Stability of the lower cusped solitary waves

where 3 depends’on the cross section area of the topography [see Miles’ Eqs. (5.2) and (5.3)‘]. A year later, VandenBroeck published his discovery of two smooth solitary wave solutions for the same physical model but from the perspective of direct numerical integration of Laplace equation with a free boundary and with nonlinear boundary conditions.2 One of his solitary waves is higher than the other and Miles’ solitary wave is considered corresponding to the lower one.’ It is common knowledge that a single layer invicid, incornpressible fluid flow over a flat bottom can support a stable solitary wave, called the free solitary wave. Vanden-Broeck pointed out that the higher solitary wave can be regarded as a perturbation of the free solitary wave and the lower solitary wave as a perturbation of the trivial solution of the flat bottom model. The perturbation is, of course, due to the presence of the bump. These smooth soiitary waves can be approximated by the two solitary wave solutions of a stationary forced KdV equation.5 When the forcing has a very shot support, the forcing may be approximately expressed in terms of the Dirac delta function. Then the two solitary wave solutions of the stationary forced KdV equation can be found analytically, but each of these solitary waves has a cusp right on the location of the delta function.” tionary fKdV, which one is stable. Because of the stability of the free solitary wave, people’s intuition might tend to suggest that the higher solitary wave is stable. In 1988, Malomed pointed out that this, as a matter of fact, is incorrect. He proved that the higher cusped solitary wave is unstable.” This is an important contribution to the fKdV studies for the case of F> F, . He further conjectured that the lower solitary wave “is, to all appearance, stable” (p. 401 of Ref. 4). In the caption of his Fig. 4, his statement is that the lower solitary wave is “presumably stable.” To the authors’ knowledge,. Malomed’s conjecture has not been rigorously proved. The purpose of our present work is to provide a numerical verification (not a mathematical proof) of his conjecture. Hence, our work helps with clarifying some past possible confusions about the stability of the cusped solitary waves. The serious difficulty of maintaining the cusp profile due to the strong dispersion was overcome by choosing proper time and space integrations in our semiimplicit spectral scheme. The two cusped solitary waves in questions are the solutions of the following BVP: