Final Report Summary - NOMATOS (Nonlinearity management of atomic and optical systems) Executive Summary:The main goal of the project was the investigation of nonlinear wave patterns and solitons in systems with temporal and spatial modulations of local strength and sign of nonlinearity possibly in combination with linear lattices. It was planned to address both continuous and discrete systems and as the applications to consider applications nonlinear optical media as well to Bose-Einstein condensates (BEC). Scientific results:In the context of study of the discrete nonlinear Schrödinger equation (NLS) we have demonstrated the existence of compactons in the presence of fast periodic modulations of the nonlinearity. It was shown that the resulting effective inter-well tunneling depends on the modulation parameters and on the field amplitude, what introduces nonlinear dispersion in the system and can lead to a prototypical realization of single- or multisite stable discrete compactons in nonlinear optical waveguide- and BEC arrays. We addressed discrete solitons in vector discrete NLS equation with fast modulation of the linear coupling. The model can be realized in bimodal BECs. There was obtained a system of averaged (autonomous) equations and identified stability regions for fundamental onsite discrete symmetric solitons, as well as for two-site in-phase and twisted modes, the in-phase ones being completely unstable. The symmetry-breaking bifurcation, which destabilizes the fundamental symmetric solitons and gives rise to their asymmetric counterparts, is investigated too. We have suggested an all-optical steering of light via spatial Bloch oscillations in a gas of three-level atoms. The phenomenon is based on application of a standing-wave control field applied to a three-level atomic medium in a planar hollow-core photonic crystal waveguide creates periodic variations of linear and nonlinear refractive indexes of the medium. By properly designing the spatial dependence of the nonlinearity, it is possible to induce long-living Bloch oscillations of spatial gap solitons, thus providing desirable change in the direction of the beam propagation without inducing appreciable diffraction. Due to the significant enhancement of the nonlinearity, such a probe beam can be reached at extremely weak light intensities. In the context of study of dissipative systems we have shown that one-dimensional NLS equation with a dissipative periodic potential, nonlinear losses, and a linear pump allow for the existence of stable nonlinear Bloch states which are attractors. The model describes a BEC with inelastic two- and three-body interactions loaded in an optical lattice with losses due to inelastic interactions of the atoms with photons. It was established the existence of localized modes supported by the parity-time – symmetric nonlinear lattices. The system reveals unusual properties: it possesses families of solutions, which can be parameterized by the propagation constant; relatively narrow localized modes appear to be stable, even when the conservative nonlinear lattice potential is absent; and finally, the system supports stable multipole solutions. We have shown that dissipation can be an effective tool for managing waves. In particular in a simple coupler where one of the waveguides is subject to controlled dissipative losses, it is possible to observe linear and nonlinear Zeno effects. The phenomenon consists in a strong enhancement of transparency of the coupler and represents an optical analogue of the quantum Zeno paradox. The system allows for observation of the cross-over between the linear and nonlinear Zeno effects, as well as effective manipulation of light transmission through the coupler. The Zeno effect was investigated also for soliton type pulses in a nonlinear directional coupler with dissipation. It is shown that localized dissipation can lead to switching of solitons between the arms. Power losses accompanying the switching can be fully compensated by using a combination of dissipative and active segments. We analyzed the mode structure of a BEC loaded in a complex parabolic potential and subjected to a constant pump. Stationary solutions for the positive and negative scattering lengths are addressed. It was shown that for a positive scattering length only the ground state appears to be stable. Further we analyzed of nonlinear modes in a complex absorbing double-well potential supported by linear gain. Families of the nonlinear modes and their bifurcations are found and their stability is described. Varying a control parameter e.g. the height of barrier between the wells results in switching from one mode to another. Apart from stationary modes we have found pulsating solutions emergent from unstable modes. We have reported exact (stable and unstable) solutions for the generalized NLS equation with inhomogeneous complex linear and nonlinear potentials are found. We also demonstrated numerically the existence of stable dissipative breathers in the presence of an additional parabolic trap. There were studied different aspects of the nonlinear phenomena in the interplay of linear and nonlinear lattices. The existence of two-dimensional matter-wave solitons stabilized solitons against decay or collapse for both attractive and repulsive interactions. We proposed how to create linear superposition