KINEMATICS OF INTERACTING SOLITONS IN TWO-DIMENSIONAL SPACE
Abstract and keywords
Abstract (English):
A simple kinematic approach to the description of interaction between solitons is developed. It is applicable to both integrable and non-integrable two-dimensional models, including those commonly used for studying surface and internal oceanic waves. This approach allows obtaining some important characteristics of the interaction between solitary waves propagating at an angle to each other. The developed theory is validated by comparison with the exact solutions of the Kadomtsev-Petviashvili equation and then applied to the observed interaction of solitary internal waves in a two-layer fluid within the two-dimensional Benjamin-Ono model.

Keywords:
Surface and internal waves, solitons, kinematic approach, Kadomtsev-Petviashvili equation, Benjamin-Ono equation, two-soliton solutions
References

1. Ablowitz, M. J., D. E. Baldwin (2012) , Nonlinear shallow ocean-wave soliton interactions on flat beaches, Phys. Rev. E, 86, p. 036305, https://doi.org/10.1103/PhysRevE.86.036305

2. Ablowitz, M. J., H. Segur (1981) , Solitons and the Inverse Scattering Transform, 425 pp., SIAM, Philadelphia, https://doi.org/10.1137/1.9781611970883

3. Anker, D., N. C. Freeman (1978) , Interpretation of three-soliton interactions in terms of resonant triads, J. Fluid Mech., 87, no. 1, p. 17-31, https://doi.org/10.1017/S0022112078002906

4. Apel, J. R., L. A. Ostrovsky, Y. A. Stepanyants, et al. (2007) , Internal solitons in the ocean and their effect on underwater sound, J. Acoust. Soc. Am., 121, p. 695-722, https://doi.org/10.1121/1.2395914

5. Grimshaw, R. (1981) , Evolution equations for long nonlinear internal waves in stratified shear flows, Stud. Appl. Math., 65, p. 159-188, https://doi.org/10.1002/sapm1981652159

6. Grimshaw, R., Y. Zhu (1994) , Oblique interaction between internal solitary waves, Stud. Appl. Math., 92, p. 249-270, https://doi.org/10.1002/sapm1994923249

7. Kadomtsev, B. B., V. I. Petviashvili (1970) , On the stability of solitary waves in weakly dispersive media, Sov. Phys. Doklady, 15, p. 539-541

8. Matsuno, Y. (1979) , Exact multi-soliton solution of the Benjamin-Ono equation, J. Phys. A, 12, p. 619-621, https://doi.org/10.1088/0305-4470/12/4/019

9. Matsuno, Y. (1980) , Interaction of the Benjamin-Ono solitons, J. Phys. A, 13, p. 1519-1536, https://doi.org/10.1088/0305-4470/13/5/012

10. Matsuno, Y. (1998) , Oblique interaction of interfacial solitary waves in a two-layer deep fluid, Proc. R. Soc. Lond. A, 454, p. 835-856, https://doi.org/10.1098/rspa.1998.0188

11. Maxworthy, T. (1980) , On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions, J. Fluid Mech., 96, no. 1, p. 47-64, https://doi.org/10.1017/S0022112080002017

12. Miles, J. W. (1977a) , Obliquely interacting solitary waves, J. Fluid Mech., 79, no. 1, p. 157-169, https://doi.org/10.1017/S0022112077000081

13. Miles, J. W. (1977b) , Resonantly interacting solitary waves, J. Fluid Mech., 79, no. 1, p. 171-179, https://doi.org/10.1017/S0022112077000093

14. Newell, A. C., L. G. Redekopp (1977) , Breakdown of Zakharov-Shabat theory and soliton creation, Phys. Rev. Lett., 38, p. 377-380, https://doi.org/10.1103/PhysRevLett.38.377

15. Satsuma, J. (1976) , Soliton solution of the two-dimensional Korteweg-deVries equation, J. Phys. Soc. Japan, 40, no. 1, p. 286-290, https://doi.org/10.1143/JPSJ.40.286

16. Small, J. (2002) , Internal tide transformation across a continental slope off Cape Sines, Portugal, J. Marine Systems, 32, p. 43-69, https://doi.org/10.1016/S0924-7963(02)00029-5

17. Tsuji, H., M. Oikawa (2001) , Oblique interaction of internal solitary waves in a two-layer fluid of infinite depth, Fluid Dyn. Res., 29, p. 251-267, https://doi.org/10.1016/S0169-5983(01)00026-0

18. Wang, C., R. Pawlowicz (2011) , Propagation speeds of strongly nonlinear near-surface internal waves in the Strait of Georgia, J. Geophys. Res., 116, p. C10021, https://doi.org/10.1029/2010JC006776

19. Wang, C., R. Pawlowicz (2012) , Oblique wave-wave interactions of nonlinear near-surface internal waves in the Strait of Georgia, J. Geophys. Res., 117, p. C0631, https://doi.org/10.1029/2012JC008022

20. Whitham, G. B. (1967) , Variational methods and applications to water waves, Proc. R. Soc. London, Ser. A, 299, p. 6-25, https://doi.org/10.1098/rspa.1967.0119

21. Wiegel, R. L. (1964) , Oceanographical Engineering, Englewood Cliffs, N.J., Prentice-Hall

22. Zakharov, V. E., S. V. Manakov, S. P. Novikov, L. P. Pitaevsky (1980) , Theory of Solitons: The Inverse Scattering Method, Nauka, Moscow (Engl. transl.: Zakharov, V. E., et al. (1984), Theory of Solitons, Consultant Bureau, New York.)

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