相关论文: The stability of Sobolev norms for the linear wave…
In this paper, we study the stability and instability of plane wave solutions to semilinear systems of wave equations satisfying the null condition. We identify a condition which allows us to prove the global nonlinear asymptotic stability…
Considered herein is the initial-value problem for the generalized periodic Camassa-Holm equation which is related to the Camassa-Holm equation and the Hunter-Saxton equation. Sufficient conditions guaranteeing the development of breaking…
We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we…
This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.
For a family of 1-d quantum harmonic oscillator with a perturbation which is $C^2$ parametrized by $E\in{\mathcal I}\subset{\Bbb R}$ and quadratic on $x$ and $-{\rm i}\partial_x$ with coefficients quasi-periodically depending on time $t$,…
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…
In this paper we consider the transverse instability for a nonlinear Schr\"odinger equation with a linear potential on $\mathbb{R} \times \mathbb{T}_L$, where $2\pi L$ is the period of the torus $\mathbb{T}_L$. Rose and Weinstein showed the…
We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schr\"odinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the…
In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ${\mathbb R} \times {\mathbb R}^d$ with $d \geq 6$. We prove the stability of solutions under the weak condition…
We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two…
In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schr\"odinger equation on the two dimensional torus. We prove that these…
Given small initial solutions of the nonlinear quantum harmonic oscillator on $\mathbb{R}$, we are interested in their long time behavior in the energy space which is an adapted Sobolev space. We perturbate the linear part by $V$ taken as…
The two-dimensional cubic nonlinear Schrodinger equation (NLS) can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. In this paper, we establish that every one-dimensional…
In this paper, we explore the orbital stability of smooth solitary wave solutions to the modified Camassa-Holm equation with cubic nonlinearity. These solutions, which exist on a nonzero constant background $k$, are unique up to translation…
In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subjects to periodic boundary conditions. More exactly, the linear wave equation $u_{tt}-u_{xx}+Mu+\varepsilon…
In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point…
This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schr{\"o}dinger equation on the torus :$$i \partial\_t u = |D|^\alpha u+|u|^2 u, \quad u(0, \cdot)=u\_0,$$where…
In this paper we study the stability of the unique continuation in the case of the wave equation with variable coefficients independent of time. We prove a logarithmic estimate in a arbitrary domain of ${\mathbb R}^{n+1}$, where all the…
We consider the damped hyperbolic equation in one space dimension $\epsilon u_{tt} + u_t = u_{xx} + F(u)$, where $\epsilon$ is a positive, not necessarily small parameter. We assume that $F(0)=F(1)=0$ and that $F$ is concave on the interval…
In the framework of variable exponent Sobolev spaces, we prove that the variational eigenvalues defined by inf sup procedures of Rayleigh ratios for the Luxemburg norms are all stable under uniform convergence of the exponents.