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EXISTENCE AND UNIQUENESS OF MINIMAL BLOW UP SOLUTIONS TO AN INHOMOGENEOUS MASS CRITICAL NLS
, 2010
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Exponential decay of eigenfunctions and generalized eigenfunctions of nonselfadjoint matrix Schrödinger operators related to NLS., preprint
, 2005
"... Abstract. We study�the decay of eigenfunctions � of the non selfadjoint −∆+µ+U W matrix operator H = −W ∆−µ−U, for µ> 0, corresponding to eigenvalues in the strip −µ < ReE < µ. 1. ..."
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Abstract. We study�the decay of eigenfunctions � of the non selfadjoint −∆+µ+U W matrix operator H = −W ∆−µ−U, for µ> 0, corresponding to eigenvalues in the strip −µ < ReE < µ. 1.
Coupled mode equations and gap solitons for the 2d GrossPitaevskii equation with a nonseparable periodic potential
 Physica D, 238(910):860 – 879
, 2009
"... equation with a nonseparable periodic potential ..."
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Solitary Wave Solutions for the Nonlinear Dirac Equations
, 2008
"... In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation i 3∑ γ j ∂jψ − mψ + F ( ¯ ψψ)ψ = 0 j=0 where m> 0 and F(s) = s  θ for 1 ≤ θ < 2. More precisely we show that there exists ε0> 0 such that for ω ∈ (m − ε0, m), there exists a s ..."
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In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation i 3∑ γ j ∂jψ − mψ + F ( ¯ ψψ)ψ = 0 j=0 where m> 0 and F(s) = s  θ for 1 ≤ θ < 2. More precisely we show that there exists ε0> 0 such that for ω ∈ (m − ε0, m), there exists a solution ψ(t, x) = e −iωt φω(x), x0 = t, x = (x1, x2, x3), and the mapping from ω to φω is continuous. We prove this result by relating the stationary solutions to the ground states of nonlinear Schrödinger equations. 1
SPECTRAL STABILITY ANALYSIS FOR SPECIAL SOLUTIONS OF SECOND ORDER IN TIME PDE’S: THE HIGHER DIMENSIONAL CASE
"... Abstract. We develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary PDEs. We apply these results to standing waves of the following problems: the KleinGordon equation, for which we consider both ground states and certain excited stat ..."
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Abstract. We develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary PDEs. We apply these results to standing waves of the following problems: the KleinGordon equation, for which we consider both ground states and certain excited states, the KleinGordonZakharov system and the beam equation. We also discuss possible applications to some nonstandard ground and excited states for the KleinGordon model as well as multidimensional traveling waves (not necessarily homoclinic to zero) for general second order equations of this type. In all cases, our abstract results provide a complete characterization of the linear stability of such solutions. 1.
BREATHING PATTERNS IN NONLINEAR RELAXATION
"... Abstract. In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes “breathing ” in the sense of the amplitude of the wave going up and down on a much faster scale than the motion of the wave – see Fig. 2 below. In this paper we investigate thi ..."
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Abstract. In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes “breathing ” in the sense of the amplitude of the wave going up and down on a much faster scale than the motion of the wave – see Fig. 2 below. In this paper we investigate this phenomenon in the simplest case of stationary waves in which the evolution corresponds to relaxation to a nonlinear ground state. The particular model is the popular δ0 impurity in the cubic nonlinear Schrödinger equation on the line. We give asymptotics of the amplitude on a finite but relevant time interval and show their remarkable agreement with numerical experiments, see Fig. 1. We stress the nonlinear origin of the “breathing patterns ” caused by the selection of the ground state depending on the initial data, and by the nonnormality of the linearized operator. 1.
Multisolitary waves for the nonlinear KleinGordon equation
, 2013
"... We consider the nonlinear KleinGordon equation inR d. We call multisolitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multisolitary waves, provided the composing boosted standing waves are stable. It is obtained by solving ..."
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We consider the nonlinear KleinGordon equation inR d. We call multisolitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multisolitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multisolitary 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.