We give here an overview of several papers concerning the domain walls dynamics in one-dimensional models of ferromagnetic nanowires. After justifying the one-dimensional models by asymptotic analysis, we first study configurations with one wall in an infinite wire, and we establish the stability of these configurations. In a second part, we prove that the same model in the case of a finite wire is irrelevant because the solution describing one wall is unstable. In addition, it cannot describe complex distributions of several walls located at arbitrary positions. In order to describe such situations, we use Carr and Pego's geometric method to obtain a metastability result for approximate solutions in a model taking into account the smallness of the exchange length.

We use bifurcation theory to prove the existence of steady two-dimensional periodic geophysical surface water waves propagating in equatorial oceanic regions.

We study asymptotic stability of solitary wave solutions in the one-dimensional Benney--Luke equation, a formally valid approximation for describing two-way water-wave propagation. For this equation, as for the full water-wave problem, the classic variational method for proving orbital stability of solitary waves fails dramatically due to the fact that the second variation of the energy-momentum functional is infinitely indefinite. We establish nonlinear stability in energy norm under the spectral stability hypothesis that the linearization admits no nonzero eigenvalues of nonnegative real part. We then verify this hypothesis for waves of small energy.

The Bäcklund transform, first developed in the context of differential geometry, has been classically used to obtain multi-soliton states in completely integrable infinite-dimensional dynamical systems. It has recently been used to study the stability of these special solutions. We offer here a dynamical perspective on the Bäcklund Transform, prove an abstract orbital stability theorem, and demonstrate its utility by applying it to the sine-Gordon equation and the Toda lattice.

In this paper, we study the standard one-dimensional (non-overdamped) Frenkel--Kontorova (FK) model describing the motion of atoms in a lattice. For this model we show that for any supersonic velocity $c>1$, there exist bounded traveling waves moving with velocity $c$. The profile of these traveling waves is a phase transition between limit states $k_-$ in $-\infty$ and $k_+$ in $+\infty$. Those limit states are some integers which reflect the assumed $1$-periodicity of the periodic potential inside the FK model. For every $c>1$, we show that we can always find $k_-$ and $k_+$ such that $k_+-k_-$ is an odd integer. Furthermore, for $c\ge \sqrt{\frac{25}{24}}$, we show that we can take $k_+-k_-=1$. These traveling waves are limits of minimizers of a certain energy functional defined on a bounded interval, when the length of the interval goes to infinity. Our method of proof uses a concentration-compactness-type argument which is based on a cleaning lemma for minimizers of this functional.

Progress has been made very recently on models for the dynamics of one or more vortex filaments in three-dimensional fluids. In this article we survey recent and previous results on this topic. We also present a collection of new situations of filaments collapse.

Extending the approach of Grillakis, Shatah, and Strauss, Bronski, Johnson, and Kapitula, and others for Hamiltonian systems, we explore relations between the constrained variational problem $$ \min_{X:C(X)=c_0} \mathcal{E}(X) , \ \ \ c_0\in \mathbb R^r , $$ and stability of solutions of a class of degenerate ``quasi-gradient'' systems $dX/dt=-M(X)\nabla \mathcal{E}(X)$ admitting constraints, including Cahn--Hilliard equations, one- and multi-dimensional viscoelasticity, and coupled conservation-law--reaction-diffusion systems arising in chemotaxis and related settings. Using the relation between variational stability and the signature of $\partial c/\partial \omega \in \mathbb R^{r\times r}$, where $c(\omega)=C(X^*_\omega)\in \mathbb R ^r$ denote the values of the imposed constraints and $\omega\in \mathbb R^r$ the associated Lagrange multipliers at a critical point $X^*_\omega$, we obtain as in the Hamiltonian case a general criterion for co-periodic stability of periodic waves, illuminating and extending a number of previous results obtained by direct Evans-function techniques. More interestingly, comparing the form of the Jacobian arising in the co-periodic theory to Jacobians arising in the formal Whitham equations associated with modulation, we recover and substantially generalize a previously mysterious ``modulational dichotomy'' observed in special cases by Oh--Zumbrun and Howard, showing that co-periodic and sideband stability are incompatible. In particular, we both illuminate and extend to general viscosity/strain-gradient effects and multidimensional deformations the result of Oh-Zumbrun of universal modulational instability of periodic solutions of the equations of viscoelasticity with strain-gradient effects, considered as functions on the whole line. Likewise, we generalize to multi-dimensions corresponding results of Howard on periodic solutions of Cahn--Hilliard equations.

By capillary fluids we mean compressible, inviscid fluids whose energy depends not only on their density but also on their density gradient. Their motion is thus governed by systems of conservation laws, either in Eulerian coordinates or in Lagrangian coordinates, that are a higher-order modification of the usual equations of gas dynamics. In both settings, we receive models that also arise in other fields, in particular in water-waves theory and quantum hydrodynamics. Those Hamiltonian systems typically admit three types of planar traveling waves, namely, heteroclinic, homoclinic, and periodic ones. The purpose here is to review the main tools and results regarding the stability of those waves, under most general assumptions on the energy law. Special attention is devoted to the correspondence between traveling waves in Eulerian coordinates and those in Lagrangian coordinates.