×

On the theory of internal waves of permanent form in fluids of great depth. (English) Zbl 0829.76012

A stationary solitary wave moving along a border between two layers of an ideal liquid with different densities is considered. The lower layer has a finite depth, while the upper one is assumed infinitely deep. The horizontal velocity of the liquid at infinity (in the vertical direction) is treated as an unknown parameter that must be found together with the shape of the interface corresponding to a stationary internal solitary wave. The problem is formulated on the basis of equations for the stream function, supplemented by the boundary conditions at the interface. By means of the hodograph transformation, the problem is mapped into a Laplace equation for an infinite strip with the Dirichlet boundary condition at the bottom, and a nonlocal nonlinear boundary condition at the top. Subsequent analysis allows one to reduce the latter problem, in the case when the above-mentioned horizontal velocity at infinity takes values slightly larger than the critical value at which a bifurcation of small solutions occurs, to the stationary Benjamin-Ono (BO) equation with a small correction. It is demonstrated that, as in the case of the pure BO equation, the solitary wave solution is given by a rational rather than by an exponential function.

MSC:

76B25 Solitary waves for incompressible inviscid fluids
76B55 Internal waves for incompressible inviscid fluids
76V05 Reaction effects in flows
35Q51 Soliton equations
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592. · Zbl 0147.46502
[2] R. E. Davis and A. Acrivos, Solitary internal waves in deep fluid, J. Fluid Mech. 29 (1967), 593-607. · Zbl 0147.46503
[3] Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082 – 1091. · Zbl 1334.76027 · doi:10.1143/JPSJ.39.1082
[4] C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), no. 1, 9 – 95. · Zbl 0468.76025 · doi:10.1007/BF00250799
[5] C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long-wave limit, Philos. Trans. Roy. Soc. London Ser. A 303 (1981), no. 1481, 633 – 669. · Zbl 0482.76029 · doi:10.1098/rsta.1981.0231
[6] C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation — a nonlinear Neumann problem in the plane, Acta Math. 167 (1991), no. 1-2, 107 – 126. · Zbl 0755.35108 · doi:10.1007/BF02392447
[7] C. J. Amick and J. F. Toland, Uniqueness of Benjamin’s solitary-wave solution of the Benjamin-Ono equation, IMA J. Appl. Math. 46 (1991), no. 1-2, 21 – 28. The Brooke Benjamin special issue (University Park, PA, 1989). · Zbl 0735.35105 · doi:10.1093/imamat/46.1-2.21
[8] C. J. Amick and R. E. L. Turner, A global theory of internal solitary waves in two-fluid systems, Trans. Amer. Math. Soc. 298 (1986), no. 2, 431 – 484. · Zbl 0631.35029
[9] C. J. Amick and R. E. L. Turner, Small internal waves in two-fluid systems, Arch. Rational Mech. Anal. 108 (1989), no. 2, 111 – 139. · Zbl 0681.76103 · doi:10.1007/BF01053459
[10] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. · Zbl 0361.35003
[11] D. P. Bennett, R. W. Brown, S. E. Stansfield, J. D. Stroughair, and J. L. Bona, The stability of internal solitary waves, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 351 – 379. · Zbl 0574.76028 · doi:10.1017/S0305004100061193
[12] N. K. Bary, A treatise on trigonometric series. Vols. I, II, Authorized translation by Margaret F. Mullins. A Pergamon Press Book, The Macmillan Co., New York, 1964. · Zbl 0129.28002
[13] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35 – 92. · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
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.