Infinite hierarchy of nonlinear Schrödinger equations and their solutions.

We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.

Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients.

We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.

Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy.

We analyze the quintic integrable equation of the nonlinear Schrödinger hierarchy that includes fifth-order dispersion with matching higher-order nonlinear terms. We show that a breather solution of this equation can be converted into a nonpulsating soliton solution on a background. We calculate the locus of the eigenvalues on the complex plane which convert breathers into solitons. This transformation does not have an analog in the standard nonlinear Schrödinger equation. We also study the interaction between the new type of solitons, as well as between breathers and these solitons.

Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions.

We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.

Etiology of bacteremia in young infants in six countries.

BACKGROUND: Neonatal illness is a leading cause of death worldwide; sepsis is one of the main contributors. The etiologies of community-acquired neonatal bacteremia in developing countries have not been well characterized. METHODS: Infants <2 months of age brought with illness to selected health facilities in Bangladesh, Bolivia, Ghana, India, Pakistan and South Africa were evaluated, and blood cultures taken if they were considered ill enough to be admitted to hospital. Organisms were isolated using standard culture techniques. RESULTS: Eight thousand eight hundred and eighty-nine infants were recruited, including 3177 0-6 days of age and 5712 7-59 days of age; 10.7% (947/8889) had a blood culture performed. Of those requiring hospital management, 782 (54%) had blood cultures performed. Probable or definite pathogens were identified in 10.6% including 10.4% of newborns 0-6 days of age (44/424) and 10.9% of infants 7-59 days of age (39/358). Staphylococcus aureus was the most commonly isolated species (36/83, 43.4%) followed by various species of Gram-negative bacilli (39/83, 46.9%; Acinetobacter spp., Escherichia coli and Klebsiella spp. were the most common organisms). Resistance to second and third generation cephalosporins was present in more than half of isolates and 44% of the Gram-negative isolates were gentamicin-resistant. Mortality rates were similar in hospitalized infants with positive (5/71, 7.0%) and negative blood cultures (42/557, 7.5%). CONCLUSIONS: This large study of young infants aged 0-59 days demonstrated a broad array of Gram-positive and Gram-negative pathogens responsible for community-acquired bacteremia and substantial levels of antimicrobial resistance. The role of S. aureus as a pathogen is unclear and merits further investigation.

Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.

We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

Solutions of the higher-order Manakov-type continuous and discrete equations.

We derive exact and approximate localized solutions for the Manakov-type continuous and discrete equations. We establish the correspondence between the solutions of the coupled Ablowitz-Ladik equations and the solutions of the coupled higher-order Manakov equations.