This work is concerned with stability properties of periodic traveling waves solutions of the focusing Schrödinger equation iu t + u xx + |u| 2 u = 0 posed in R, and the modified Korteweg-de Vries equationOur principal goal in this paper is the study of positive periodic wave solutions of the equation φ ω + φ 3 ω − ωφ ω = 0, called dnoidal waves. A proof of the existence of a smooth curve of solutions with a fixed fundamental period L, ω ∈ (2π 2 /L 2 , +∞) → φ ω ∈ H ∞ per ([0, L]), is given. It is also shown that these solutions are nonlinearly stable in the energy space H 1 per ([0, L]) and unstable by perturbations with period 2L in the case of the Schrödinger equation.

We consider the coupled Schrödinger–Korteweg–de Vries system which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δi, ci, we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.

No abstract

We study the nonlinear Schrödinger equation (NLS) on a star graph G. At the vertex an interaction occurs described by a boundary condition of delta type with strength α ∈ R. We investigate the orbital instability of the standing waves e iωt Φ(x) of the NLS-δ equation with attractive power nonlinearity on G when the profile Φ(x) has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein -von Neumann, and the analytic perturbations theory, avoiding the variational techniques standard in the stability study. We also prove the orbital stability of the unique standing wave solution to the NLS-δ equation with repulsive nonlinearity.2010 Mathematics Subject Classification. Primary: 35Q55, 81Q35, 37K40, 37K45; Secondary: 47E05.

This paper sheds new light on the stability properties of solitary wave solutions associated with models of Korteweg-de Vries and Benjamin&Bona&Mahoney type, when the dispersion is very lower. Via an approach of compactness, analyticity and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of linear instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and linear instability of the ground states solutions for both models is obtained for some specific regimen of parameters. Via a Lyapunov strategy and a variational analysis we obtain the stability of the blow-up of solitary waves for the critical fractional KdV equation.The arguments presented in this investigation has prospects for the study of the instability of traveling waves solutions of other nonlinear evolution equations.