiewicz, J. & Rowlands, G. 1987 On the two surface dimensional dynamics of periodic and solitary wavetrains on a water surface over arbitrary depth. Phys. Fluids 30, 2330-2338.
25. Infeld, E., Senatorski, A. & Skorupski, A. A. 1994 Decay of Kadomtsev-Petviashvili solitons. Phys. Rev. Lett. 72, 1345-1347
26. Infeld, E., Senatorski, A. & Skorupski, A. A. 1995 Numerical simulations of Kadomtsev-Petviashvili soliton interactions. Phys. Rev. E51, 3183-3191.
27. Jones, C. A. & Roberts, P. H. 1982 Motions in a Bose condensate. IV. Axisymmetric solitary waves. J. Phys. A15, 2599-2619.
28. Kadomtsev, B. B. & Pctviashvili, V. 1.1970 On the stability of solitary waves in weakly dispersive media. Dokl. Akad. Nauk SSSR 192, 753-756. (English transi. Sov. Phys. Dokl. 15 (1970), 539-554.)
29. Kodama, Y. 1988 A method for solving the dispersionless KP equation and its exact solutions. Phys. Lett: A129, 223-226.
30. Kodama, Y. & Gibbons, J. 1989 A method for solving the dispersionless KP hierarchy and its exact solutions. II. Phys. Lett. A 135, 167-170.
31. Koplik, J. & Levine, H. 1993 Vortex reconnection in superfluid helium, Phys. Rev. Lett. 71, 1375-1378.
32. Kuznetsov, E. A. & Musher, C. L. 1986 Influence of the collapse of sound waves on the structure of current: free shock waves in magnetized plasma. Zh. Eksp. Tear. Fiz. 91, 1605-1618. (English transi. Sov. Phys. JETP 64 (1986), 947-962.)
33. Kuznetsov, E. A. & Rasmussen, J. J. 1995 Instability of two dimensional solitons and vortices in defocusing media. Phys. Rev. E 51, 4479-4484.
34. Kuznetsov, E. A. & Turitsyn, S. K. 1982 Two dimensional and three dimensional solitons in weakly dispersive media. 1982 Int. Conf. on Plasma Physics, University of Göteborg, p. 235.
35. Kuznetsov, E. A., Spector, M. D. & Falkovich, V. E. 1984 On the stability of nonlinear waves in integrable models. Physica D10, 379-386.
36. Lighthill, M. J. 1965 Contributions to the theory of waves in nonlinear dispersive systems. J. Inst. Math. Appl. 1, 269-306.
37. McLean, J. W. 1982 Instabilities of finite amplitude water wave. J. Fluid Mech. 114, 315-330, 331-334.
38. Manakov, S. V. 1981 The inverse scattering transform for the time dependent Schrodinger and Kadomtsev-Petviashvili equations. Physica D 3, 420-427.
39. Manakov, S. V., Zakharov, V. E., Bordag, L. A., Its, A. R. &: Matveev, V. B. 1977 Two dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys. Lett. A 63, 205-206.
40. Miles, J. W. 1976 Kortcwcg-dc Vries equation modified by viscosity. Phys. Fluids 19, 1063.
41. Milne-Thomson, L. M. 1950 Jacobian elliptic function tables. New York: Dover.
42. Ovchinnikov, Y. N. & Sigal, I. M. 1998 Long time behaviour of Ginzburg-Landau vortices. Nonlinearity 11, 1295-1309.
43. Pelinovsky, D. E. fe Stepanyants, Yu. A. 1993 Self-focusing instability of plane solitons and chains of two-dimensional solitons in positive-dispersion media. Zh. Eksp. Tear. Fiz. 104, 3387-3400. (English transi. Russian Phys. JETP 77 (1993), 602-608.)
44. Petviashvili, V. & Pokhotelov, O. 1992 Solitary waves in plasma and the atmosphere. London: Gordon & Breach.
45. Saffman, P. G. & Yucn, H. C. 1985 Three dimensional waves on deep water. In Advances in nonlinear waves (cd. L. Debnath), pp. 1-30. Boston and London: Pitman.
46. Satsuma, J. 1976 N soliton solution of the two dimensional Kortewcg-de Vries equation. J. Phys. Soc. Jap. 40, 286-290.
47. Senatorski, A. & Infold, E. 1998 Breakup of two dimensional into three dimensional KadomtsevPetviashvili solitons. Phys. Rev. E57, 6050-6055.
48. Stuart, J. T. & DiPrima, R. C. 1978 The Eckhaus and Benjamin-Fcir resonance mechanisms. Proc. R. Soc. Lond. A 362, 43-56.
49. Zaitsev, A. A. 1983 Formation of stationary nonlinear waves by superposition of solitons. Dokl. Acad. Nauk. SSSR 272, 583-587. (English transi. Sov. Phys. Doklady 28 (1983), 720-722.)
50. Zakharov, V. E. 1966 The instability of waves in nonlinear dispersive media. Zh. Eksp. Tear. Fiz. 51, 1107-1114.
51. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 86-94.
52. Zakharov, V. E., Manakov, S. V., Novikov, S. P. & Pitayevski, L. P. 1980 Teoriya solitonov. Nauka Izdat. (English transi. Theory of solitons (1984). New York: Plenum.)
53. Zakharov, V. E. & Shabat, A. B. 1971 Exact theory of two dimensional self focusing and one dimensional self modulation of waves in nonlinear media. Zh. Eksp. Tear. Fiz. 61, 118-134. (English transi. Sov. Phys. JETP 34 (1972), 62-69.)