Analytic Theory of Global Bifurcation

Analyticity of periodic traveling free surface water waves with vorticity

“…By the implicit function theorem for real-analytic maps [3] we now conclude the existence of some ε > 0 and some ϕ ∈ A((−ε, ε), O) such that in a sufficiently small neighbourhood of (h 0 , 0) ∈ X × R all solutions of Φ(h, a) = 0 are given by (h, a) = (ϕ(a), a). Taking into account (3.8), by uniqueness we deduce that τ a h 0 = ϕ(a) for a ∈ (−ε, ε).…”

Global Bifurcation for the Whitham Equation

“…The assertion then follows from the analytic version of the Crandall-Rabinowitz theorem for bifurcation from a simple eigenvalue [6,Thm 8.4.1]. The fact that the solutions are 2π/k-periodic can be seen by restricting attention to the subspaces {ϕ ∈ C α even (S) : ϕ is 2π/k-periodic}, and, corresponding to the case k = 0, it is instantly verified that ϕ = µ − 1 is a solution.…”

“…The following result is immediate (cf. [6]) if we are able to show that, in some small neighborhood |ε| < δ, µ(ε) is not identically equal to a constant.…”

Regularity of rotational travelling water waves

“…Thus, it follows from the implicit function theorem for real analytical mappings (see [22]) that there is 3 > 0 and a unique 4 ∈ A ((−3, 3), O) such that F(a, 4(a)) = 0 for all a ∈ (−3, 3). Taking into account (4.7), by uniqueness, we deduce that t a h 0 = 4(a) for a ∈ (−3, 3).…”