Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The asymptotic approximations are uniformly valid for unbounded complex values of the argument, and are applied to inhomogeneous Airy equations having polynomial and exponential

Recent generalizations of the Camassa–Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well‐posedness in Sobolev spaces for the family under consideration. Moreover, we prove that wave breaking phenomena

The aim of the paper is to construct nonlocal reverse‐space nonlinear Schrödinger (NLS) hierarchies through nonlocal group reductions of eigenvalue problems and generate their inverse scattering transforms and soliton solutions. The inverse scattering problems are formulated by Riemann‐Hilbert problems which determine generalized matrix Jost eigenfunctions. The Sokhotski‐Plemelj formula is used to

This current study deals with the long‐time dynamics of a nonlinear system of coupled parabolic equations with memory. The system describes the thermodiffusion phenomenon where the fluxes of mass diffusion and heat depend on the past history of the chemical potential and the temperature gradients, respectively, according to Gurtin‐Pipkin's law. Inspired by the works of Chueshov and Lasiecka on the

Based on the inverse scattering method, the formulae of one higher‐order pole solitons and multiple higher‐order poles solitons of the nonlinear Schrödinger equation (NLS) equation are obtained. Their denominators are expressed as , where is a matrix frequently constructed for solving the Riemann‐Hilbert problem, and the asterisk denotes complex conjugate. We take two methods for proving is invertible

A Bäcklund transformation (BT), which involves both independent and dependent variables, is established and studied for the short pulse (SP) equation. Based it, the nonlinear superposition formulae for 2‐, 3‐, and 4‐BT are presented. The general result for the composition of ‐BTs is achieved and given in terms of determinants. As applications, various solutions including loop solitons, breather solutions

We consider the dynamical stability of periodic and quasiperiodic stationary solutions of integrable equations with 2 2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use the squared eigenfunction connection between the Lax spectrum and the stability spectrum to show that the subset

One‐ and two‐dimensional solitons of a multicomponent nonlocal nonlinear Schrödinger (NLS) system are constructed. The model finds applications in nonlinear optics, where it may describe the interaction of optical beams of different frequencies. We asymptotically reduce the model, via multiscale analysis, to completely integrable ones in both Cartesian and cylindrical geometries; we thus derive a

We compute the kinetic energy of the Langevin particle using different approaches. We build stochastic differential equations that describe this physical quantity based on both the Itô and Stratonovich stochastic integrals. It is shown that the Itô equation possesses a unique solution, whereas the Stratonovich one possesses infinitely many, all but one absent of physical meaning. We discuss how this

In this paper, we employ a difference equation approach to study the Plancherel‐Rotach asymptotics of ‐orthogonal polynomials about their largest zeros. Our method for ‐difference equations is an analogue to the turning point problem for Hermite differential equations. It works well in the toy problems of Stieltjes‐Wigert polynomials and ‐Hermite polynomials.

We study stable blow‐up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger‐type equation with a nonlocal, convolution‐type nonlinearity: First, we consider the ‐critical case in dimensions and obtain that a generic blow‐up has a self‐similar structure and exhibits not only the square root blowup rate , but also the log‐log correction (via asymptotic analysis and

We investigate a partial differential equation which models solid‐solid phase transitions. This model is for martensitic phase transitions driven by configurational force and its counterpart is for interface motion by mean curvature. Mathematically, this equation is a second‐order nonlinear degenerate parabolic equation. And in multidimensional case, its principal part cannot be written into divergence

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

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

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 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

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

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.

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

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 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

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

In this paper, we reflect upon control intervention practices habitually exerted by healthcare authorities in tropical areas that suffer from incidental outbreaks of dengue fever, in particular, the city of Cali, Colombia. Such control interventions, principally based on the insecticide spraying, are carried out sporadically in order to overcome an ongoing epidemic or at least to reduce its size. It

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

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on

A new vector long wave‐short wave‐type model is proposed by resorting to the zero‐curvature equation. Based on the resulting Riccati equations related to the Lax pair and the gauge transformations between the Lax pairs, multifold Darboux transformations are constructed for the vector long wave‐short wave‐type model. This method is general and is suitable for constructing the Darboux transformations

A mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions having an unboundedly growing amplitude and a limited phase mismatch. The paper investigates the behavior of such solutions when the parameters of the excitation take

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically

We construct the multi‐breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and nontrivial phase‐modulating elliptic solutions are presented, and their

We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. We use the unified transform method, also known as the Fokas method, and the Deift‐Zhou nonlinear steepest descent method to construct solutions in a sector close to the boundary whose leading behavior is described by a single exponential plane wave. Furthermore, we compute the

Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points, which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic

In this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background satisfies a “small norm” condition . On the other hand, Ohta and Yang recently showed that the defocusing AL system, which is modulationally

Matrix Szegő biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szegő polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szegő matrix and the associated Szegő recursion relations for the matrix orthogonal polynomials

Novel soliton solutions of a two‐dimensional (2D) nonlocal nonlinear Schrödinger (NLS) system are revealed by asymptotically reducing the system to a completely integrable Davey–Stewartson (DS) set of equations. In so doing, the reductive perturbation method in addition to a multiple scales scheme are utilized to derive both the DS‐I and DS‐II systems, depending on the strength of the nonlocality,

We consider a coefficient identification problem for a mathematical model with free boundary related to ductal carcinoma in situ (DCIS). This inverse problem aims to determine the nutrient consumption rate from additional measurement data at a boundary point. We first obtain a global‐in‐time uniqueness of our inverse problem. Then based on the optimization method, we present a regularization algorithm

In this paper, we introduce a new property of two‐dimensional integrable hydrodynamic chains—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Infinitely many local three‐dimensional conservation laws for the Benney commuting hydrodynamic chains are constructed. As a by‐product, we established a new method for computation

A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterized by the Dyson parameter β, and the Laguerre weight , in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable . Previous work has established the corresponding functional

We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimizing frame, also known as the Normal, Parallel, or Bishop frame. Such systems have previously been studied using the Frenet–Serret frame. However, the Rotation Minimizing frame has many advantages, and can be used to study a wider class of examples.

We study the response of nonlinear wave systems in bounded domains at or near resonance. There are typically two qualitatively distinct types of response which may be observed relating to whether or not higher harmonics are themselves resonant. We introduce a variety of nonlinear model problems at or near resonance and study the subsequent response. We explain how the features of this problem such

We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. As a preliminary analysis, we first consider the case when the filaments are arranged as straight and parallel lines, and explicitly solve the system to show that the motion of the lines resemble that of point vortices moving in a plane. Then, we consider the motion of a

In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial

We analyze the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave for spatially periodic solutions of the one‐dimensional, nonisentropic gas dynamics equations. The case of an entropy wave with a sawtooth profile is of interest because the oscillations of the reflected sound waves are nondispersive with frequency independent of their wavenumber, leading to an unusual

We consider symmetry‐integrable evolution equations in 1 + 1 dimensions of order 3 and order 5. We show that there exist only three equations in this class that are invariant under the Möbius transformation, and we name those Schwarzian equations. We report an interesting relation between the recursion operators of the Schwarzian equations and the corresponding adjoint operators that generate hierarchies

We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be nonlocal in the non‐Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first‐order PDEs