We study optical spectral singularities of a weakly nonlinear -symmetric bilinear planar slab of optically active material. In particular, we derive the lasing threshold condition and calculate the laser output intensity. These reveal the following unexpected features of the system: (1) for the case that the real part of the refractive index η of the layers are equal to unity, the presence of the lossy layer decreases the threshold gain; (2) for the more commonly encountered situations when is much larger than the magnitude of the imaginary part of the refractive index, the threshold gain coefficient is a function of η that has a local minimum. The latter is in sharp contrast to the threshold gain coefficient of a homogeneous slab of gain material which is a decreasing function of η. We use these results to comment on the effect of nonlinearity on the prospects of using this system as a coherence perfect absorption-laser.

]]>This paper presents a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters. A regular perturbation procedure typically provides asymptotic solutions valid for bounded *t* values as a positive parameter ε tends to zero. One can eliminate secular terms by introducing a slowly-varying amplitude obtained as a solution of an amplitude equation on intervals where is bounded. With sufficient stability hypotheses, the results may even hold for all . These ideas are illustrated for a number of nontrivial problems involving ordinary differential equations.

The subjects of the work are pairs of linearly coupled -symmetric dimers. Two different settings are introduced, namely, *straight-coupled dimer*s, where each gain site is linearly coupled to one gain and one loss site, and *cross-coupled dimers*, with each gain site coupled to two lossy ones. The latter pair with equal coupling coefficients represents a -*hypersymmetric* quadrimer. We find symmetric and antisymmetric solutions in these systems, chiefly in an analytical form, and explore the existence, stability, and dynamical behavior of such solutions by means of numerical methods. We thus identify bifurcations occurring in the systems, including spontaneous symmetry breaking and saddle-center bifurcations. Simulations demonstrate that evolution of unstable branches typically leads to blowup. However, in some cases, unstable modes rearrange into stable ones.

The patterning of many developing tissues is orchestrated by gradients of signaling morphogens. Included among the molecular events that drive the formation of morphogen gradients are a variety of elaborate regulatory interactions. Such interactions are thought to make gradients robust, i.e., insensitive to change in the face of genetic or environmental perturbations. However just how this is accomplished is a major unanswered question. Recently extensive numerical simulations suggest that robustness of signaling gradients can be achieved through morphogen degradation mediated by cell surface bound nonsignaling receptor molecules (or *nonreceptors* for short) such as heparan sulfate proteoglycans. The present paper provides a mathematical validation of the results from the aforementioned numerical experiments. Extension of a basic extracellular model to include reversible binding with nonreceptors synthesized at a prescribed rate and mediated morphogen degradation shows that the signaling gradient diminishes with increasing concentration of cell-surface nonreceptors. Perturbation and asymptotic solutions obtained for (i) low (receptor and nonreceptor) occupancy, and (ii) high nonreceptor concentration permit more explicit delineation of the effects of nonreceptors on signaling gradients and facilitate the identification of scenarios in which the presence of nonreceptors may or may not be effective in promoting robustness.

The Miyata–Choi–Camassa (MCC) system of equations describing long internal nonhydrostatic and nonlinear waves at the interface between two layers of inviscid fluids of different densities bounded by top and bottom walls is mathematically ill-posed despite the fact that physically stable internal waves are observed matching closely those of MCC. A regularization to the MCC equations that yields a computationally simple well-posed system for time-dependent evolution is proposed here. The regularization is accomplished by keeping the full hyperbolic part of MCC and exchanging spatial and temporal derivatives in one of the linearized dispersive terms. Solitary waves of MCC over a wide range of parameters are used as a benchmark to check the accuracy of the model. Our model includes the possibility of a background shear, and we show that, contrary to the no shear case, solitary waves can cross the midlevel between the top and the bottom walls and may have different polarity from the case with no background shear. Time-dependent solutions of the regularization stable model are presented, including interactions of its solitary waves, and classical and modified Korteweg-de Vries equations for small amplitude waves with the inclusion of background shear are derived. Throughout the paper, the Boussinesq approximation is taken, although the results can be extended to the non-Boussinesq case.

]]>Wave packet ansätze are introduced into an -dimensional Manakov-type system and key invariants are isolated. Reduction is made to a novel coupled Ermakov–Painlevé II system and an algorithm presented for the derivation of wave packet representations via the classical Ermakov nonlinear superposition principle. Application of the procedure in the context of certain transverse wave motions in a generalized Mooney–Rivlin hyperelastic material is likewise shown to lead an Ermakov–Painlevé II reduction.

]]>A dynamical model equation for interfacial gravity-capillary (GC) waves between two semi-infinite fluid layers, with a lighter fluid lying above a heavier one, is derived. The model proposed is based on the fourth-order truncation of the kinetic energy in the Hamiltonian of the full problem, and on weak transverse variations, in the spirit of the Kadomtsev-Petviashvilli equation. It is well known that for the interfacial GC waves in deep water, there is a critical density ratio where the associated cubic nonlinear Schrödinger equations changes type. Our numerical results reveal that, when the density ratio is below the critical value, the bifurcation diagram of plane solitary waves behaves in a way similar to that of the free-surface GC waves on deep water. However, the bifurcation mechanism in the vicinity of the minimum of the phase speed is essentially similar to that of free-surface gravity-flexural waves on deep water, when the density ratio is in the supercritical regime. Different types of lump solitary waves, which are fully localized in both transverse and longitudinal directions, are also computed using our model equation. Some dynamical experiments are carried out via a marching-in-time algorithm.

]]>The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi-squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum *Q*-function and for the complementary Marcum *P*-function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.