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We examine the stability of vortex-like configuration of magnetization in magnetic materials, so-called the magnetic skyrmion. These correspond to critical points of the Landau-Lifshitz energy with the Dzyaloshinskii-Moriya (DM) interactions. In an earlier work of the D\"oring and Melcher, it is known that the skyrmion is a ground state when the coefficient of the DM term is small. In this paper, we prove that there is an explicit critical value of the coefficient above which the skyrmion is unstable, while stable below this threshold. Moreover, we show that in the unstable regime, the infimum of energy is not bounded below, by giving an explicit counterexample with a sort of helical configuration. This mathematically explains the occurrence of phase transition observed in some experiments., Comment: 12 pages

We obtain probabilistic local well-posedness in quasilinear regimes for the Schrödinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low frequency interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation to Schrödinger-type equations. In addition, we discuss ill-posedness results for this equation.

In this paper, we study the transverse stability of the line Schr��dinger soliton under a full wave guide Schr��dinger flow on a cylindrical domain $\mathbb R\times\mathbb T$. When the nonlinearity is of power type $|��|^{p-1}��$ with $p>1$, we show that there exists a critical frequency $��_{p} >0$ such that the line standing wave is stable for $0 ��_{p}$. Furthermore, we characterize the ground state of the wave guide Schr��dinger equation. More precisely, we prove that there exists $��_{*} \in (0, ��_{p}]$ such that the ground states coincide with the line standing waves for $��\in (0, ��_{*}]$ and are different from the line standing waves for $��\in (��_{*}, \infty)$., To appear in JDDE

In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schr��dinger equation on the product space $\mathbb{R} \times \mathbb{T}$. We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the ``auxiliary equation'' and then on the application of the Crandall-Rabinowitz argument

We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schroedinger equation and prove some corresponding laws., Comment: 25 pages

We construct a singular solution of a stationary nonlinear Schr\"{o}dinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schr��dinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L^{2}_{x}H^{s}_{y}(\mathbb{R}^{2})$, with $s>\frac{1}{2}$.

For the 3D Navier-Stokes-Maxwell problem on the whole space and in the presence of external time-periodic forces, first we study the existence of time-periodic small solutions, and then we prove their asymptotic stability. We use new type of spaces to account for averaged decay in time., Comment: 40 pages

In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schrödinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in [10, 23] and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\mathbb R^2)$ into the critical Orlicz space ${\mathcal L}(\mathbb R^2)$ settled in [4]. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.

We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posedness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions à la Leray, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. Our results extend Giga-Yoshida (1984) [8] ones to the space dimension two, and improve them in terms of requiring less regularity on the velocity fields.

We study global dynamics for the focusing nonlinear Klein-Gordon equation with the energy-critical nonlinearity in two or higher dimensions when the energy equals the threshold given by the ground state of a mass-shifted equation, and prove that the solutions are divided into scattering and blowup. In short, the Kenig-Merle scattering/blowup dichotomy extends to the threshold energy in the case of mass-shift for the critical nonlinear Klein-Gordon equation., Comment: 16 pages

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the "subcritical" nonlinear Schr\"odinger equation., Comment: 33 pages, submitted