Generalized Korteweg—de Vries Equation for Internal Waves in Two-Layer Fluid

The derivation of the fifth-order Korteweg—de Vries equation is presented for internal waves in two-layer fluid with surface tension on the interface between the layers. The fluid motion is not supposed to be potential, therefore similar derivation can be used for consideration of wave motion in viscous fluid, in rotated fluid or for the shear flows with nonzero vorticity. Explicit expressions are obtained for the coefficients of the equation depending on the parameters of the background medium: widths of the layers, densities of the fluids, coefficient of surface tension. It is shown that for some combinations of the parameters of background medium the coefficients of the quadratic nonlinear and lowest order dispersive terms in the derived generalized equation can vanish and change their signs. Especially interesting is the situation when these terms become small simultaneously, and the coefficients at the nonlinear dispersive terms are also small. This is possible when the widths of the layers are almost equal. In the vicinity of such a double critical point the derived equation reduces to Gardner-Kawahara equation, which possesses solitary wave solutions with oscillating tails. Such a property makes this equation attractive theoretically and from the point of view of practical applications in the problems of flows in thin surface films of immiscible fluids. The characteristics of the flow in the presence of solitons significantly differ from those in the laminar flows, and this can lead to either negative or positive effects. On the base of the derived generalized equation and its solutions one can propose a method of control over a flow.

1. Ablowitz M. J., Segur H. Solitons and inverse scattering transform. Moscow, Mir, 1987. 480 p. (in Russian).

2. Ostrovsky L. A. Nonlinear internal waves in a rotating ocean. Okeanologiya. 1978, 18, 2, 181–191. (in Russian).

3. Ostrovsky L. A., Stepanyants Yu. A. Nonlinear surface and internal waves in rotating fluids. Nonlinear Waves. 3. Physics and Astrophysics, Springer-Verlag, 1990, 106—128.

4. Grimshaw R. H. J., Ostrovsky L. A., Shrira V. I., Stepanyants Yu. A. Long nonlinear surface and internal gravity waves in a rotating ocean. Surveys in Geophysics. 1998, 19, 4, 289—338.

5. Holloway P., Pelinovsky E., Talipova T. A generalised Korteweg–de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res. 1999, 104, 18, 333—350.

6. Pelinovskii E. N., Polukhina O. E., Lamb K. C. Nonlinear internal waves in the ocean stratified in density and current. Oceanology. 2000, 40, 6, 757—766.

7. Grimshaw R., Pelinovsky E., Poloukhina O. Higher-order Korteweg–de Vries models for internal solitary waves in a stratified shear flow with a free surface. Nonlinear Processes in Geophysics. 2002, 9, 221—235.

8. Ostrovsky L. A., Stepanyants Y. A. Internal solitons in laboratory experiments: Comparison with theoretical models. Chaos. 2005, 15, 037111-1-28.

9. Apel J., Ostrovsky L. A., Stepanyants Y. A., Lynch J. F. Internal solitons in the ocean and their effect on underwater sound. J. Acoust. Soc. Am. 2007, 121, 2, 695—722.

10. Kakutani T., Matsuuchi K. Effect of viscosity on long gravity waves. J. Phys. Soc. Japan. 1975, 39, 237— 246.

11. Lee Ch.-Y., Beardsley R. C. The generation of long nonlinear internal waves in a weakly stratified shear flows. J. Geophys. Res. 1974, 79, 3, 453—457.

12. Kakutani T., Yamasaki N. Solitary waves on a two-layer fluid. J. Phys. Soc. Japan. 1978, 45, 2, 674—679.

13. Brekhovskikh L .M., Goncharov V. V. Introduction to the continuum mechanics. Moscow, Nauka, 1982. 335 p. (in Russian).

14. Koop C., Butler G. An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 1981, 112, 225—251.

15. Djordjevic V. D., Redekopp L. G. The fission and disintegration of internal solitary waves moving over twodimensional topography. J. Phys. Oceanogr. 1978, 8, 6, 1016—1024.

16. Kawahara T. Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan. 1972, 33, 260—264.

17. Gorshkov K. A., Ostrovskii L. A., Papko V. V. Interactions and bound states of solitons as classical particles. Zh. Eksp. Teor. Fiz. 1976, 71, 2, 585—593. (in Russian).

18. Gorshkov K. A., Ostrovsky L. A., Papko V. V., Pikovsky A. S. On the existence of stationary multisolitons. Phys. Lett. A. 1979, 74, 3—4, 177—179.

19. Kawahara T., Takaoka M. Chaotic motions in oscillatory soliton lattice. J. Phys. Soc. Japan. 1988, 57, 11, 3714—3732.