Related papers: Asymptotic stability of scalar multi-D inviscid sh…
We consider perturbations of the one-dimensional cubic Schr\"odinger equation, under the form $i \, \partial_t \psi + \partial_x^2 \psi + |\psi|^2 \psi - g( |\psi|^2 ) \psi = 0$. Under hypotheses on the function g that can be easily…
In this paper, we study the asymptotic stability of two wave equations coupled by velocities of anti-symmetric type via only one damping. We adopt the frequency domain method to prove that the system with smooth initial data is…
We consider the following hypothesis: some of KdV equation shock-like waves are invariant with respect to the combination of the Galilean symmetry and KdV equation higher symmetries. Also we demonstrate our approach on the example of…
We consider a noncommutative scalar field with a covariantly constant noncommutative parameter in a curved space-time background. For a potential as a noncommutative polynomial it is shown that the stability conditions are unaffected by the…
Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schr\"odinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, that span a…
We report the experimental observation of intermittency in a regime dominated by random shock waves on the surface of a fluid. We achieved such a nondispersive surface-wave field using a magnetic fluid subjected to a high external magnetic…
This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic…
This paper is concerned with the asymptotic stability of a composite wave of two viscous shocks under spatially periodic perturbations for the 1-D full compressible Navier-Stokes equations. It is proved that as time increases, the solution…
We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$. This model has solitary waves with speeds $-1<c<1$. When $|c|$ approaches 1, Bona and Sachs showed…
This paper proves the existence of unstable shocks of the Burgers-Hilbert equation conjectured in arXiv:2006.05568. More precisely, we construct smooth initial data with finite $H^9$-norm such that the solution in self-similar coordinates…
Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote…
We investigate the stability of traveling front solutions to nonlinear diffusive-dispersive equations of Burgers type, with a primary focus on the Korteweg-de Vries-Burgers (KdVB) equation, although our analytical findings extend more…
We complete a full classification of non-degenerate traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under (piecewise) smooth perturbations. A striking feature of our analysis is…
We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a nonnegative coefficient. Some examples on nonstandard time scales are also…
We demonstrate that, in contrast with what was previously believed, multi-hump solitary waves can be stable. By means of linear stability analysis and numerical simulations, we investigate the stability of two- and three-hump solitary waves…
In this work, we consider a system of two wave equations coupled by velocities in one-dimensional space, with one boundary fractional damping. First, we show that the system is strongly asymptotically stable if and only if the coupling…
We consider a nonlinear Schr\"odinger equation with double power nonlinearity \begin{align*} i\partial_t u+\Delta u-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \end{align*} where $1<p<q<1+4/(N-2)_+$. Due to the…
Coherent structures, such as solitary waves, appear in many physical problems, including fluid mechanics, optics, quantum physics, and plasma physics. A less studied setting is found in geophysics, where highly viscous fluids couple to…
The present contribution contains a quite extensive theory for the stability analysis of plane periodic waves of general Schr{\"o}dinger equations. On one hand, we put the one-dimensional theory, or in other words the stability theory for…
We propose the existence of an infinite-parameter family of solutions in AdS that oscillate on any number of non-commensurate frequencies. Some of these solutions appear stable when perturbed, and we suggest that they can be used to map out…