Self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation

We consider the one-dimensional Landau-Lifshitz-Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated to a discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter. We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schr\"odinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1d-Schr\"odinger map equation

Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.Comment: 48 pages, 11 figure

Recent results for the Landau-Lifshitz equation

We give a survey on some recent results concerning the Landau-Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. We revisit the Cauchy problem for the anisotropic Landau-Lifshitz equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. We also examine two approximations of the Landau-Lifshitz equation given by of the Sine-Gordon equation and cubic Schr\"odinger equations, arising in certain singular limits of strong easy-plane and easy-axis anisotropy, respectively. Concerning localized solutions, we review the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical framework. Finally, we survey results concerning the existence, uniqueness and stability of self-similar solutions (expanders and shrinkers) for the isotropic Landau-Lifshitz equation with Gilbert term. Since expanders are associated with a singular initial condition with a jump discontinuity, we also review their well-posedness in spaces linked to the BMO space

Stabilité des solitons de l'équation de Landau-Lifshitz à anisotropie planaire

Séminaire Laurent Schwartz - EDP et applicationsCet exposé présente plusieurs résultats récents quant à la stabilité des solitons sombres de l'équation de Landau-Lifshitz à anisotropie planaire, en particulier, quant à la stabilité orbitale des trains (bien préparés) de solitons gris et à la stabilité asymptotique de ces mêmes solitons

Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

International audienceWe consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time

Stability in the energy space for chains of solitons of the Landau-Lifshitz equation

International audienceWe prove the orbital stability of sums of solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy, under the assumptions that the (non-zero) speeds of the solitons are different, and that their initial positions are sufficiently separated and ordered according to their speeds

Self-similar shrinkers of the one-dimensional Landau-Lifshitz-Gilbert equation

International audienceThe main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere, at an exponential rate.In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles

Exotic traveling waves for a quasilinear Schr\"odinger equation with nonzero background

We study a defocusing quasilinear Schr\"odinger equation with nonzero conditions at infinity in dimension one. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross--Pitaevskii equation, and can also be derived by considering the effects of surface tension in superfluids. When the quasilinear term is neglected, the resulting equation is the classical Gross-Pitaevskii equation, which possesses a well-known stable branch of subsonic traveling waves solution, given by dark solitons. Our goal is to investigate how the quasilinear term affects the traveling-waves solutions. We provide a complete classification of finite energy traveling waves of the equation, in terms of the two parameters: the speed and the strength of the quasilinear term. This classification leads to the existence of dark and antidark solitons, as well as more exotic localized solutions like dark cuspons, compactons, and composite waves, even for supersonic speeds. Depending on the parameters, these types of solutions can coexist, showing that finite energy solutions are not unique. Furthermore, we prove that some of these dark solitons can be obtained as minimizers of the energy, at fixed momentum, and that they are orbitally stable