Parity-time () symmetry, initially proposed in the context of Quantum Mechanics and Quantum Field Theory, has recently been studied and demonstrated in optical and electronic systems where laboratory demonstrations are possible. The model considered here consists of two nonlinearly coupled van der Pol (VDP) oscillators, originally studied in [1]. This dimer serves as an experimental realization of a class of nonlinear systems, where the anharmonic component has gain for one oscillator and loss of equal strength for the other, so that Hermiticity is broken while symmetry is preserved. The existence of spontaneous symmetry breaking at some critical value of the gain/loss parameter is proven by use of modulation theory in the weakly nonlinear regime, and by use of asymptotic methods to demonstrate relaxed oscillations for a strongly nonlinear coupling. We then prove similar phenomena in an infinite chain composed of such VDP dimers in the long-wave limit. Finally, we perform initial studies of asymmetric transport properties in the VDP arrays.

]]>In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials: the coefficients of the formal power series appearing in the solutions near 0, 1, or ∞ (in the variables , or , respectively). The result, which seems to be true for all values of the parameter τ, according to our numerical experiments, is here proved for small values of the parameter τ.

]]>The interaction of flexural-gravity waves with a thin circular-arc-shaped permeable plate submerged beneath the ice-covered surface of water with uniform finite depth is considered under the assumption of linear theory. The problem is reduced to a second kind hypersingular integral equation for the potential difference across the plate which is solved approximately by an expansion–collocation method. Utilizing the solution, the reflection and the transmission coefficients and the hydrodynamic forces are evaluated numerically. The focus of the paper is to illustrate the effect of a porous curved plate submerged in finite depth water with an ice-cover on the normally incident waves. Numerical results for a circular-arc-shaped plate for different configurations are derived and represented graphically. Also, by choosing an appropriate set of parameters, the known results for a circular-arc-shaped rigid plate submerged in deep water and a semicircular porous plate submerged in finite depth water with a free surface are recovered as special cases.

]]>Human immunodeficiency virus (HIV) can infect various types of cell populations such as CD4^{+} T cells and macrophages. The heterogeneity of these target cells implies different birth, death, infection rates, and so on. To investigate the within-host dynamics of HIV which can infect *n* different types of target cells, a theoretical model with infection-age structure for each type of target cells and a general nonlinear incidence rate is proposed in this manuscript. The model, in the form of a hyperbolic system of partial differential equations (PDE) for infected target cells coupled with several ordinary differential equations, is shown to be biologically reasonable with the establishment of existence, positivity, and boundedness of solutions. Although the PDE form poses novel challenges to theoretical investigation, rigorous analysis is performed to show the uniform persistence of the virus when the basic reproduction number is greater than one. Furthermore, by constructing suitable Lyapunov functionals, we show that the infection-free steady state is globally asymptotically stable when the basic reproduction number is less than unity, while the positive steady state is globally asymptotically stable when the basic reproduction number is greater than one.

We describe a computational method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic–quintic complex Ginzburg–Landau equation about numerically determined stationary solutions. We compare the results of the method to a formula for an edge bifurcation obtained using the small dissipation perturbation theory of Kapitula and Sandstede. This comparison highlights the importance for analytical studies of perturbed nonlinear wave equations of using a pulse ansatz in which the phase is not constant, but rather depends on the perturbation parameter. In the presence of large dissipative effects, we discover variations in the structure of the spectrum as the dispersion crosses zero that are not predicted by the small dissipation theory. In particular, in the normal dispersion regime we observe a jump in the number of discrete eigenvalues when a pair of real eigenvalues merges with the intersection point of the two branches of the continuous spectrum. Finally, we contrast the method to computational Evans function methods.

]]>We study a class of partial differential equations (PDEs) in the family of the so-called Euler–Poincaré differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses nonsmooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particle system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when nonhomogeneous “constant stream” boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic-like to dispersive with a nontrivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two-particle dynamics under the resulting lack of Lipschitz continuity. Curiously, we find that for the conically shaped kernels the motion of the related two-dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two-dimensional system to the one-dimensional completely integrable Shallow-Water equation [1], while retaining the full dependence on two spatial dimensions for the single channel solutions. Finally, by comparing with an operator-splitting pseudospectral method we illustrate the performance of the particle algorithms with respect to their Eulerian counterpart for this class of nonsmooth kernels.

]]>In this paper, we consider an initial-value problem for Burgers' equation with variable coefficients

where *x* and *t* represent dimensionless distance and time, respectively, while , are given continuous functions of *t* ( > 0). In particular, we consider the case when the initial data has algebraic decay as , with as and as . The constant states and are problem parameters. We focus attention on the case when (with ) and . The method of matched asymptotic coordinate expansions is used to obtain the large-*t* asymptotic structure of the solution to the initial-value problem over all parameter values.

In this paper, we establish the orbital stability of a class of spatially periodic wave train solutions to multidimensional nonlinear Klein–Gordon equations with periodic potential. We show that the orbit generated by the one-dimensional wave train is stable under the flow of the multidimensional equation under perturbations which are, on one hand, coperiodic with respect to the translation or Galilean variable of propagation, and, on the other hand, periodic (but not necessarily coperiodic) with respect to the transverse directions. That is, we show their transverse orbital stability. The class of periodic wave trains under consideration is the family of subluminal rotational waves, which are periodic in the momentum but unbounded in their position.

]]>Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave prop agating with the constant velocity. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with radians angle on the crest. A conformal map is used that maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase in the numerical precision and subsequent increase in the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one square-root branch point per horizontal spatial period λ of Stokes wave located at the distance from the real line. The increase in the scaled wave height from the linear limit to the critical value marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called the Stokes wave of the greatest height). Here, *H* is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least 10^{−26}. The number of poles in tables increases from a few for near-linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with

Errors in nonlinear lightwave systems are often associated with rare, noise-induced, large deviations of the signal. We present a method to determine the most probable manner in which such rare events occur by solving a sequence of constrained optimization problems. These results then guide importance-sampled Monte Carlo simulations to determine the events' probabilities. The method applies to a general class of intensity-based optical detectors and to arbitrarily shaped and multiple pulses.

]]>The work we describe addresses the process of whitecapping. We first argue that, when the winds are strong enough, the ocean surface must develop an alternative means to dissipate energy when its flux from large to small scales becomes too large. We then show that the resulting Phillips' spectrum, which holds at small or meter length scales, is dominated by sharp crested waves. We next idealize such a sea locally by a family of close to maximum amplitude Stokes waves and show, using highly accurate simulation algorithms based on a conformal map representation, that perturbed Stokes waves develop the universal feature of an overturning plunging jet. We analyze both the cases when surface tension is absent and present. In the latter case, we show the plunging jet is regularized by capillary waves that rapidly become nonlinear Crapper waves in whose trough pockets whitecaps may be spawned. We are careful not to claim this as the definitive mechanism for whitecaps because three-dimensional effects, although qualitatively discussed, are not included in the analysis.

]]>We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half-plane with conductivity , . The representations are obtained via the so-called unified transform method or Fokas method, involving a Riemann–Hilbert problem on the complex plane when *p* is even and on a two-sheeted Riemann surface when *p* is odd. They are given in terms of the Dirichlet and Neumann data on the boundary of the domain. For even exponent *p*, we also show how to make the conversion from one type of conditions to the other by using the global relation that follows from the closedness of some differential form. The method used to derive our integral representations could be applied in any bounded simply connected domain of the right half-plane with a smooth boundary.

We introduce a new class of Green–Naghdi type models for the propagation of internal waves between two (1 + 1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, and irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green–Naghdi model, and in particular to manage high-frequency Kelvin–Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups, and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well-posedness, and stability results. These results apply in particular to the original Green–Naghdi model as well as to the Saint–Venant (hydrostatic shallow water) system with surface tension.

