Related papers: Odd Periodic Waves for some Klein-Gordon Type Equa…
In this paper, we investigate the spectral stability of periodic traveling waves in the two dimensional gravity-capillary water wave problem. We derive a stability criterion based on an index function, whose sign determines the spectral…
We study the instability of standing waves for nonlinear Schr\"{o}dinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a…
We study existence and stability of standing waves for coupled nonlinear Hartree type equations \[ -i\frac{\partial}{\partial t}\psi_j=\Delta \psi_j+\sum_{k=1}^m \left(W\star |\psi_k|^p \right)|\psi_j|^{p-2}\psi_j, \] where…
We study the focusing power nonlinear wave equation with any power, in Minkowski space of any spacetime dimension. We present a complete understanding of the local stability and scattering theory (both in high regularity spaces) for…
We consider a system of two coupled non-linear Klein-Gordon equations. We show the existence of standing waves solutions and the existence of a Lyapunov function for the ground state.
We study local well-posedness and orbital stability/instability of standing waves for a first order system associated with a nonlinear Klein-Gordon equation on a star graph. The proof of the well-posedness uses a classical fixed point…
This paper studies the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler…
We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $ x $) of a $ 2 $-dimensional ocean with infinite depth under the action of…
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…
Euler's equations govern the behavior of gravity waves on the surface of an incompressible, inviscid, and irrotational fluid of arbitrary depth. We investigate the spectral stability of sufficiently small-amplitude, one-dimensional Stokes…
We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital…
We prove the existence of time-periodic solutions to non-linear massive Klein-Gordon equations in Anti-de Sitter as well as their orbital stability over exponentially long times for certain values of the mass corresponding to completely…
In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated…
This paper presents a comprehensive analysis of several aspects of the sinh-Gordon equation within a periodic setting. Our investigation proceeds in three main stages. First we establish the existence of periodic solutions for a fixed wave…
In this paper, we consider the degenerate semi-linear Schr\"odinger and Korteweg-deVries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which…
We analyze the Benney model for interaction of short and long waves in resonant water wave interactions. Our particular interest is in the periodic traveling waves, which we construct and study in detail. The main results are that, for all…
This paper is concerned with the Klein-Gordon-Maxwell system in a bounded spatial domain. We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We…
In this paper, we determine the transverse instability of periodic standing wave solutions for the generalized Schr\"odinger equation with fractional power nonlinearity. The existence of periodic waves is determined by using a constrained…
We give a new purely algebraic approach to odd unitary groups using odd form rings. Using these objects, we prove the stability theorems for odd unitary $K_1$-functor without using the corresponding result from linear $K$-theory under the…
We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi $ with periodic boundary conditions. Our result includes all values of…