Thresholds for breather solutions on the Discrete Nonlinear Schr\"odinger Equation with saturable and power nonlinearity

We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear Schr\"odinger Equation with saturable and power nonlinearity. Theoretical and numerical results are proved concerning the existence and nonexistence of periodic solutions by a variational approach and a fixed point argument. In the variational approach we are restricted to DNLS lattices with Dirichlet boundary conditions. It is proved that there exists parameters (frequency or nonlinearity parameters) for which the corresponding minimizers satisfy explicit upper and lower bounds on the power. The numerical studies performed indicate that these bounds behave as thresholds for the existence of periodic solutions. The fixed point method considers the case of infinite lattices. Through this method, the existence of a threshold is proved in the case of saturable nonlinearity and an explicit theoretical estimate which is independent on the dimension is given. The numerical studies, testing the efficiency of the bounds derived by both methods, demonstrate that these thresholds are quite sharp estimates of a threshold value on the power needed for the the existence of a breather solution. This it justified by the consideration of limiting cases with respect to the size of the nonlinearity parameters and nonlinearity exponents.Comment: 26 pages, 10 figure

Global existence and compact attractors for the discrete nonlinear Schrödinger equation

AbstractWe study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations

Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation

We study various properties of solutions of an extended nonlinear Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric evolution problems -- including vortex filament dynamics -- and governs propagation of short pulses in optical fibers and nonlinear metamaterials. For the periodic initial-boundary value problem, we derive conservation laws satisfied by local in time, weak $H^2$ (distributional) solutions, and establish global existence of such weak solutions. The derivation is obtained by a regularization scheme under a balance condition on the coefficients of the linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS model. Next, we investigate conditions for the existence of traveling wave solutions, focusing on the case of bright and dark solitons. The balance condition on the coefficients is found to be essential for the existence of exact analytical soliton solutions; furthermore, we obtain conditions which define parameter regimes for the existence of traveling solitons for various linear dispersion strengths. Finally, we study the modulational instability of plane waves of the ENLS equation, and identify important differences between the ENLS case and the corresponding NLS counterpart. The analytical results are corroborated by numerical simulations, which reveal notable differences between the bright and the dark soliton propagation dynamics, and are in excellent agreement with the analytical predictions of the modulation instability analysis.Comment: 27 pages, 5 figures. To be published in Journal of Physics A: Mathematical and Theoretica

Lower and upper estimates on the excitation threshold for breathers in DNLS lattices

We propose analytical lower and upper estimates on the excitation threshold for breathers (in the form of spatially localized and time periodic solutions) in DNLS lattices with power nonlinearity. The estimation depending explicitly on the lattice parameters, is derived by a combination of a comparison argument on appropriate lower bounds depending on the frequency of each solution with a simple and justified heuristic argument. The numerical studies verify that the analytical estimates can be of particular usefulness, as a simple analytical detection of the activation energy for breathers in DNLS lattices.Comment: 10 pages, 3 figure

Reconstruction nailing for ipsilateral femoral neck and shaft fractures

The surgical management of ipsilateral fractures of the femoral neck and shaft presents a difficult and challenging problem for the orthopaedic surgeon. The purpose of the present study was to report the mid-term results and complications in a series of patients who sustained ipsilateral femoral neck and shaft fractures and treated in our trauma department with a single reconstruction nail for both fractures. Eleven patients were included in the study with an average age of 46.4 years. The mean follow-up was 47 months (range, 15–75 months). There were no cases of a missed diagnosis at initial presentation. The mean time to union was 4.5 months for the neck fracture and 8.2 months for the shaft. There were no cases of avascular necrosis of the femoral head or non-union of the neck fracture. The mean Harris Hip Score was (85 ± 4.3). Complications included two cases of shaft fracture non-union and one case of peroneal nerve palsy. Heterotopic ossification at the tip of the greater trochanter was evident in two cases without causing any functional deficit. The current study suggests that reconstruction nailing produces satisfactory clinical and functional results in the mid-term. The complications involved only the femoral shaft fracture and were successfully treated with a single operative procedure