]]>We prove that conformally parameterized surfaces in Euclidean space of curvature *c* admit a symmetry reduction of their Gauss–Codazzi equations whose general solution is expressed with the sixth Painlevé function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko et al. 1997) can be recovered by such a reduction.

Presented are the results of numerical experiments on calculation of probability distribution functions (PDFs) for surface elevations of water waves arising during the evolution of statistically homogeneous wave field. Extreme waves or freak waves are an integral part of ocean waving, and PDFs are compared both for nonlinear and linear models. Obviously, linear model demonstrates the Rayleigh distribution of surface elevations while PDFs for nonlinear equation have tails for large elevations similar to Rayleigh distribution, but with much larger σ.

]]>Large-scale simulations in spherical geometries require associated quadrature formulas. Classical approaches based on tabulated weights are limited to specific quasi-uniform distributions of relatively low numbers of nodes. By using a radial basis function-generated finite differences (RBF-FD)-based approach, the proposed algorithm creates quadrature weights for *N* arbitrarily scattered nodes in only operations.

We explore the possibility of multi-site breather states in a nonlinear Klein–Gordon lattice to become nonlinearly unstable, *even if* they are found to be spectrally stable. The mechanism for this nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the Klein–Gordon lattice with a soft (Morse) and a hard (ϕ^{4}) potential. Compared to the case of the nonlinear Schrödinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the multi-site breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, multi-site breather states are observed to be nonlinearly stable.

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is nonlocal, and the proper choice of integration constants should be the one dictated by the associated inverse scattering transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation , in this paper we establish the following. 1. The nonlocal term arising from its evolutionary form corresponds to the asymmetric integral . 2. Smooth and well-localized initial data evolve in time developing, for , the constraint , where . 3. Because no smooth and well-localized initial data can satisfy such constraint at , the initial () dynamics of the Pavlov equation cannot be smooth, although, because it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results should be successfully used in the study of the nonlocality of other basic examples of integrable dispersionless PDEs in multidimensions.

]]>The purpose of this paper is to construct the inverse scattering transform for the focusing Ablowitz-Ladik equation with nonzero boundary conditions at infinity. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann-Hilbert problem on a doubly connected curve in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz-Ladik potential.

]]>We compute the scattering data of the Benjamin–Ono equation for arbitrary rational initial conditions with simple poles. Specifically, we obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient.

]]>The three-wave, resonant interaction equations appear in many physical applications. These partial differential equations (PDEs) are known to be completely integrable, and have been solved with initial data that decay rapidly in space, using inverse scattering theory. We present a new way to solve these equations, which makes no use of inverse scattering theory, and which can be used with a wide variety of boundary conditions. A “general solution” of these PDEs would involve six free, real-valued functions of space. At this time, our “nearly general solution” accepts five free, real-valued functions of space, and embeds them in convergent series in a deleted neighborhood of a pole.

]]>In this paper, we discuss Airy solutions of the second Painlevé equation (P_{II}) and two related equations, the Painlevé XXXIV equation () and the Jimbo–Miwa–Okamoto σ form of P_{II} (S_{II}), are discussed. It is shown that solutions that depend only on the Airy function have a completely different structure to those that involve a linear combination of the Airy functions and . For all three equations, the special solutions that depend only on are *tronquée* solutions, i.e., they have no poles in a sector of the complex plane. Further, for both and S_{II}, it is shown that among these *tronquée* solutions there is a family of solutions that have no poles on the real axis.

The anti-self-dual Yang-Mills equations are known to have reductions to many integrable differential equations. A general Bäcklund transformation (BT) for the anti-self-dual Yang-Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.

]]>A technique to manufacture *solvable* variants of the “goldfish” many-body problem is introduced, and several many-body problems yielded by it are identified and discussed, including cases featuring *multiperiodic* or *isochronous* dynamics.

A map from the initial conditions to the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with *n* interfaces. The existence of this map allows changing the problem at hand from an interface problem to a boundary value problem which allows for an alternative to the approach of finding a closed-form solution to the interface problem.