Russian Journal of Earth Sciences Russian Journal of Earth Sciences Russian Journal of Earth Sciences 1681-1208 46531 10.2205/2020ES000735 ОРИГИНАЛЬНЫЕ СТАТЬИ ORIGINAL ARTICLES ОРИГИНАЛЬНЫЕ СТАТЬИ Kinematics of interacting solitons in two-dimensional space Kinematics of interacting solitons in two-dimensional space Ostrovsky L A Ostrovsky L A Stepanyants Y A Stepanyants Y A Yury.Stepanyants@usq.edu.au University of Colorado Boulder and Institute of Applied Physics RAS ru University of Colorado Boulder and Institute of Applied Physics RAS ru University of Southern Queensland and Nizhny Novgorod State Technical University n.a. R. E. Alekseev ru University of Southern Queensland and Nizhny Novgorod State Technical University n.a. R. E. Alekseev ru 20 4 29 10 2021 https://rjes.ru/en/nauka/article/46531/view

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.

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.

 Surface and internal waves solitons kinematic approach Kadomtsev-Petviashvili equation Benjamin-Ono equation two-soliton solutions Surface and internal waves solitons kinematic approach Kadomtsev-Petviashvili equation Benjamin-Ono equation two-soliton solutions The authors declare that they do not have conflicts of interest. Y. S. acknowledges the funding of this study from the State task program in the sphere of scientific activity of the Ministry of Science and Higher Education of the Russian Federation (project No. FSWE-2020-0007) and the grant of the President of the Russian Federation for state support of leading Scientific Schools of the Russian Federation (grant No. NSH-2485.2020.5).

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 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 Ablowitz, M. J., H. Segur (1981) , Solitons and the Inverse Scattering Transform, 425 pp., SIAM, Philadelphia, https://doi.org/10.1137/1.9781611970883 Ablowitz, M. J., H. Segur (1981) , Solitons and the Inverse Scattering Transform, 425 pp., SIAM, Philadelphia, https://doi.org/10.1137/1.9781611970883 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 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 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 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 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 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 Grimshaw, R., Y. Zhu (1994) , Oblique interaction between internal solitary waves, Stud. Appl. Math., 92, p. 249-270, https://doi.org/10.1002/sapm1994923249 Grimshaw, R., Y. Zhu (1994) , Oblique interaction between internal solitary waves, Stud. Appl. Math., 92, p. 249-270, https://doi.org/10.1002/sapm1994923249 Kadomtsev, B. B., V. I. Petviashvili (1970) , On the stability of solitary waves in weakly dispersive media, Sov. Phys. Doklady, 15, p. 539-541 Kadomtsev, B. B., V. I. Petviashvili (1970) , On the stability of solitary waves in weakly dispersive media, Sov. Phys. Doklady, 15, p. 539-541 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 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 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 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 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 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 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 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 Miles, J. W. (1977a) , Obliquely interacting solitary waves, J. Fluid Mech., 79, no. 1, p. 157-169, https://doi.org/10.1017/S0022112077000081 Miles, J. W. (1977a) , Obliquely interacting solitary waves, J. Fluid Mech., 79, no. 1, p. 157-169, https://doi.org/10.1017/S0022112077000081 Miles, J. W. (1977b) , Resonantly interacting solitary waves, J. Fluid Mech., 79, no. 1, p. 171-179, https://doi.org/10.1017/S0022112077000093 Miles, J. W. (1977b) , Resonantly interacting solitary waves, J. Fluid Mech., 79, no. 1, p. 171-179, https://doi.org/10.1017/S0022112077000093 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 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 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 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 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 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 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 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 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 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 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 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 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 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 Wiegel, R. L. (1964) , Oceanographical Engineering, Englewood Cliffs, N.J., Prentice-Hall Wiegel, R. L. (1964) , Oceanographical Engineering, Englewood Cliffs, N.J., Prentice-Hall 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.) 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.)