A note on Berestycki-Cazenave's classical instability result for nonlinear Schr\"odinger 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 Schr\"odinger equations

High speed excited multi-solitons in nonlinear Schr\"odinger equations

We consider the nonlinear Schr\"odinger equation with a general nonlinearity. In dimension higher than 2, this equation admits travelling wave solutions with a fixed profile which is 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

Standing waves in nonlinear Schrödinger equations

In the theory of nonlinear Schrödinger equations, it is expected that the solutions will either spread out because of the dispersive effect of the linear part of the equation or concentrate at one or several points because of nonlinear effects. In some remarkable cases, these behaviors counterbalance and special solutions that neither disperse nor focus appear, the so-called standing waves. For the physical applications as well as for the mathematical properties of the equation, a fundamental issue is the stability of waves with respect to perturbations. Our purpose in these notes is to present various methods developed to study the existence and stability of standing waves. We prove the existence of standing waves by using a variational approach. When stability holds, it is obtained by proving a coercivity property for a linearized operator. Another approach based on variational and compactness arguments is also presented. When instability holds, we show by a method combining a Virial identity and variational arguments that the standing waves are unstable by blow-up

A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations

International audienceIn 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

Minimal mass blow up solutions for a double power nonlinear Schr\"odinger equation

We consider a nonlinear Schr\"odinger equation with double power nonlinearity, where one power is focusing and mass critical and the other mass sub-critical. Classical variational arguments ensure that initial data with mass less than the mass of the ground state of the mass critical problem lead to global in time solutions. We are interested by the threshold dynamic and in particular by the existence of finite time blow up minimal solutions. For the mass critical problem, such an object exists thanks to the explicit conformal symmetry, and is in fact unique. For the focusing double power nonlinearity, we exhibit a new class of minimal blow up solutions with blow up rates deeply affected by the double power nonlinearity. The analysis adapts the recent approach developed by Rapha\"el and Szeftel for the construction of minimal blow up elements

Stability of multi-solitons for the derivative nonlinear Schrödinger equation

International audienceThe nonlinear Schrödinger equation with derivative cubic nonlinearity admits a family of solitons, which are orbitally stable in the energy space. In this work, we prove the orbital stability of multi-solitons configurations in the energy space, under suitable assumptions on the speeds and frequencies of the composing solitons. The main ingredients of the proof are modulation theory, energy coercivity and monotonicity properties

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 nonlinearity, 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 instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension N>2