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Motivated by the work of J.K.~Jang et al., Nat.~Commun.~{\bf 6}, 7370 (2015), where the authors experimentally tweeze cavity solitons in a passive loop of optical fiber, we study the amenability to tweezing of cavity solitons as the properties of a localized tweezer are varied. The system is modeled by the Lugiato-Lefever equation, a variant of the complex Ginzburg-Landau equation. We produce an effective, localized, trapping tweezer potential by assuming a Gaussian phase-modulation of the holding beam. The potential for tweezing is then assessed as the total (temporal) displacement and speed of the tweezer are varied, and corresponding phase diagrams are presented. As the relative speed of the tweezer is increased we find two possible dynamical scenarios: successful tweezing and release of the cavity soliton. We also deploy a non-conservative variational approximation (NCVA) based on a Lagrangian description which reduces the original dissipative partial differential equation to a set of coupled ordinary differential equations for the cavity soliton parameters. We illustrate the ability of the NCVA to accurately predict the separatrix between successful and failed tweezing. This showcases the versatility of the NCVA to provide a low-dimensional description of the experimental realization of the temporal tweezing., Comment: 27 pages, 9 figures

We study solutions to the evolution equation $u_t=\Delta u-u +\sum_{k\geqslant 1}q_ku^k$, $t>0$, in $\mathbf{R}^d$. Here the coefficients $q_k\geqslant 0$ verify $ \sum_{k\geqslant 1}q_k=1< \sum_{k\geqslant 1}kq_k<\infty$. First, we deal with existence, uniqueness, and the asymptotic behavior of the solutions as $t\to +\infty$. We then deduce results on the long time behavior of the associated branching process, with state space the set of all finite configurations of $\mathbf{R}^d$, under the assumption that $\sum_{k\geq 1} k^2q_k<\infty$. It turns out that the distribution of the branching process behaves when the time tends to infinity like that of the Brownian motion on the set of all finite configurations of $\mathbf{R}^d$. However, due to the lack of conservation of the total mass of the initial non linear equation, a deformation with a multiplicative coefficient occurs. Finally, we establish asymptotic properties of the occupation time of this branching process., Comment: 22 pages

We study the spontaneous temporal symmetry breaking instability in a coherently-driven passive optical Kerr resonator observed experimentally by Xu and Coen in Opt.~Lett.~{\bf 39}, 3492 (2014). We perform a detailed stability analysis of the Lugiato-Lefever model for the optical Kerr resonators and analyze the temporal bifurcation structure of stationary symmetric and the emerging asymmetric states as a function of the pump power. For intermediate pump powers a pitchfork loop is responsible for the destabilization of symmetric states towards stationary asymmetric ones while at large pump powers we find the emergence of periodic asymmetric solutions via a Hopf bifurcation. From a theoretical perspective, we use local bifurcation theory in order to analyze the most unstable eigenmode of the system. We also explore a non-conservative variational approximation capturing, among others, the evolution of the solution's amplitude, width and center of mass. Both methods provide insight towards the pitchfork bifurcations associated with the symmetry breaking., Comment: 13 pages, 8 figures

Recently, Galley [Phys. Rev. Lett. {\bf 110}, 174301 (2013)] proposed an initial value problem formulation of Hamilton's principle applied to non-conservative systems. Here, we explore this formulation for complex partial differential equations of the nonlinear Schrodinger (NLS) type, examining the dynamics of the coherent solitary wave structures of such models by means of a non-conservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems; namely, the perturbed variational approximation and a generalization of the so-called Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models studied herein. We showcase the relevance of the NCVA method by exploring test case examples within the NLS setting including combinations of linear and density dependent loss and gain. We also present an example applied to exciton polariton condensates that intrinsically feature loss and a spatially dependent gain term., Comment: 11 pages, 3 figures

We find interpretation using optimal mass transport theory for eigenvalue problems obtained as limits of the eigenvalue problems for the fractional $p-$Laplacian operators as $p\to +\infty$. We deal both with Dirichlet and Neumann boundary conditions., Comment: 20 pages

In this paper we obtain bounds for the decay rate for solutions to the nonlocal problem $\partial_t u(t,x) = \int_{\R^n} J(x,y)[u(t,y) - u(t,x)] dy$. Here we deal with bounded kernels $J$ but with polynomial tails, that is, we assume a lower bound of the form $J(x,y) \geq c_1|x-y|^{-(n + 2\sigma)}$, for $|x - y| > c_2$. Our estimates takes the form $\|u(t)\|_{L^q(\R^n)} \leq C t^{-\frac{n}{2\sigma} (1 - \frac{1}{q})}$ for $t$ large.

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \, dy$. Here we consider a kernel $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$ where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on $\rr^d$. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\psi$. Indeed, in the linear case $a(x) = Ax$ we obtain an explicit expression for the first eigenvalue in the whole $\rr^d$ and it is positive when the the determinant of the matrix $A$ is different from one. As an application of our results, we observe that, when the first eigenvalue is positive, there is an exponential decay for the solutions to the associated evolution problem. As a tool to obtain the result, we also study the behaviour of the principal eigenvalue of the nonlocal Dirichlet problem in the ball $B_R$ and prove that it converges to the first eigenvalue in the whole space as $R\to \infty$.

We study the behavior of $p$-Dirichlet optimal design problem with volume constraint for $p$ large. As the limit as $p$ goes to infinity, we find a limiting free boundary problem governed by the infinity-Laplacian operator. We establish a necessary and sufficient condition for uniqueness of the limiting problem and, under such a condition, we determine precisely the optimal configuration for the limiting problem. Finally, we establish convergence results for the free boundaries.

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.

We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition.

Let $\Omega$ be a bounded smooth domain in $\RR^N$. We consider the problem $u_t= \Delta u + V(x) u^p$ in $\Omega \times [0,T)$, with Dirichlet boundary conditions $u=0$ on $\partial \Omega \times [0,T)$ and initial datum $u(x,0)= M \phi (x)$ where $M \geq 0$, $\phi$ is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of $M$. As a consequence of these estimates we find that, for $M$ large, the blow up set concentrates near the points where $\phi^{p-1}V$ attains its maximum.