Solitons in quadratic media

Abstract

In this paper, we investigate the properties of solitonic structures arising in quadratic media. First, we recall the derivation of systems governing the interaction process for waves propagating in such media and we check the local and global well-posedness of the corresponding Cauchy problem. Then, we look for stationary states in the context of normal or anomalous dispersion regimes, that lead us to either elliptic or non-elliptic systems and we address the problem of orbital stability. Finally, some numerical experiments are carried out in order to compute localized states for several regimes and to study dynamic stability as well as long-time asymptotics.

On a nonlinear Schrödinger system arising in quadratic media

“…Noticing that P 2 Q ∈ R + (see [2]), we can show that the orbit of every ground state contains an element (P ,Q), withP ,Q > 0. More precisely:…”

“…Now, observe that since (P, Q) is a solution of (9), A(0) = B(0) = 0 (see [2], (5.2)). Hence, putting α 0 = α ′ (0) and λ 0 = λ ′ (0),…”

“…Here, ǫ 0 is the vacuum permittivity, E represents the electric field and ω 0 its angular frequency (see [2] for a rigourous derivation of (1) from the Maxwell-Faraday equation and Ampère's Law). In fact, the quadratic Schrödinger system (1) governs the dynamics of propagation in χ (2) media in other physical contexts, namely in nonlinear optics (see for instance [10], [11], [12]).…”

On the Existence of Ground States for a Nonlinear Klein-Gordon-Maxwell Type System

“…To prove the existence of a ground state, we adapt a similar argument as in [8]. Here we briefly explain the strategy.…”

On the dynamics of a quadratic Schrödinger system in dimension $n = 5$

“…From the mathematical point of view, the study of nonlinear Schrödinger systems with quadratic interaction has been increasing in recent years. To cite a few, we refer the reader to [3], [4], [7], [8], [9], [12], [14], [15], [20], and references therein. An almost complete study of system (1.1) in L 2 (R n ) and H 1 (R n ) was undertaken in [9] (see also [7], [8]).…”