Abstract
A mathematical theory is mounted for a complex system of equations derived by Gear and Grimshaw that models the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratified fluid. For the model in question, the Cauchy problem is of interest, and is shown to be globally well-posed in suitably strong function spaces. Our results make use of Kato's theory for abstract evolution equations together with somewhat delicate estimates obtained using techniques from harmonic analysis. In weak function classes, a local existence theory is developed. The system is shown to be susceptible to the dispersive blow-up phenomenon investigated recently by Bona and Saut for Korteweg-de Vries-type equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alkylas, T. R., Benney, D. J.: Direct resonance in nonlinear wave systems. Stud. Appl. Math.63, 209–226 (1980)
Alkylas, T. R., Benney, D. J.: The evolution of waves near direct-resonance conditions. Stud. Appl. Math.67, 107–123 (1982)
Benjamin, T. B.: Internal waves of finite amplitude and permanent form. J. Fluid Mech.25, 241–270 (1966)
Benjamin, T. B.: Internal waves of permanent form in fluids of great depth. J. Fluid Mech.29, 559–592 (1967)
Bona, J. L., Sachs, R.: Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun. Math. Phys.118, 15–29 (1988)
Bona, J. L., Saut, J.-C.: Dispersive blow-up for the generalized Korteweg-de Vries equation. To appear in J. Diff'l Equations (1991a)
Bona, J. L., Saut, J.-C.: The general intermediate long wave equation and related systems. In preparation (1991b)
Bona, J. L., Smith, R.: The initial value problem for the Korteweg-de Vries equation. Phil. Trans. R. Soc. Lond.A278, 555–604 (1975)
Erkart, C.: Internal waves in the ocean. Phys. Fluid4, 791–799 (1961)
Fornberg, R. B., Whitham, G. B.: A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond.A289, 373–404 (1978)
Gear, J. A.: Strong interactions between solitary waves belonging to different wave modes. Stud. Appl. Math.72, 95–124 (1985)
Gear, J. A., Grimshaw, R.: Weak and strong interactions between internal solitary waves. Stud. Appl. Math.70, 235–258 (1984)
Ginibre, J., Tsutsumi, Y.: Uniqueness for the generalized Korteweg-de Vries equation. SIAM J. Math. Anal.20, 1388–1425 (1989)
Grimshaw, R.: Evolution equations for long, nonlinear internal waves in stratified shear flows. Stud. Appl. Math.65, 159–188 (1981)
Kato, T.: Quasilinear equations of evolution with applications to partial differential equations, Lect. Notes in Math.448, pp. 27–50. Berlin, Heidelberg, New York:Springer 1975
Kato, T.: On the Korteweg-de Vries equation. Manuscripta Math.28, 89–99 (1979)
Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Adv. Math. Supplementary Studies in Applied Math. Vol.8, pp. 93–128 (1983)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Applied Math.41, 891–907 (1988)
Kenig, C. E., Ponce, G., Vega, L.: On the (generalized) Korteweg-de Vries equation. Duke Math. J.59, 585–610 (1989)
Kenig, C. E., Ponce, G., Vega, L.: Oscillatory integrals and regularity of dispersive equations. Indiana U. Math. J.40, 33–69 (1991a)
Kenig, C. E., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc.4, 323–347 (1991b)
Kenig, C. E., Ruiz, A.: A strong type (2,2) estimate for the maximal function associated to the Schrödinger equation. Trans. Am. Math. Soc.280, 239–246 (1983)
Kubota, T., Ko, D. R. S., Dobbs, L. D.: Weakly-nonlinear, long internal waves in stratified fluids of finite depth. AIAA J. Hydronautics12, 157–165 (1980)
Lions, J.-L.: Quelques methodes de résolution des problèmes aux limites non linéaires. Paris: Dunod 1969
Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications. Vol. 1. Paris: Dunod 1968
Liu, A. K., Kubota, T., Ko, D. R. S.: Resonant transfer of energy between nonlinear waves in neighboring pycnoclines. Stud. Appl. Math.63, 25–45 (1980)
Liu, A. K., Pereira, N. R., Ko, D. R. S.: Weakly interacting internal solitary waves in neighboring pycnoclines. J. Fluid Mech.122, 187–194 (1982)
Saut, J.-C.: Sur quelques généralisations de l'équation de Korteweg-de Vries. J. Math. Pures Appl.58, 21–61 (1975)
Saut, J.-C., Temam, R.: Remarks on the Korteweg-de Vries equation. Israel J. Math.24, 78–87 (1976)
Stein, E. M.: Oscillatory integrals in Fourier Analysis. Beijing lectures in Harmonic Analysis, pp. 307–355. Princeton NJ: Princeton University Press 1986
Strichartz, R. S.: Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations. Duke Math. J.44, 705–714 (1977)
Vega, L.: El multiplicador de Schrödinger: La function maximal y los operadores de restriction. Doctoral Thesis Universidad Autonoma, Madrid, Spain 1987
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Rights and permissions
About this article
Cite this article
Bona, J.L., Ponce, G., Saut, JC. et al. A model system for strong interaction between internal solitary waves. Commun.Math. Phys. 143, 287–313 (1992). https://doi.org/10.1007/BF02099010
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02099010