In this paper we give a systematic and simple account that put in evidence that many breather solutions of integrable equations satisfy suitable variational elliptic equations, which also implies that the stability problem reduces in some sense to (i) the study of the spectrum of explicit linear systems (spectral stability), and (ii) the understanding of how bad directions (if any) can be controlled using low regularity conservation laws. We exemplify this idea in the case of the modified Korteweg-de Vries (mKdV), Gardner, and the more involved sine-Gordon (SG) equation. Then we perform numerical simulations that confirm, at the level of the spectral problem, our previous rigorous results in [8,10], where we showed that mKdV breathers are H 2 and H 1 stable, respectively. In a second step, we also discuss the Gardner case, a relevant modification of the KdV and mKdV equations, recovering similar results. Then we discuss the Sine-Gordon case, where the spectral study of a fourth-order linear matrix system is the key element to show stability. Using numerical methods, we confirm that all spectral assumptions leading to the H 2 × H 1 stability of SG breathers are numerically satisfied, even in the ultra-relativistic, singular regime. In a second part, we study the periodic mKdV case, where a periodic breather is known from the work of Kevrekidis et al. [41]. We rigorously show that these breathers satisfy a suitable elliptic equation, and we also show numerical spectral stability. However, we also identify the source of nonlinear instability in the case described in [41], and we conjecture that, even if spectral stability is satisfied, nonlinear stability/instability depends only on the sign of a suitable discriminant function, a condition that is trivially satisfied in the case of non-periodic breathers. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.

In this article we prove that 2-soliton solutions of the sine-Gordon equation (SG) are orbitally stable in the natural energy space of the problem. The solutions that we study are the 2-kink, kink-antikink and breather of SG. In order to prove this result, we will use Bäcklund transformations implemented by the Implicit Function Theorem. These transformations will allow us to reduce the stability of the three solutions to the case of the vacuum solution, in the spirit of previous results by Alejo and the first author [3], which was done for the case of the scalar modified Korteweg-de Vries equation. However, we will see that SG presents several difficulties because of its vector valued character. Our results improve those in [5], and give a first rigorous proof of the stability in the energy space of SG 2-solitons.

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Quantitative receptor autoradiography was used to study possible alterations of the densities of multiple serotonin (5-HT) receptor subtypes and of serotonin transporter in the brain of 5-HT(2C) receptor knockout mice. The radioligands employed were [(3)H]citalopram, [(3)H]WAY100,635, [(3)H]8-OH-DPAT, [(3)H]GR125743, [(3)H]sumatriptan, [(3)H]MDL100,907, [(125)I](+/-)DOI, [(3)H]mesulergine, [(3)H]5-HT, [(3)H]GR113808, and [(3)H]5-CT. As expected, radioligands that label 5-HT(2C) receptors showed a complete absence of labeling in mutant mice choroid plexus and significantly reduced densities in other brain regions expressing 5-HT(2C) receptors. With the rest of the radioligands, no significant alterations in the densities of labeled sites were found in any brain region. In situ hybridization showed no changes in 5-HT(2A) receptor and serotonin transporter mRNA levels, whereas 5-HT(2C) receptor mRNA levels were reduced in certain brain regions. The present results indicate that the mouse serotonergic system does not exhibit compensatory up- or down-regulation of the majority of its components (serotonin transporter and most 5-HT receptor subtypes) in response to the absence of 5-HT(2C) receptors.

We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H 1 × L 2 . It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space.In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in [24,52] and Virial estimates around odd perturbations of the vacuum solution, in the spirit of [32], we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and [32], using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and φ 4 at the level of linear Bäcklund transformations. Date: March 23, 2020. * M.A. Alejo was partially supported by CNPq grant no. 305205/2016-1. * * C. M. work was funded in part by Chilean research grants FONDECYT 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. * * * J. M. P. was partially supported by Chilean research grants FONDECYT 1191412 and project France-Chile ECOS-Sud C18E06.