Related papers: A breather construction for a semilinear curl-curl…
We consider the nonlinear Schr\"odinger equation with non-local derivatives in a two-dimensional periodic domain. For certain orders of derivatives, we find a new type of breather solution dominating the field evolution at low nonlinearity…
In this paper, we study the following $p(x)$-curl systems: \begin{eqnarray*} \begin{cases} \nabla\times(|\nabla\times \mathbf{u}|^{p(x)-2}\nabla\times \mathbf{u})+a(x)|\mathbf{u}|^{p(x)-2}\mathbf{u}=\lambda f(x,\mathbf{u})+\mu…
We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground…
In this work we derive some blow-up results for semilinear wave equations both in de Sitter and anti-de Sitter spacetimes. By requiring suitable conditions on a time-dependent factor in the nonlinear term, we prove the blow-up in finite…
We study the quasilinear equation $(P)\qquad - {\rm div} (a(x,u,\nabla u)) +A_t(x,u,\nabla u) + |u|^{p-2}u\ =\ g(x,u) \qquad \hbox{in $\R^N$,} $ with $N\ge 3$ and $p > 1$. Here, we suppose $A : \R^N \times \R \times \R^N \to \R$ is a given…
We study oscillons in D+1 space-time dimensions using a spherically symmetric ansatz. From Gaussian initial conditions, these evolve by emitting radiation, approaching ``quasi-breathers'', near-periodic solutions to the equations of motion.…
We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We…
We consider the following half wave Schr{\"o}dinger equation,$$\left(i \partial_{t}+\partial_{x }^2-\left|D_{y}\right|\right) U=|U|^{2} U$$on the plane $\mathbb{R}_x \times \mathbb{R}_y$. We prove the existence of modified wave operators…
We are concerned with numerical approximations of breather solutions for the cubic Whitham equation which arises as a water-wave model for interfacial waves. The model combines strong nonlinearity with the non-local character of the…
We study the existence of polychromatic solutions of cubically nonlinear Maxwell equations in the whole space and with dispersive media, i.e., with a time delayed polarization. Due to the complex nature of the dielectric function, the…
We construct series solutions to all orders for breathers of Klein-Gordon equations, in powers of an amplitude parameter epsilon, under a sign condition on the coefficients of the expansion of the nonlinearity. All terms may be computed…
We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more…
Oscillons are localized field configurations oscillating in time with lifetimes orders of magnitude longer than their oscillation period. In this paper, we simulate non-travelling oscillons produced by deforming the breather solutions of…
We consider Maxwell's equations for Kerr-type optical materials, which are magnetically inactive and have a nonlinear response to electric fields. This response consists of a linear plus a cubic term, which are both inhomogeneous with…
In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t)|_{t=0} = (u_0,u_1)\in…
We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is…
In this reviewing paper, we are interested in the proof of estimating the lifespan of classical solutions of semilinear wave equations with the critical exponent from above especially in low space dimensions. There are a few ways to show…
In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be…
For Maxwell's equations with nonlinear polarization we prove the existence of time-periodic breather solutions travelling along slab or cylindrical waveguides. The solutions are TE-modes which are localized in space directions orthogonal to…
This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations…