Stefan Le Coz

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrodinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in H^1_rad and unstable in H^1 under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blow-up in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

A Note on Berestycki-Cazenave's Classical Instability Result for Nonlinear Schrodinger Equations

In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schrödinger equations. 2000 Mathematics Subject Classification. 35Q55,(35B35,35A15).

High-speed excited multi-solitons in nonlinear Schrödinger equations

Abstract We consider the nonlinear Schrodinger equation in R d i ∂ t u + Δ u + f ( u ) = 0 . For d ⩾ 2 , this equation admits traveling wave solutions of the form e i ω t Φ ( x ) (up to a Galilean transformation), where Φ is a fixed profile, solution to − Δ Φ + ω Φ = f ( Φ ) , but not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable.

INSTABILITY FOR STANDING WAVES OF NONLINEAR KLEIN-GORDON EQUATIONS VIA MOUNTAIN-PASS ARGUMENTS

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 stability/instability. For a power-type nonlin- earity, we prove that the ground states of the associated stationary equation are minimizers of the functional action on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N = 2 the classical in- stability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension N > 3.

Multi-solitary waves for the nonlinear Klein-Gordon equation

We consider the nonlinear Klein-Gordon equation in ℝ d . We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.

On the Cauchy Problem and the Black Solitons of a Singularly Perturbed Gross--Pitaevskii Equation

We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we show the persistence of the stationary black soliton of the unperturbed problem as a solution. We also prove the existence of another branch of non-trivial stationary waves. Depending on the attractive or repulsive nature of the Dirac perturbation and of the type of stationary solutions, we prove orbital stability via a variational approach, or linear instability via a bifurcation argument.