of nonlinear matter waves in optical lattices achieved by means of spatial variation of the interatomic interaction which suppresses the nonlinear overlapping terms, which otherwise would destroy the superposition, and at the same time retaining all the nonlinearity necessary for each component state to exist. The superposition state is shown to be long lived and can be split into constituent parts by accelerating the lattice. We explored the BEC superfluidity in toroidal traps with nonlinear lattices. The existence of several types of stable periodic waves, ranging from almost uniform to very fragmented chains of weakly interacting and equally spaced solitons, is demonstrated. We show that these waves may support persistent atomic currents and sound waves with spectra of Bogoliubov type. Fragmented condensates can be viewed as arrays of Josephson junctions and the current as a BEC manifestation of the dc-Josephson effect. We found that under critical velocities, linear defects that are static with respect to the lattice preserve the BEC superfluidity. Further, we studied bulk and surface dipolar BECs, loaded into a deep optical lattice, focusing on development of the modulational instability of matter wave. We found a region where the two fundamental modes are simultaneously unstable allowing for enhanced mobility across the lattice. We have found symmetric and anti-symmetric localized modes and performed detail study of their stability. We explored propagation of solitons in dipolar BEC in trap potential with a barrier potential. The regimes of soliton transmission, reflection, splitting and fusion in dependence on the ratio between the local and dipolar nonlocal interactions were obtained. The predicted phenomena can be useful for the design of a matter wave splitter and interferometers using the matter wave solitons. We also studied inhomogeneous dark states of atomic-molecular BECs in trapping potentials. We have shown that in principle a trap does not allow for existence of the dark states unless it has a particular configuration. We have found a class of trapping potentials allowing for the existence of localized stable dark states. The respective atomic and molecular distributions are computed, and their stability and dynamics were discussed. We presented a higher-dimensional similarity transformation reducing the (3+1)-dimensional inhomogeneous NLS equation with variable coefficients and parabolic potential to the (1+1)-dimensional NLS equation with constant coefficients. This transformation allowed us to relate certain class of localized exact solutions of the (3+1)-dimensional case to the variety of solutions of integrable NLS equation of the (1+1)-dimensional case. As an example, we illustrated our technique using two lowest-order rational solutions of the NLS equation as seeding functions to obtain rogue wavelike solutions localized in three dimensions that have complicated evolution in time including interactions between two time-dependent rogue wave solutions. We have shown that interplay of linear attractive (repulsive) boundary with inhomogeneous repulsive (attractive) nonlinearity may result in stationary nonlinear surface modes. Several models describing interaction of inhomogeneous BECs condensates with a rigid surface and allowing for exact solutions are considered and their stability is established. Finally, we addressed gap solitons in nonlinear χ2- media with periodically modulated linear refractive index and quadratic nonlinearity. It was shown that modes may possess different symmetries of the fundamental field, while require the second harmonic to be symmetric. We found that only small amplitude gap solitons can be stable and the respective modes can bifurcate only from an edge of a total gap when it coincides with the band-edge of the linear spectrum of the fundamental field. Moreover, in this case stable modes can be excited with their centers belonging to a slab with either higher or lower refractive index. Impact:The project had significant scientific impact on the research in the nonlinear Physics carried out at the University of Lisbon. There were created new lines of investigation, like study of nonlinear parity-time symmetric structures, exploration of nonlinear phenomena due to light matter interactions, symmetry breaking in dissipative systems, and creation of linear superposition of nonlinear modes. The visiting scientist, Prof. Abdullaev, was actively involved in collaboration with all members the Nonlinear Dynamics and Waves Group, as well as most importantly with the PhD student, offering young researchers new scientific insights on the physical problems. Further impact was given to strengthen of the relations between Lisbon group with the researches currently working in other Portuguese universities (in particular Porto and Minho), as well as to the extension of the international links of the Nonlinear Waves and Dynamics. Practically all collaborations resulted in join publications, as well as in elaboration of future join projects.Web-page:We did not create a special webpage. Meantime, seminars, conferences, publications were announced through the official web-page of the Center for Theoretical and Computational Physics, http://cftc.cii.fc.ul.pt/ as well as on the personal Webpage of the Principal Investigator, http://sites.google.com/site/vvkonotop/.