Anna Geyer
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View article: On Ocean Currents with Constant Vorticity: Explicit Solutions and an Application to the ACC
On Ocean Currents with Constant Vorticity: Explicit Solutions and an Application to the ACC Open
We study the three‐dimensional, divergence‐free, incompressible Euler equations in the ‐plane approximation for off‐equatorial oceanic flows of constant vorticity, where the fluid domain is bounded by a free surface and a flat bed. The maj…
View article: Stability of smooth periodic traveling waves in the Degasperis–Procesi equation
Stability of smooth periodic traveling waves in the Degasperis–Procesi equation Open
We derive a precise energy stability criterion for smooth periodic waves in the Degasperis–Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation, the number of negative eigenvalues of an associated Hessian operator changes in t…
View article: On the transverse stability of smooth solitary waves in a two-dimensional Camassa-Holm equation
On the transverse stability of smooth solitary waves in a two-dimensional Camassa-Holm equation Open
We consider the propagation of smooth solitary waves in a two-dimensional generalization of the Camassa--Holm equation. We show that transverse perturbations to one-dimensional solitary waves behave similarly to the KP-II theory. This conc…
View article: Stability of smooth periodic traveling waves in the Degasperis-Procesi equation
Stability of smooth periodic traveling waves in the Degasperis-Procesi equation Open
We derive a precise energy stability criterion for smooth periodic waves in the Degasperis--Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation, the number of negative eigenvalues of an associated Hessian operator changes in …
View article: Weakly nonlinear waves in stratified shear flows
Weakly nonlinear waves in stratified shear flows Open
We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, the…
View article: Stability of smooth periodic travelling waves in the Camassa–Holm equation
Stability of smooth periodic travelling waves in the Camassa–Holm equation Open
We solve the open problem of spectral stability of smooth periodic waves in the Camassa–Holm equation. The key to obtaining this result is that the periodic waves of the Camassa–Holm equation can be characterized by an alternative Hamilton…
View article: Stability of smooth periodic traveling waves in the Camassa-Holm equation
Stability of smooth periodic traveling waves in the Camassa-Holm equation Open
Smooth periodic travelling waves in the Camassa--Holm (CH) equation are revisited. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common…
View article: Stability of smooth periodic traveling waves in the Camassa-Holm\n equation
Stability of smooth periodic traveling waves in the Camassa-Holm\n equation Open
Smooth periodic travelling waves in the Camassa--Holm (CH) equation are\nrevisited. We show that these periodic waves can be characterized in two\ndifferent ways by using two different Hamiltonian structures. The standard\nformulation, com…
View article: Spectral instability of the peaked periodic wave in the reduced Ostrovsky equations
Spectral instability of the peaked periodic wave in the reduced Ostrovsky equations Open
We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. We dis…
View article: Spectral instability of the peaked periodic wave in the reduced Ostrovsky equation
Spectral instability of the peaked periodic wave in the reduced Ostrovsky equation Open
We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. The ma…
View article: Spectral instability of the peaked periodic wave in the reduced\n Ostrovsky equation
Spectral instability of the peaked periodic wave in the reduced\n Ostrovsky equation Open
We show that the peaked periodic traveling wave of the reduced Ostrovsky\nequations with quadratic and cubic nonlinearity is spectrally unstable in the\nspace of square integrable periodic functions with zero mean and the same\nperiod. The…
View article: Shallow water models for stratified equatorial flows
Shallow water models for stratified equatorial flows Open
Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full govern…
View article: Linear instability and uniqueness of the peaked periodic wave in the reduced Ostrovsky equation
Linear instability and uniqueness of the peaked periodic wave in the reduced Ostrovsky equation Open
Stability of the peaked periodic waves in the reduced Ostrovsky equation has remained an open problem for a long time. In order to solve this problem we obtain sharp bounds on the exponential growth of the $L^2$ norm of co-periodic perturb…
View article: Linear instability and uniqueness of the peaked periodic wave in the\n reduced Ostrovsky equation
Linear instability and uniqueness of the peaked periodic wave in the\n reduced Ostrovsky equation Open
Stability of the peaked periodic waves in the reduced Ostrovsky equation has\nremained an open problem for a long time. In order to solve this problem we\nobtain sharp bounds on the exponential growth of the $L^2$ norm of co-periodic\npert…
View article: Traveling wave solutions of a highly nonlinear shallow water equation
Traveling wave solutions of a highly nonlinear shallow water equation Open
Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasil…
View article: Shallow water equations for equatorial tsunami waves
Shallow water equations for equatorial tsunami waves Open
We present derivations of shallow water model equations of Korteweg–de Vries and Boussinesq type for equatorial tsunami waves in the f -plane approximation and discuss their applicability. This article is part of the theme issue ‘Nonlinear…
View article: Symmetric solutions of evolutionary partial differential equations
Symmetric solutions of evolutionary partial differential equations Open
We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms …
View article: On the number of limit cycles for perturbed pendulum equations
On the number of limit cycles for perturbed pendulum equations Open
Agraïments: The second author is supported by the project J3452 "Dynamical Systems Methods in Hydrodynamics" of the Austrian Science Fund (FWF).
View article: On the number of limit cycles for perturbed pendulum equations
On the number of limit cycles for perturbed pendulum equations Open
We consider perturbed pendulum-like equations on the cylinder of the form $ \ddot x+\sin(x)= \varepsilon \sum_{s=0}^{m}{Q_{n,s} (x)\, \dot x^{s}}$ where $Q_{n,s}$ are trigonometric polynomials of degree $n$, and study the number of limit c…
View article: On the wave length of smooth periodic traveling waves of the Camassa–Holm equation
On the wave length of smooth periodic traveling waves of the Camassa–Holm equation Open
This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height a (or "peak-to-peak amplitude"). Our main result …
View article: On the wave length of smooth periodic traveling waves of the Camassa-Holm equation
On the wave length of smooth periodic traveling waves of the Camassa-Holm equation Open
This paper is concerned with the wave length $λ$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$ (or "peak-to-peak amplitude"). Our main res…
View article: On the wave length of smooth periodic traveling waves of the\n Camassa-Holm equation
On the wave length of smooth periodic traveling waves of the\n Camassa-Holm equation Open
This paper is concerned with the wave length $\\lambda$ of smooth periodic\ntraveling wave solutions of the Camassa-Holm equation. The set of these\nsolutions can be parametrized using the wave height $a$ (or "peak-to-peak\namplitude"). Ou…