Related papers: Stability of multipeakons
In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peakons. The 3CH model is proven integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system…
The purpose of this paper is to prove that, for a large class of nonlinear evolution equations known as scalar viscous balance laws, the spectral (linear) instability condition of periodic traveling wave solutions implies their orbital…
We study the stability of a four parameter family of spatially periodic traveling wave solutions of the generalized Benjamin-Bona-Mahony equation to two classes of perturbations: periodic perturbations with the same periodic structure as…
de la Pe\~na 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics are stable. Though the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation.…
In this work, we present a stability criteria for the solitary wave solutions to a BBM system that contains coupled nonlinear terms. Using the idea by Bona, Chen and Karakashian and exploiting the accurate point spectrum information of the…
This paper is devoted to periodic travelling waves solving Lie-Poisson equations based on the Virasoro group. We show that the reconstruction of any such solution can be carried out exactly, regardless of the underlying Hamiltonian (which…
In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schr\"odinger equations with potentials. We show that the standing waves are orbitally stable for all frequencies in the $L^{2}$- subcritical and…
Long-time evolution of a weakly perturbed wavetrain near the modulational instability threshold is investigated within the framework of the compact Zakharov equation for unidirectional deep-water waves, recently derived by Zakharov &…
It is known from the previous works that the peakon solutions of the Novikov equation are orbitally and asymptotically stable in $H^1$. We prove, via the method of characteristics, that these peakon solutions are unstable under…
We discuss the recent paper by A. Dar\'os and L.K. Arruda (On the instability of elliptic traveling wave solutions of the modified Camassa-Holm equation, J. Diff. Equat., 266 (2019), 1946-1968). Our intention is to correct some…
We give a necessary and sufficient condition for strong stability of low dimensional Hamiltonian systems, in terms of the iterates of a closed orbit and the Conley-Zehnder index. Applications to Mathieu equation and stable harmonic…
In this paper, we study the nonlinear wave modulation of arbitrary amplitude periodic traveling wave solutions of the Camassa-Holm (CH) equation. Slow modulations of wave trains is often described through Whitham's theory of modulations,…
By combining results of Mizumachi on the stability of solitons for the Toda lattice with a simple rescaling and a careful control of the KdV limit we give a simple proof that small amplitude, long-wavelength solitary waves in the…
We consider singular solutions of a system of two cross-coupled Camassa-Holm (CCCH) equations. This CCCH system admits peakon solutions, but it is not in the two-component CH integrable hierarchy. The system is a pair of coupled Hamiltonian…
We present a study on the nonlinear dynamics of a disturbance to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. The associated Navier-Stokes equations are reduced to a set of coupled generalized Camassa-Holm type…
We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all…
In this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional…
Periodic waves are investigated in a system composed of a Kuramoto-Sivashinsky - Korteweg-de Vries (KS-KdV) equation, which is linearly coupled to an extra linear dissipative equation. The model describes, e.g., a two-layer liquid film…
We consider the nonlinear Dirac equation in 1+1 dimension with scalar-scalar self interaction $ \frac{g^2}{\kappa+1} ({\bar \Psi} \Psi)^{\kappa+1}$ and with mass $m$. Using the exact analytic form for rest frame solitary waves of the form…
Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the…