Bifurcation of Gap Solitons in Coupled Mode Equations in d Dimensions

We consider a system of first order coupled mode equations in $${\mathbb {R}}^d$$ R d describing the envelopes of wavepackets in nonlinear periodic media. Under the assumptions of a spectral gap and a generic assumption on the dispersion relation at the spectral edge, we prove the bifurcation of standing gap solitons of the coupled mode equations from the zero solution. The proof is based on a Lyapunov–Schmidt decomposition in Fourier variables and a nested Banach fixed point argument. The reduced bifurcation equation is a perturbed stationary nonlinear Schrödinger equation. The existence of solitary waves follows in a symmetric subspace thanks to a spectral stability result. A numerical example of gap solitons in $${\mathbb {R}}^2$$ R 2 is provided.

Which Exterior Powers are Balanced?

A signed graph is a graph whose edges are given $\pm 1$ weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal $\pm 1$ matrix. For a signed graph $\Sigma$ on $n$ vertices, its exterior $k$th power, where $k=1,\ldots,n-1$, is a graph $\bigwedge^{k} \Sigma$ whose adjacency matrix is given by\[ A(\mbox{$\bigwedge^{k} \Sigma$}) = P_{\wedge}^{\dagger} A(\Sigma^{\Box k}) P_{\wedge}, \]where $P_{\wedge}$ is the projector onto the anti-symmetric subspace of the $k$-fold tensor product space $(\mathbb{C}^{n})^{\otimes k}$ and $\Sigma^{\Box k}$ is the $k$-fold Cartesian product of $\Sigma$ with itself. The exterior power creates a signed graph from any graph, even unsigned. We prove sufficient and necessary conditions so that $\bigwedge^{k} \Sigma$ is balanced. For $k=1,\ldots,n-2$, the condition is that either $\Sigma$ is a signed path or $\Sigma$ is a signed cycle that is balanced for odd $k$ or is unbalanced for even $k$; for $k=n-1$, the condition is that each even cycle in $\Sigma$ is positive and each odd cycle in $\Sigma$ is negative.

FIRST-ORDER PAIRING TRANSITION AND PHASE SEPARATION IN THE ATTRACTIVE HUBBARD MODEL

The normal state properties of the Hubbard model are studied by means of the Dynamical Mean-Field Theory. Even in the gauge-symmetric subspace, a first-order transition occurs between a Fermi-liquid phase and a strong-coupling bound-pairs phase, which can be thought as a "disordered" superconductor. The transition is of first order for all densities different from n = 1, and it is accompanied by a region of phase separation between the two phases at different densities.

ANTIPHASE SYNCHRONIZATION IN SYMMETRICALLY COUPLED SELF-OSCILLATORS

We consider the antiphase synchronization in symmetrically coupled self-oscillators. As model, two Chua's circuits coupled via a capacity are used. Linear analysis in the vicinity of the symmetric subspace gives the stability conditions for antiphase oscillations. Numerical oscillations demonstrate controlled antiphase synchronization at different values of the parameters of the system.

Self-adjoint subspace extensions satisfying λ-linear boundary conditions

SynopsisLet S be a symmetric subspace in a Hilbert space ℋ2 with finite equal deficiency indices and let S* be its adjoint subspace in ℋ2. We consider those self-adjoint subspace extensions ℋ of S into some larger Hilbert spaces ℋ2 = (ℋ × ℂm)2 which satisfy H⋂({0} × ℂm)2 = {{0,0}}. These extensions H are characterized in terms of inhomogeneous boundary conditions for S*; they are associated with eigenvalue problems for S* depending on λ-linear boundary conditions, which we also characterize.