Math. Proc. Camb. Phil. Soc. (1983), 94, 351 351 Printed in Great Britain The stability of internal solitary waves BY D. P. BENNETT, R. W. BROWN, S. E. STANSFIELD, J. D. STROUGHAIR Physics Department, Case Western Reserve University, Cleveland, Ohio AND J. L. BONA Department of Mathematics, University of Chicago, Chicago, Illinois (Received 12 July 1982) 1. Introduction A theory is developed relating to the stability of solitary-wave solutions of the so- called Benjamin-Ono equation. This equation was derived by Benjamin (5) as a model for the propagation of internal waves in an incompressible non-diffusive heterogeneous fluid for which the density is non-constant only within a layer whose thickness is much smaller than the total depth. In his article, Benjamin wrote in closed form the one- parameter family of solitary-wave solutions of his model equation whose stability will be the focus of attention presently. The theory developed herein deals with the full nonlinear problem, and so allowance is made for small but finite perturbations of the solitary-wave solutions of the Benjamin-Ono equation. The outcome of our study is a result about stability which is practically satisfactory, in that it bears upon the size and shape of the wave in question. The analysis follows generally the lines of argument developed by Benjamin (6) and Bona (10) in their proof of the stability of the solitary-wave solutions of both the Korteweg-de Vries equation and the alternative model put forward by Peregrine (29) and Benjamin et al.(9). Nevertheless, certain details are quite different, and difficulties arise relating to the non-local character of the underlying evolution equation. A linearized stability analysis of the waves in question here has been given recently by Chen and Kaup (16). It is noteworthy that they found a secular instability, growing linearly in time. Apparently this reflects the fact that a small perturbation of a given solitary wave <f> might lead to a solution of the Benjamin-Ono equation, the bulk of which propagates at a speed different from the speed at which <f> propagates. This would account for an initial linear growth in the difference between <j> and the solution of the Benjamin-Ono equation corresponding to the perturbed solitary wave (cf. Section 2 and Appendix A). In any case, an infinitesimal analysis of this problem, while suggestive in various ways, would not be conclusive. The paper is organized as follows. A statement of the problem at hand, together with various mathematical preliminaries, is provided in Section 2. A reduction of the problem to the estimation of certain integrals is given in Section 3, while Section 4 is devoted to providing the required estimates. With the aid of these estimates, the proof of stability is completed in Section 5. The last section reviews the earlier accomplish- ments aDd suggests possible extensions of the theory. As noted above, Appendix A is devoted to commentary concerning linearized stability analysis. In Appendix B, 12 psp 94