Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier--Bloch decomposition and the Implicit Function Theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a non-degeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross--Pitaevskii equation and the coupled-mode equations are obtained for a finite-time interval.Comment: 32 pages, 16 figure

Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $\phi^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis

Computation of the radiation amplitude of oscillons

The radiation loss of small amplitude oscillons (very long-living, spatially localized, time dependent solutions) in one dimensional scalar field theories is computed in the small-amplitude expansion analytically using matched asymptotic series expansions and Borel summation. The amplitude of the radiation is beyond all orders in perturbation theory and the method used has been developed by Segur and Kruskal in Phys. Rev. Lett. 58, 747 (1987). Our results are in good agreement with those of long time numerical simulations of oscillons.Comment: 22 pages, 9 figure

Separatrix splitting at a Hamiltonian 02iω0^2 i\omega bifurcation

We discuss the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a double zero one. It is well known that an one-parametric unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable trajectories of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies non-existence of single-round homoclinic orbits and divergence of series in the normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with behaviour of analytic continuation of the system in a complex neighbourhood of the equilibrium

Qualitative Analysis of Causal Anisotropic Viscous Fluid Cosmological Models

The truncated Israel-Stewart theory of irreversible thermodynamics is used to describe the bulk viscous pressure and the anisotropic stress in a class of spatially homogeneous viscous fluid cosmological models. The governing system of differential equations is written in terms of dimensionless variables and a set of dimensionless equations of state is utilized to complete the system. The resulting dynamical system is then analyzed using standard geometric techniques. It is found that the presence of anisotropic stress plays a dominant role in the evolution of the anisotropic models. In particular, in the case of the Bianchi type I models it is found that anisotropic stress leads to models that violate the weak energy condition and to the creation of a periodic orbit in some instances. The stability of the isotropic singular points is analyzed in the case with zero heat conduction; it is found that there are ranges of parameter values such that there exists an attracting isotropic Friedmann-Robertson-Walker model. In the case of zero anisotropic stress but with non-zero heat conduction the stability of the singular points is found to be the same as in the corresponding case with zero heat conduction; hence the presence of heat conduction does not apparently affect the global dynamics of the model.Comment: 35 pages, REVTeX, 3 Encapsulated PostScript Figure