Small-amplitude steady water waves with critical layers: Non-symmetric waves

Abstract

The problem for two-dimensional steady water waves with vorticity is considered. Using methods of spatial dynamics, we reduce the problem to a finite dimensional Hamiltonian system. The reduced system describes all smallamplitude solutions of the problem and, as an application, we give a proof of the existence of non-symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one.

Solitary waves on constant vorticity flows with an interior stagnation point

“…where λ ∈ Λ , and soΨ ∈X (λ) 1 andΦ ∈X (λ) 2 for all x ∈ R and λ ∈ Λ . Then theorem 3.1 proved in Kozlov & Lokharu (2019) yields the following assertion for the decomposed system (3.3)-(3.8a,b). THEOREM 3.1.…”

“…Moreover, we find that Substituting these expressions into formulae for , and , we see that (2.12 a ) and (2.12 b ) form an infinite-dimensional reversible dynamical system on the manifold defined by (2.12 c ) and (2.12 d ); the mapping is the reverser. Nonlinearity of the boundary condition (2.12 d ) is inessential in view of its reducibility to a homogeneous one by a proper change of variables; see Groves & Wahlén (2008) and Kozlov & Lokharu (2019) for details.…”

“…(2016) have critical layers, the geometry of free-surface profiles is still simple; it is symmetric about every crest and trough, whereas monotone in between (just like that of the classical Stokes waves). Examples of more complicated wave profiles are known (see Ehrnström, Escher & Wahlén 2011; Aasen & Varholm 2017; Kozlov & Lokharu 2017, 2019). However, the vorticity distribution must be at least linear in order to construct them.…”

Global Bifurcation of Waves with Multiple Critical Layers

“…There are, of course, several papers on stagnant waves that do not fit neatly into our categorizations above. For instance, a recent paper [21] constructs small-amplitude nonsymmetric waves with critical layers using a spatial-dynamics approach, as opposed to the bifurcation-theoretic nature of the above results. There is also a preprint [19] on solitary waves with constant vorticity and a critical layer connecting with the bed, again using spatial dynamics.…”

Symmetric solutions of evolutionary partial differential equations

“…Indeed, as follows from the result [25] by Li and the principle (P1) presented below, for a large set of equations the sets of spatially symmetric and travelling solutions completely coincide; an example of this behaviour is the Whitham equation, see [3]. Note that this is not the case for the free-boundary Euler equations: although we give in Section 4 the Euler equations as an example of systems belonging to principle (P1), and even though one can show that large classes of its steady solutions are symmetric [5,32], there are also non-symmetric steady solutions [21].…”