Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

Abstract

The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic extensible rod, and the fluid beneath is supposed to be in steady two-dimensional irrotational motion under gravity. When the wavelength has been normalized to be 2π , and when gravity and the density of the undeformed membrane are prescribed, there are two free parameters in the problem: the speed of the wave and the drift velocity of the membrane.It is observed that the problem, when linearized about uniform horizontal flow, has at most two independent solutions for any values of the parameters. When the linearized problem has only one normalized solution, it is shown that the full nonlinear problem has a sheet of solutions consisting of a family of curves bifurcating from simple eigenvalues. Here one of the problem's parameters is used to index a family of bifurcation problems in which the other is the bifurcation parameter.When the linearized problem has two solutions, with wave numbers k and l such that max{k, l}/min{k, l} / ∈ Z, it is shown that there are three two-dimensional sheets of bifurcating solutions. One consists of "special" solutions with minimal period 2π/k; another consists of "special" solutions with minimal period 2π/ l; and the third, apart from those on the curves where it intersects the "special" sheets, consists of "general" solutions with minimal period 2π .The two sheets of "special" solutions are rather similar to those that occur when the linearized problem has only one solution. However, points where the first sheet or the second sheet intersects the third sheet are period-multiplying (or symmetry-breaking) secondary bifurcation points on primary branches of "special" solutions. This phenomenon is analogous to that of Wilton ripples, which arises in the classical water-wave problem when the surface tension has special values. In the case of Wilton ripples, the coefficient of surface tension and the wave speed are the problem's two parameters. In the present context, there are two speed parameters, meaning that the membrane elasticity does not need to be highly specified for this symmetry-breaking phenomenon to occur.

Trimodal Steady Water Waves

“…The first part of the proof is analogous to that of [9, Theorem 4.8] (see also [1]). The second part involves calculating the determinant of 3 × 3-matrix, the entries of which depend transcendentally on the parameters of the problem.…”

Steady Water Waves with Multiple Critical Layers

“…Our approach is based on the Lyapunov-Schmidt reduction and the implicit function theorem. The same method was recently used to study the bifurcation of steady irrotational water waves under a heavy elastic membrane [1].…”

Periodic traveling interfacial hydroelastic waves with or without mass

“…Even if (θ, γ; c) yield a traveling wave solution (z, γ), we cannot expect that 2π-periodic θ to yield periodic z via (18). We would like for any 2π-periodic (θ, γ) that solve some equations analogous to (16) and (17) to correspond directly to a periodic traveling wave solution (z, γ) of (2), (5), and (10). Hence, in a manner closely analogous to [9], we modify the mappings in (16) and (17) to ensure this.…”

“…The second author, Siegel, and Liu have shown that the initial value problems for these Cosserat-type hydroelastic waves are well-posed in Sobolev spaces [8], [16]. Toland and Baldi and Toland have proved existence of periodic traveling hydroelastic water waves with and without mass including studying secondary bifurcations [25], [26], [10], [11]. A number of authors have also computed traveling hydroelastic water waves, finding results in 2D and 3D, computations of periodic and solitary waves, comparison with weakly nonlinear models, and comparison across different modelling assumptions for the bending force [13], [14], [17], [18], [19], [28], [29].…”

“…The authors will treat the cases of two-dimensional kernels in a subsequent paper. This will involve studying secondary bifurcations as in [10], and also studying Wilton ripples [22], [23], [30], [3], [27].…”