On the existence and qualitative theory of stratified solitary water waves

Stratified Large-Amplitude Steady Periodic Water Waves with Critical Layers

“…which is of the form R = −P − ψ M where P is the pressure and we choose M := γ . Using Yih's equation we find that in , the function R satisfies: Note, this is the equation derived in [1], Proposition 4.1. However, since ψ Y < 0 on ∪ B, the righthand side of (82) is strictly positive.…”

“…carefully verify that the added stratification term does not cause too many complications. Indeed, in past literature, one major difficulty of proving nodal properties for stratified waves lies in the fact that the zeroth order term generally has the wrong sign (for example, see [1,3]).…”

“…These waves are solutions to the incompressible Euler equations and the only exterior force acting on them is gravity. In this paper, we will only study periodic waves; for work on solitary waves, we refer to [1,2]. The term stratification here refers to the presence of a non-constant density function ρ(X, Y ), and throughout this paper, we will choose ρ to depend linearly on the streamfunction.…”

Pressure Transfer Functions for Interfacial Fluids Problems

“…Very recently, two of the authors provided the first construction of large-amplitude solitary waves with arbitrary (smooth and stable) stratification and background current [10,11]. In the present paper, however, we merely focus on periodic traveling waves.…”

Existence and amplitude bounds for irrotational water waves in finite depth

“…These bifurcation methods, however, cannot give a qualitative description of the obtained solutions as they rely upon topological arguments. Only based on a thorough analysis of the underlying equations, a priori estimates and bounds on physical quantities have been derived in different settings of the water-wave problem, both with and without vorticity [8,9].…”