The semiclassical (small dispersion) limit of the focusing nonlinear Schrödinger equation with periodic initial conditions (ICs) is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of ICs, referred to as “periodic single‐lobe” potentials, share the same qualitative features, which also coincide

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as are found. These asymptotics are parameterized by monodromy data of the associated linear ODE, The parameterization allows one to derive connection formulae for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulae are also considered

We classify zeroth‐order conservation laws of systems from the class of two‐dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the

Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work, we extend this methodology to a hierarchy of nonclassical orthogonal polynomials on disk slices and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being

We generalize a previously published numerical approach for the one‐dimensional (1D) nonlinear Schrödinger (NLS) equation based on a multidomain spectral method on the whole real line in two ways: first, a fully explicit fourth‐order method for the time integration, based on a splitting scheme and an implicit Runge‐Kutta method for the linear part, is presented. Second, the 1D code is combined with

We consider the Isobe‐Kakinuma model for two‐dimensional water waves in the case of a flat bottom. The Isobe‐Kakinuma model is a system of Euler‐Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe‐Kakinuma model in the long wave regime. Numerical analysis for large

Given a local one parameter Lie group of transformations G , we determine the most general scalar partial differential equation in (1 + 1)‐independent variables of a given order admitting G as nonclassical symmetry in the Bluman and Cole sense.

In this paper, the Hirota's bilinear method and Kadomtsev‐Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi‐)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram‐type determinants. When N is even, soliton, line breather and (semi‐)rational solutions on the constant background are

We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy‐type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile

Kernel density estimation on a finite interval poses an outstanding challenge because of the well‐recognized bias at the boundaries of the interval. Motivated by an application in cancer research, we consider a boundary constraint linking the values of the unknown target density function at the boundaries. We provide a kernel density estimator (KDE) that successfully incorporates this linked boundary

Electromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source

Measure‐valued weak solutions for conservation laws with discontinuous flux are proposed and explicit formulae have been derived. We propose convergent discontinuous flux‐based numerical schemes for the class of hyperbolic systems that admit nonclassical ‐shocks, by extending the theory of discontinuous flux for nonlinear conservation laws to scalar transport equation with a discontinuous coefficient

We construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stäckel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stäckel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of

General soliton solutions to a reverse‐time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear

The complex Hamiltonian systems with real‐valued Hamiltonians are generalized to deduce quasi‐periodic solutions for a hierarchy of derivative nonlinear Schrödinger (DNLS) equations. The DNLS hierarchy is decomposed into a family of complex finite‐dimensional Hamiltonian systems by separating the temporal and spatial variables, and the complex Hamiltonian systems are then proved to be integrable in

The Ostrovskyi (Ostrovskyi‐Vakhnenko/short pulse) equations are ubiquitous models in mathematical physics. They describe water waves under the action of a Coriolis force as well as the amplitude of a “short” pulse in an optical fiber. In this paper, we rigorously construct ground traveling waves for these models as minimizers of the Hamiltonian functional for any fixed L2 norm. The existence argument

In this paper, we establish a formula determining the value of the Cauchy principal value integrals of the real and purely imaginary Ablowitz‐Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann‐Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are

One of the essential tasks of the theory of water waves is a construction of simplified mathematical models, which are applied to the description of complex events, such as wave breaking, appearing of freak waves in the assumption of weak nonlinearity. The Zakharov equation and its simplification, such as nonlinear Schrodinger equations and Dysthe equations, are among them. Recently, for unidirectional

In this work, we study the behavior of saturation fronts for two‐phase flow through a long homogeneous porous column . In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses traveling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, pseudo‐differential operators, and Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results.

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more

This paper focuses on the construction of rational solutions for the ‐Painlevé system, also called the Noumi‐Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each

Using a difference‐equation method in a previous paper, we study the associated Pollaczek polynomials defined by a three‐term recurrence relation. Two asymptotic approximations are derived for these polynomials; one holds for with and , and the other holds for with t in a neighborhood of . An asymptotic formula is also provided for their largest zeros.

A simple formula is proven for an upper bound for amplitudes of hyperelliptic (finite‐gap or N‐phase) solutions of the derivative nonlinear Schrödinger equation. The upper bound is sharp, viz, it is attained for some initial conditions. The method used to prove the upper bound is the same method, with necessary modifications, used to prove the corresponding bound for solutions of the focusing NLS equation

We study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials

In this paper, we develop and analyze a malaria model with seasonality of mosquito life‐history traits: periodic‐mosquitoes per capita birth rate, ‐mosquitoes death rate, ‐probability of mosquito to human disease transmission, ‐probability of human to mosquito disease transmission, and ‐mosquitoes biting rate. All these parameters are assumed to be time dependent leading to a nonautonomous differential

Chlamydia trachomatis is a bacterium that causes eye infection and blindness in humans. It has an unusual life cycle involving two developmental forms. Within a cytoplasmic inclusion, the reticulate body (RB) repeatedly divides by binary fission and asynchronously differentiates into the infectious elementary body (EB). Upon the death of the mammalian cell that host many such inclusions, only the EB

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. But just how this is accomplished

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

Receptor-mediated BMP degradation has been seen to play an important role in allowing for the formation of relatively stable P Mad patterns. To the extent that receptors act as a "sink" for BMPs, one would predict that the localized over-expression of signaling receptors would cause a net flux of freely diffused BMPs toward the ectopic, i.e., abnormally high concentration, receptor site. One possible