×

Nonlinear concentric water waves of moderate amplitude. (English) Zbl 07881962


MSC:

76-XX Fluid mechanics
35-XX Partial differential equations

Software:

Matlab; Mathematica

References:

[1] Whitham, G. B., Linear and Nonlinear Waves, 1999, John Wiley & Sons Inc · Zbl 0940.76002
[2] Iordansky, S. V., On the asymptotics of an axisymmetric divergent wave in a heavy fluid, Dokl. Akad. Sci. USSR, 125, 1211-1214, 1959 · Zbl 0089.19905
[3] Lugovtsov, A. A.; Lugovtsov, B. A., Study of axisymmetric long waves in the Korteweg - de Vries approximation, (Dynamics of a Continuous Medium, vol. 1, 1969, Nauka: Nauka Novosibirsk), 195-198, (in Russian)
[4] Maxon, S.; Viecelli, J., Cylindrical solitons, Phys. Fluids, 17, 1614-1616, 1974
[5] Miles, J. W., An axisymmetric Boussinesq wave, J. Fluid Mech., 84, 181-191, 1978 · Zbl 0365.76023
[6] Johnson, R. S., Water waves and Korteweg - de Vries equations, J. Fluid Mech., 97, 701-719, 1980 · Zbl 0441.76012
[7] Druma, V. S., Analytical solution of the axially symmetric KdV equation, Izv. Akad. Nauk MSSR, 3, 14-16, 1976
[8] Calogero, F.; Degasperis, A., Solution by the spectral transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation, Lett. Nuovo Cimento, 23, 150-154, 1978
[9] Nakamura, A.; Chen, H.-H., Soliton solutions of the cylindrical KdV equation, J. Phys. Soc. Japan, 50, 711-718, 1981
[10] Stepanyants, Y. A., Experimental investigation of cylindrically diverging solitons In an electric lattice, Wave Motion, 3, 335-341, 1981
[11] Ko, K.; Kuel, H. H., Cylindrical and spherical KdV solitary waves, Phys. Fluids, 22, 1343-1348, 1979 · Zbl 0418.76016
[12] Dorfman, A. A.; Pelinovsky, E. N.; Stepanyants, Y. A., Finite-amplitude cylindrical and spherical waves in weakly dispersive media, Sov. Phys. J. Appl. Mech. Tech. Phys., 2, 206-211, 1981
[13] Johnson, R. S., A note on an asymptotic solution of the cylindrical Korteweg-de Vries equation, Wave Motion, 30, 1-16, 1999 · Zbl 1067.35503
[14] Grimshaw, R., Initial conditions for the cylindrical Korteweg-de-Vries equation, Stud. Appl. Math., 143, 176-191, 2019 · Zbl 1423.35341
[15] Hu, W.; Zhang, Z.; Guo, Q.; Stepanyants, Y., Solitons and lumps in the cylindrical Kadomtsev-Petviashvili equation. I. Axisymmetric solitons and their stability, Chaos, 34, Article 013138 pp., 2024 · Zbl 1540.35342
[16] Lipovskii, V. D., On the nonlinear internal wave theory in fluid of finite depth, Izv. Akad. Nauk SSSR, Ser. Fiz., 21, 864-871, 1985
[17] Weidman, P. D.; Velarde, M. G., Internal solitary waves, Stud. Appl. Math., 86, 167-184, 1992 · Zbl 0744.76032
[18] Johnson, R. S., Ring waves on the surface of shear flows: a linear and nonlinear theory, J. Fluid Mech., 215, 145-160, 1990 · Zbl 0698.76016
[19] Khusnutdinova, K. R.; Zhang, X., Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech., 794, 17-44, 2016 · Zbl 1445.86003
[20] Hooper, C.; Khusnutdinova, K.; Grimshaw, R. H.J., Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current, J. Fluid Mech., 927, A37, 2021 · Zbl 1481.76056
[21] Tseluiko, D.; Alharthi, N. S.; Barros, R.; Khusnutdinova, K., Internal ring waves in a three - layer fluid on a current with a constant vertical shear, Nonlinearity, 36, 3431-3466, 2023 · Zbl 1524.76119
[22] Koop, C.; Butler, G., An investigation of internal solitary waves in a two-fluid system, J. Fluid Mech., 112, 225-251, 1981 · Zbl 0479.76036
[23] Grimshaw, R. H.J.; Pelinovsky, E.; Talipova, T.; Kurkina, O., Internal solitary waves: propaga- tion, deformation and disintegration, Nonlinear Process. Geophys., 17, 633-649, 2010
[24] Ostrovsky, L. A.; Pelinovsky, E.; Shrira, V.; Stepanyants, Y., Beyond the KdV: post-explosion development, Chaos, 25, Article 097620 pp., 2015 · Zbl 1374.35355
[25] Garbuzov, F. E.; Beltukov, Y. M.; Khusnutdinova, K. R., Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and cubic nonlinearities, Theoret. Math. Phys., 202, 319-333, 2020 · Zbl 1440.74182
[26] Kodama, Y., Normal forms for weakly dispersive wave equations, Phys. Lett. A, 112, 193-196, 1985
[27] Fokas, A. S.; Liu, Q. M., Asymptotic integrability of water waves, Phys. Rev. Lett., 77, 2347-2351, 1996 · Zbl 0982.76511
[28] Marchant, T. R.; Smyth, N. F., Soliton interactions for the extended Korteweg-de Vries equation, IMA J. Appl. Math., 56, 157-176, 1996 · Zbl 0857.35113
[29] Marchant, T. R.; Smyth, N. F., An undular bore solution for the higher-order Korteweg-de Vries equation, J. Phys. A, 39, L563, 2006 · Zbl 1170.35517
[30] Bager, S.; Smyth, N., Whitham shocks and resonant dispersive shock waves governed by the higher order Korteweg-de Vries equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 479, Article 20220580 pp., 2023
[31] Hooper, C. G.; Ruiz, P. D.; Huntley, J. M.; Khusnutdinova, K. R., Undular bores generated by fracture, Phys. Rev. E, 104, Article 044207 pp., 2021
[32] Horikis, T. P.; Frantzeskakis, D. J.; Marchant, T. R.; Smyth, N. F., Higher-dimensional extended shallow water equations and resonant soliton radiation, Phys. Fluids, 6, Article 104401 pp., 2021
[33] Khusnutdinova, K. R.; Stepanyants, Y. A.; Tranter, M. R., Soliton solutions to the fifth-order Korteweg-de Vries equation and their applications to surface and internal water waves, Phys. Fluids, 30, Article 022104 pp., 2018
[34] Horikis, T. P.; Frantzeskakis, D. J.; Marchant, T. R.; Smyth, N. F., Extended shallow water wave equations, Wave Motion, 112, Article 102934 pp., 2022 · Zbl 1524.76073
[35] Choi, W.; Camassa, R., Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313, 83-103, 1996 · Zbl 0863.76015
[36] Matsuno, Y., Hamiltonian structure for two-dimensional extended Green-Naghdi equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 472, Article 20160127 pp., 2016 · Zbl 1371.76032
[37] Chwang, A. T.; Wu, T. Y., Cylindrical solitary waves, Lecture Notes in Phys., 64, 80-90, 1977
[38] Serre, F., Contribution a l’etude des ecoulements permanents et variables dans les canaux, Houille Blanche, 3, 374-388, 1953
[39] Green, A. E.; Naghdi, P. M., A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78, 237-246, 1976 · Zbl 0351.76014
[40] Su, C. H.; Gardner, C. S., Korteweg-de Vries equation and generalisations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Math. Phys., 10, 536-539, 1969 · Zbl 0283.35020
[41] Matsuno, Y., Hamiltonian formulation of the extended Green-Naghdi equations, Physica D, 301-302, 1-7, 2015 · Zbl 1364.76031
[42] Choi, W., High-order strongly nonlinear long wave approximation and solitary wave solution, J. Fluid Mech., 945, A15, 2022 · Zbl 1493.76023
[43] Ramirez, C.; Renouard, D.; Stepanyants, Y. A., Propagation of cylindrical waves in a rotating fluid, Fluid Dyn. Res., 30, 169-196, 2002
[44] The MathWorks Inc., 2022 MATLAB, Version: 9.13.0 (R2022b) Natick, MA (available at: www.mathworks.com).
[45] Kassam, A.; Trefethen, L. N., Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 2005, Society for Industrial and Applied Mathematics, Vol. 26, No. 4, pp. 1214-1233 · Zbl 1077.65105
[46] Boole, G., A Treatise on the Calculus of Finite Differences, 1980, Macmillan and Co
[47] Orszag, S. A., On the elimination of aliasing in finite difference scheme by filtering high wave components, J. Atm. Sci., 28, 1074, 1971
[48] Derevyanko, S., The \(( n + 1 ) / 2\) rule for dealiasing in the split-step Fourier methods for n-wave interactions, IEEE Photon. Technol. Lett., 20, 1929-1931, 2008
[49] Berry, M. V., Minimal model for tidal bore revisited, New J. Phys., 21, Article 073021 pp., 2019
[50] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving the Korteweg - de Vries equation, Phys. Rev. Lett., 19, 1095-1097, 1967 · Zbl 1103.35360
[51] Gardner, C. S., The Korteweg - de Vries equation as a Hamiltonian system, J. Math. Phys., 12, 1548-1551, 1971 · Zbl 0283.35021
[52] Zakharov, V. E.; Faddeev, L. D., A Korteweg - de Vries equation: A completely integrable Hamiltonian system, Func. Anal. Appl., 5, 280-287, 1971 · Zbl 0257.35074
[53] Magri, F., A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19, 1156-1162, 1978 · Zbl 0383.35065
[54] Klein, C.; Matveev, V. B.; Smirnov, A. O., Cylindrical Kadomtsev-Petviashvili equation: old and new results, Theoret. Math. Phys., 152, 1132-1145, 2007 · Zbl 1131.35072
[55] Wolfram Research Inc., 2021 Mathematica, Version 12.3.1 (Champaign, IL) (available at: www.wolfram.com).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.