Vortex Rossby waves in cyclones in the tropical atmosphere are believed to play a role in the observed eyewall replacement cycle, a phenomenon in which concentric rings of intense rainbands develop outside the wall of the cyclone eye, strengthen and then contract inward to replace the original eyewall. In this paper, we present a two-dimensional configuration that represents the propagation of forced Rossby waves in a cyclonic vortex and use it to explore mechanisms by which critical layer interactions could contribute to the evolution of the secondary eyewall location. The equations studied include the nonlinear terms that describe wave-mean-flow interactions, as well as the terms arising from the latitudinal gradient of the Coriolis parameter. Asymptotic methods based on perturbation theory and weakly nonlinear analysis are used to obtain the solution as an expansion in powers of two small parameters that represent nonlinearity and the Coriolis effects. The asymptotic solutions obtained give us insight into the temporal evolution of the forced waves and their effects on the mean vortex. In particular, there is an inward displacement of the location of the critical radius with time which can be interpreted as part of the secondary eyewall cycle.

]]>We study the Hankel determinants associated with the weight

where , , , is analytic in a domain containing [ − 1, 1] and for . In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as and . We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ-function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.

]]>The unified transform method introduced by Fokas can be used to analyze initial-boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann-Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of *L*^{2}-RH problems, we consider the construction of quarter plane solutions which are *C*^{1} in time and *C*^{3} in space.

Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line are investigated via the Fokas method. It is shown that the solution can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex *k*-plane, whose jump matrix is defined in terms of the matrix spectral functions and that depend on the initial data and all boundary values, respectively. If there exist spectral functions satisfying the global relation, it can be proved that the function defined by the above Riemann–Hilbert problem solves the coupled nonlinear Schrödinger equation and agrees with the prescribed initial and boundary values. The most challenging problem in the implementation of this method is to characterize the unknown boundary values that appear in the spectral function . For a particular class of boundary conditions so-called linearizable boundary conditions, it is possible to compute the spectral function in terms of and given boundary conditions by using the algebraic manipulation of the global relation. For the general case of boundary conditions, an effective characterization of the unknown boundary values can be obtained by employing perturbation expansion.

It has been shown in our paper [1] that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the *quasi-potential*. This class of fluid flows is characterized by three-component velocity field having two-component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi-potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two-component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.

A generalized two-component model with peakon solutions is proposed in this paper. It allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions. The generalized two-component system is shown to possess Lax pair and infinitely many conservation laws. Bi-Hamiltonian structures and peakon interactions are discussed in detail for typical representative equations of the generalized system. In particular, a new type of *N*-peakon solution, which is not in the traveling wave type, is obtained from the generalized system.

The Whitham modulation theory for periodic traveling waves of PDEs generated by a Lagrangian produces first-order dispersionless PDEs that are, generically, either hyperbolic or elliptic. In this paper, degeneracy of the Whitham equations is considered where one of the characteristic speeds is zero. In this case, the Whitham equations are no longer valid. Reformulation and rescaling show that conservation of wave action morphs into the Korteweg–de Vries (KdV) equation on a longer time scale thereby generating dispersion in the Whitham modulation equations even for finite amplitude waves.

]]>We consider systems of differential equations with quadratic nonlinearities having applications for biochemistry and population dynamics, which may have a large dimension *n*. Due to the complexity of these systems, reduction algorithms play a crucial role in study of their large time behavior. Our approach aims to reduce a large system to a smaller one consisting of *m* differential equations, where . Under some restrictions (that allow us to separate slow and fast variables in the system) we obtain a new system of differential equations, involving slow variables only. This reduction is feasible from a computational point of view for large *n* that allows us to investigate sensitivity of dynamics with respect to random variations of parameters. We show that the quadratic systems are capable to generate all kinds of structurally stable dynamics including chaos.

The paper investigates the integrable discretization of a negative order AKNS equation. Two semidiscrete and one fully discrete versions of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented. Dynamics of one-soliton and two-soliton solutions of spacial-discrete system are characterized.

]]>We examine the impact of small parity-time () symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing perturbations offer potentially novel means for the control of light in honeycomb lattices.

]]>The problem of a weak shock, reflected and diffracted by a wedge, is studied for the two-dimensional compressible Euler system. Some recent developments are overviewed and a perspective is presented within the context of a real gas, modeled by the van der Waals equation of state. The regular reflection configuration and the detachment criterion are studied in the light of real gas effects. Some basic features of the phenomenon and the nature of the self-similar flow pattern are explored using asymptotic expansions. The analysis presented here predicts several inviscid flow properties of the real gases undergoing shock reflection–diffraction phenomenon.

]]>On reexamining the hydrodynamic instability, Yu [1] showed that when the fully dynamical interactions are duly accounted for, and proper mathematical analysis is carried out, the positive feedback between the wave and evolving current can initiate and sustain rip current circulations with scales comparable to field observations on alongshore uniform beaches. In this study, we extend that analysis to consider nonplanar beaches, and to include a new branch of unstable modes that correspond to alongshore propagating horizontal circulations with the magnitudes of the flow growing in time. The latter has not previously been studied. These propagating unstable modes have typical time periods of tens of minutes and alongshore propagation speeds of a few centimeters per second. The physical implications of their spatial and slow time oscillations are discussed, as of relevance to occurrence and recurrence of transient rips, alongshore migration of rip currents and very low frequency pulsations in surf zone eddy circulations.

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