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High-order Davies’ approximation for a solitary wave solution in Packham’s complex plane. (English) Zbl 1317.30056

Summary: This paper considers a progressive solitary wave of permanent form in an ideal fluid of constant depth and explores P. Davies’ approximation [Proc. R. Soc. Lond., Ser. A 208, 475–486 (1951; Zbl 0043.23810)] with high-order corrections to Levi-Civita’s surface condition for the logarithmic hodograph variable. Using a complex plane that was originally introduced by B. A. Packham [Proc. R. Soc. Lond., Ser. A 213, 238–249 (1952; Zbl 0046.42711)], it is shown that a singularity at infinity can be regularized. Therefore, the solutions in Packham’s complex plane under high-order Davies’ approximation maintain two critical properties of a solitary wave, the correct exponential decay in the outskirt of wave and the harmonic property of a solution, that are often violated in classical long wave approximations. After introducing an accurate numerical method to compute solitary wave solutions in Packham’s complex plane, we compare high-order Davies’ approximate solutions with fully nonlinear solutions as well as long wave approximate solutions. The results demonstrate that high-order Davies’ approximation produces rapidly converging series solutions even for relatively large amplitude waves and that Davies’ approximate solutions compare much better with the fully nonlinear solutions than the long wave approximate solutions.

MSC:

30E10 Approximation in the complex plane
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
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References:

[1] J. G. B. Byatt-Smith, {\it An exact integral equation for steady surface waves}, Proc. R. Soc. Lond. A, 315 (1970), pp. 405-418.
[2] D. Clamond and D. Dutykh, {\it Fast accurate computation of the fully nonlinear solitary surface gravity waves}, Comput. & Fluids, 84 (2013), pp. 35-38. · Zbl 1290.76018
[3] T. V. Davies, {\it The theory of symmetrical gravity waves of finite amplitude.} I, Proc. R. Soc. Lond. A, 208 (1951), pp. 475-486. · Zbl 0043.23810
[4] T. V. Davies, {\it Gravity waves of finite amplitude. III. Steady, symmetrical, periodic waves in a channel of finite depth}, Quart. Appl. Math., 10 (1952), pp. 57-67. · Zbl 0046.19901
[5] D. Dutykh and D. Clamond, {\it Efficient computation of steady solitary gravity waves}, Wave Motion, 51 (2014), pp. 86-99. · Zbl 1524.76046
[6] W. A. B. Evans and M. J. Ford, {\it An exact integral equation for solitary waves (with new numerical results for some “internal” properties)}, Proc. R. Soc. Lond. A, 452 (1996), pp. 373-390. · Zbl 0870.65136
[7] J. Fenton, {\it A ninth-order solution for the solitary wave}, J. Fluid Mech., 53 (1972), pp. 257-271. · Zbl 0236.76016
[8] K. O. Friedrichs and D. H. Hyers, {\it The existence of solitary waves}, Comm. Pure Appl. Math., 7 (1954), pp. 517-550. · Zbl 0057.42204
[9] P. Henrici, {\it Applied Computational Complex Analysis}, Vol. 3, John Wiley & Sons, New York, 1986. · Zbl 0578.30001
[10] J. K. Hunter and J.-M. Vanden-Broeck, {\it Accurate computations for steep solitary waves}, J. Fluid Mech., 136 (1983), pp. 63-71. · Zbl 0525.76014
[11] D. J. Korteweg and G. de Vries, {\it On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves}, Phil. Mag., 39 (1895), pp. 422-443. · JFM 26.0881.02
[12] H. Lamb, {\it Hydrodynamics}, 6th ed., Dover, Mineola, NY, 1945. · JFM 26.0868.02
[13] C. W. Lenau, {\it The solitary wave of maximum amplitude}, J. Fluid Mech., 26 (1966), pp. 309-320. · Zbl 0141.44203
[14] R. R. Long, {\it Solitary waves in the one- and two-fluid systems}, Tellus, 8 (1956), pp. 460-471.
[15] M. S. Longuet-Higgins and J. D. Fenton, {\it On the mass, momentum, energy and circulation of a solitary wave.} II, Proc. R. Soc. Lond. A., 340 (1974), pp. 471-493. · Zbl 0306.76025
[16] L. M. Milne-Thomson, {\it Theoretical Hydrodynamics}, 5th ed., Dover, Mineola, NY, 1968. · Zbl 0164.55802
[17] S. Murashige, {\it Numerical use of exterior singularities for computation of gravity waves in shallow water}, J. Engrg. Math., 77 (2012), pp. 1-18. · Zbl 1276.76014
[18] S. Murashige, {\it Davies’ surface condition and singularities of deep water waves}, J. Engrg. Math., 85 (2014), pp. 19-34. · Zbl 1359.76065
[19] B. A. Packham, {\it The theory of symmetrical gravity waves of finite amplitude. I. The solitary wave}, Proc. R. Soc. Lond. A, 213 (1952), pp. 234-249. · Zbl 0046.42711
[20] S. A. Pennell and C. H. Su, {\it A seventeenth-order series expansion for the solitary wave}, J. Fluid Mech., 149 (1984), pp. 431-443. · Zbl 0565.76021
[21] S. A. Pennell, {\it On a series expansion for the solitary wave}, J. Fluid Mech., 179 (1987), pp. 557-561. · Zbl 0622.76020
[22] G. G. Stokes, {\it The outskirts of the solitary wave}, Math. and Phys. Papers, 5 (1905), p. 163.
[23] J, Strutt (Lord Rayleigh), {\it On waves}, Phil. Mag. 5th Ser., 1 (1876), pp. 257-279.
[24] M. Tanaka, {\it The stability of solitary waves}, Phys. Fluids, 29 (1986), pp. 650-655. · Zbl 0605.76025
[25] J.-M. Vanden-Broeck, {\it Gravity-Capillary Free-Surface Flows}, Cambridge University Press, Cambridge, UK, 2010. · Zbl 1202.76002
[26] T. Y. Wu, J. Kao, and J. E. Zhang, {\it A unified intrinsic functional expansion theory for solitary waves}, Acta Mech Sin., 21 (2005), pp. 1-15. · Zbl 1200.76037
[27] T. Y. Wu, X. Wang, and W. Qu, {\it On solitary waves. Part \(2\): A unified perturbation theory for higher order waves}, Acta Mech Sin., 21 (2005), pp. 515-530. · Zbl 1200.76038
[28] T. Y. Wu and S. Murashige, {\it On tsunami and the regularized solitary wave theory}, J. Engrg. Math., 70 (2011), pp. 137-146. · Zbl 1254.76038
[29] H. Yamada, {\it On the highest solitary wave}, Rep. Res. Inst. Appl. Mech. Kyushu Univ., 5 (1957), pp. 53-67.
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