On the theory of internal waves of permanent form in fluids of great depth

Reversible Bifurcation of Homoclinic Solutions in Presence of an Essential Spectrum

“…We also assume that the upper layer is bounded by a rigid horizontal top. This case was treated, using different formulations by Amick [1] and Sun [12]. Since the spatial dynamics formulation we use is very similar to the one used in the second example, we only present the details for the second example.…”

“…We assume that the upper layer is bounded by a rigid horizontal top and the bottom one is infinitely deep. We are interested in travelling waves of horizontal velocity c. The existence of solitary waves with polynomial decay at infinity has been obtained independently by Amick [1] and Sun [12]. The solitary wave (see Figure 1), corresponding to a homoclinic solution in a dynamical system approach, is approximated at first order by the Benjamin-Ono solitary wave (see (1.4) below).…”

“…2 (R)) in the subset of even functions, thanks to a result of [1]. Actually we need to adapt the implicit function theorem as in the proof of Theorem 4.9, since we fix ε small enough, but not 0 (G is not defined for ε = 0).…”

Asymptotic Behavior and Symmetry of Internal Waves in Two-Layer Fluids of Great Depth

“…In addition, the interface y*='*(x*) is analytic using a general theory for regularity of solutions in elliptic free boundary problems by Kinderlehrer et al [18]. Therefore, the results obtained here give a mathematical justification of the symmetry assumption made for the solitary wave solutions of the exact equations in [3,20] and show that the solitary wave solution in [3,20] possesses the properties stated above, some of which cannot be obtained there. The method and ideas to obtain the symmetry of solution of the exact equations are essentially from the work by Craig and Sternberg [13].…”

“…As mentioned in a survey paper by Benjamin et al [8], the existence of solitary internal wave solution of the exact governing equations for fluids of infinite depth was still unsettled until recently. For a two-fluid flow of infinite depth bounded below by a horizontal rigid bottom, it was shown by Amick [3] and Sun [20] using different methods that the exact governing equations of a two-fluid flow have solitary wave solutions and the first order approximations of such solutions are the solutions of the Benjamin Ono equation. The solution is a supercritical solution and decays algebraically to the equilibrium at infinity.…”