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Stefan Le Coz

Publications and source records attributed to Stefan Le Coz.

27 records · Page 2Linked to original sources
arXiv2014

Minimal mass blow up solutions for a double power nonlinear Schrödinger equation

We consider a nonlinear Schrödinger 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 Raphaël and Szeftel for the construction of minimal blow up elements.

math.AP↗
arXiv2013

Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations

We look for solutions to generic nonlinear Schrödinger equations build upon solitons and kinks. Solitons are localized solitary waves and kinks are their non localized counter-parts. We prove the existence of infinite soliton trains, i.e. solutions behaving at large time as the sum of infinitely many solitons. We also show that one can attach a kink at one end of the train. Our proofs proceed by fixed point arguments around the desired profile. We present two approaches leading to different results, one based on a combination of dispersive estimates and Strichartz estimates, the other based only on Strichartz estimates.

math.AP↗
arXiv2013

Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations

We study *infinite soliton trains* solutions of nonlinear Schrödinger equations (NLS), i.e. solutions behaving at large time as the sum of infinitely many solitary waves. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighborhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

math.AP↗
arXiv2013

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.

math.AP↗
arXiv2013

Multi-Speeds solitary waves solutions for nonlinear Schrödinger systems

We prove the existence of a new type of solutions to a nonlinear Schrödinger system. These solutions, which we call "multi-speeds solitary waves", are behaving at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backwards in time and using energy methods to obtain uniform estimates.

math.AP↗
arXiv2011

High speed excited multi-solitons in nonlinear Schrödinger equations

We consider the nonlinear Schrödinger 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.

math.AP↗
arXiv2007

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.

math.AP↗