Related papers: Global bifurcation for the Whitham equation
The present paper is concerned with the existence of traveling wave solutions of the asymptotic model, derived by the authors in a previous work, to approximate the unidirectional evolution of a collision-free plasma in a magnetic field.…
We consider singular perturbations of eigenvalue problems. We prove that to these problems correspond simple eigenvalues and we study their asymptotic behavior. As a result, we prove global bifurcation results for non uniformly and fully…
We prove the existence of a periodic traveling wave of extreme form of the Whitham equation that has a convex profile between consecutive stagnation points, at which it is known to feature a cusp of exactly $C^{1/2}$ regularity. The…
Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a…
In this article we derive rigorously a nonlinear, steady, bifurcation through spectral bifurcation (i.e., eigenvalues of the linearized equation crossing the imaginary axis) for a class of hyperbolic-parabolic model in a strip. This is…
This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work \cite{ConstantinStrauss}, we first obtain…
We use the $\mathbb T^2$-equivariant degree to establish the existence of unbounded branches of rotating spiral wave solutions with any number of arms for the complex Ginzburg Landau equation GLe on the planar unit disc, leveraging the…
New travelling wave solutions to the Fornberg-Whitham equation are investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.
We investigate the global bifurcation structure of the radial nodal solutions to the coupled elliptic equations \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u+u=u^3+\beta uv^2\mbox{ in }B_1 ,\nonumber -{\Delta}v+v=v^3+\beta…
We investigate the Cauchy problem for a 2x2-system of weakly coupled semi-linear fractional wave equations with polynomial nonlinearities posed in R+ x RN. Under appropriate conditions on the exponents and the fractional orders of the time…
The Whitham equation is a model for the evolution of surface waves on shallow water that combines the unidirectional linear dispersion relation of the Euler equations with a weakly nonlinear approximation based on the KdV equation. We show…
We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation $\delta$ from a global periodic traveling wave with small amplitude $\epsilon$. We use a modified energy method to prove the existence time of smooth…
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and…
We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod without any use of the…
We study a two-fluid description of high and low temperature components of the electron velocity distribution of an idealized tokamak plasma. We refine previous results on the laminar steady-state solution. On the one hand, we prove global…
The Whitham equation is a nonlocal, nonlinear partial differential equation that models the temporal evolution of spatial profiles of surface displacement of water waves. However, many laboratory and field measurements record time series at…
Analytic global bifurcation theory is used to construct a large variety of families of steady periodic two-dimensional gravity water waves with real-analytic vorticity distributions, propagating in an incompressible fluid. The waves that…
In this paper, we prove the existence of two-dimensional, traveling, capillary-gravity, water waves with compactly supported vorticity. Specifically, we consider the cases where the vorticity is a $\delta$-function (a point vortex), or has…
The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep water under gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a…
The paper deals with a problem of interaction between hydrodynamics and mechanics of nonlinear elastic bodies. The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy…