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The main aim of this research study is to examine optical soliton phenomena in a (2+1)-dimensional Schrödinger class nonlinear model that degenerates from Biswas-Milovic equation (BME) using Riccati modified extended simple equation method (RMESEM). This model has particular relevance in the fiber optics domain. The proposed anstaz RMESEM uses a complex structured wave transformation to produce nonlinear ordinary differential equation (NODE) and constraint conditions for Kerr law nonlinearity form of the model. The resulting NODE is assumed to have a closed form solution that converts it into a system of nonlinear algebraic equations via substitution in order to determine fresh variety of optical soliton solutions. The final visualizations of the obtained optical soliton solutions in the form of 3D, contour, and 2D forms demonstrate that the model develops Hopf bifurcation, rogue and internal envelope solitons as a result of the elastic and inelastic collision of optical periodic solitons while the norms of the obtained optical soliton reveal dark and bright kink structures. Using phase portraits and time-series maps, we also study bifurcating and chaotic behavior, observing its presence in the perturbed dynamical system and obtaining favorable results indicating Hopf bifurcation and periodicity. We use a generalized trigonometric function to perturb the planner system for the first time in order carry out chaotic analysis. Furthermore, our results are analyzed and linked to the soliton dynamics in BME, demonstrating the effectiveness of the suggested method as an effective method of identifying novel soliton phenomena within such nonlinear settings.

The covariant formulation of teleparallel gravity theories must include the spin connection, which has 6 degrees of freedom. One can, however, always choose a gauge such that the spin connection is put to zero. In principle this gauge may affect counting of degrees of freedom in the Hamiltonian analysis. We show for general teleparallel theories of gravity, that fixing the gauge such that the spin connection vanishes in fact does not affect the counting of degrees of freedom. This manifests in the fact that the momenta of the Lorentz transformations which generate the spin connection are fully determined by the momenta of the tetrads.

The higher category theory can be employed to generalize the B F action to the so-called 3 B F action, by passing from the notion of a gauge group to the notion of a gauge 3-group. The theory of scalar electrodynamics coupled to Einstein–Cartan gravity can be formulated as a constrained 3 B F theory for a specific choice of the gauge 3-group. The complete Hamiltonian analysis of the 3 B F action for the choice of a Lie 3-group corresponding to scalar electrodynamics is performed. This analysis is the first step towards a canonical quantization of a 3 B F theory, an important stepping stone for the quantization of the complete scalar electrodynamics coupled to Einstein–Cartan gravity formulated as a 3 B F action with suitable simplicity constraints. It is shown that the resulting dynamic constraints eliminate all propagating degrees of freedom, i.e., the 3 B F theory for this choice of a 3-group is a topological field theory, as expected.

We prove that vector fields described by the generalized Proca class of theories do not admit consistent coupling with a gravitational sector defined by a scalar–tensor theory of the degenerate type. Under the assumption that there exists a frame in which the Proca field interacts with gravity only through the metric tensor, our analysis shows that at least one of the constraints associated with the degeneracy of the scalar–tensor sector is inevitably lost whenever the vector theory includes coupling with the Christoffel connection.

This volume contains the proceedings of the conference on tropical geometry and integrable systems, held July 3-8, 2011, at the University of Glasgow, United Kingdom. One of the aims of this conference was to bring together researchers in the field of tropical geometry and its applications, from apparently disparate ends of the spectrum, to foster a mutual understanding and establish a common language which will encourage further developments of the area. This aim is reflected in these articles, which cover areas from automata, through cluster algebras, to enumerative geometry. In addition, two survey articles are included which introduce ideas from researchers on one end of this spectrum to researchers on the other. This book is intended for graduate students and researchers interested in tropical geometry and integrable systems and the developing links between these two areas.

Steel curved girder bridges are largely used today in motorways and railways. They are often composed of thin-walled crosssections, entirely made of steel or with an upper concrete slab. The deck may have I-girders or box cross-sections: in any case curved girders are subjected to twisting moment, associated with bending, even for dead loads. Moreover, in thin-walled sections the influence of non-uniform torsion becomes sizable with respect to Saint Venant torsion, modifying the state of tangential stresses in the section and introducing axial stresses due to warping being prevented. Open sections of I-girder bridges are especially subject to these phenomena and warping can be significant not only for curved bridges but also for eccentrically applied traffic loads in straight ones. In this paper a method for the evaluation of the effects of non-uniform torsion, based on an energetic approach, is proposed; the method is simple and fast, with a reduced computational burden with respect to finite elements. The solution of curved girder bridges is performed by the Hamiltonian Structural Analysis method, which is implemented for straight and curved girders with thin-walled cross-sections. A parametric analysis is proposed for single and multi-span bridges, with variation in type of loads applied, crosssection parameters, geometric curvature of bridge deck and stiffness ratios of the cross-section. The parametric study is presented through dimensionless diagrams of internal forces for the evaluation of the global behaviour of curved girder bridges together with indications on the stress state of the cross-section. The interpretation of the results of the analyses performed can be useful to designers for the conceptual design stages and for optimization of the geometric and mechanical parameters of I-girder and box girder cross-sections.

Consider a general linear Hamiltonian system

The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.

Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it The formulas relating the growth

In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references.