The patterning of many developing tissues is orchestrated by gradients of morphogens through a variety of elaborate regulatory interactions. Such interactions are thought to make gradients robust, that is, resistant to changes induced by genetic or environmental perturbations; but just how this might be done is a major unanswered question. Recently extensive numerical simulations suggest that robustness of signaling gradients cannot be attained by negative feedback (of the Hill's function type) on signaling receptors but can be achieved through binding with nonsignaling receptors (or nonreceptors for short) such as heparan sulfate proteoglycans with the resulting complexes degrading after endocytosis. These were followed by a number of analytical and numerical studies in support of the aforementioned observations. However, evidence of feedback regulating signaling gradients has been reported in literature. The present paper undertakes a different approach to the role of feedback in robust signaling gradients. The overall goal of the project is to investigate the effectiveness of feedback mechanisms on ligand synthesis, receptor synthesis, nonreceptor synthesis, and other regulatory processes in the morphogen gradient system. As a first step, we embark herein a proof-of-concept examination of a new spatially uniform feedback process that is distinctly different from the conventional spatially nonuniform Hill function approach.

]]>A class of multicomponent integrable systems associated with Novikov algebras, which interpolate between Korteweg–de Vries (KdV) and Camassa–Holm-type equations, is obtained. The construction is based on the classification of low-dimensional Novikov algebras by Bai and Meng. These multicomponent bi-Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated with the Novikov algebras. The related bilinear forms generating cocycles of first, second, and third order are classified. Several examples, including known integrable equations, are presented.

]]>We consider the initial-value problem for the regularized Boussinesq-type equation in the class of periodic functions. Validity of the weakly nonlinear solution, given in terms of two counterpropagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the timescales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis.

]]>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.

]]>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.

For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev–Stirling numbers, a special case of the Jacobi–Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.

]]>We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg-de Vries type.

]]>For the one-dimensional nonlinear Schrödinger equations with parity-time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT-symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT-symmetric potential, symmetry breaking of PT-symmetric solitary waves does not occur. Based on these findings, it is conjectured that one-dimensional PT-symmetric potentials cannot support continuous families of non-PT-symmetric solitary waves.

]]>We analyze the semilocal convergence of Steffensen's method, using a novel technique, which is based on recurrence relations, for solving systems of nonlinear equations. This technique allows analyzing the convergence of Steffensen's method to solutions of equations, where the function involved can be both differentiable and nondifferentiable. Moreover, this technique also allows enlarging the domain of starting points for Steffensen's method from certain predictions with the simplified Steffensen method.

]]>We study initial boundary value problems for the sine-Gordon equation on the half-line via the Fokas method, known as an extension of the inverse scattering transform. The method is based on the simultaneous analysis of both parts of the Lax pair and the global algebraic relation that couples known and unknown boundary values. One of most difficult steps of the method is to characterize the unknown boundary values that appear in the spectral functions. We derive the Dirichlet to Neumann map by using the global relation and the asymptotics of the eigenfunctions. Furthermore, employing perturbation expansion, we present an effective characterizations of the unknown boundary value in terms of the given initial and boundary values, and we then derive the first few terms of the expansions of the Neumann boundary value up to the third order.

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