We study the existence of weak solutions to an initial‐boundary value problem for a new phase‐field model, which consists of a degenerate parabolic equation coupled to linear elasticity equations. This model is used to describe the evolution of interfaces in elastically deformable solid materials which moves by configurational forces, such as martensitic phase transformations in shape memory alloys

The aim of this paper is to study the global unique solvability on Sobolev solution perturbated around diffusion waves to the Cauchy problem of conservative form of Hsieh's equations. Furthermore, convergence rates are also obtained as one of the diffusion parameters goes to zero. The difficulty is created due to conservative nonlinearity to enclose the uniform (in diffusion parameter) higher order

A parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent

We equip a knot K with a set of colored bonds, that is, colored intervals properly embedded into . Such a construction can be viewed as a structure that topologically models a closed protein chain including any type of bridges connecting the backbone residues. We introduce an invariant of such colored bonded knots that respects the HOMFLYPT relation, namely, the HOMFLYPT skein module of colored bonded

The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half‐line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations and (ii) by expressing the linearized classical Boussinesq system

The aim of this paper is to prove real analytic properties of all streamlines of two‐dimensional steady rotational gravity water waves in two‐layer flows. Provided that there are no stagnation points in the flow, we show that each streamline, including the free surface and the interface, is a real analytic curve if the height function is in , which corresponds to an arbitrarily bounded and measurable

This paper studies the blow‐up solutions for the Schrödinger equation with a Hartree‐type nonlinearity together with a power‐type subcritical perturbation. The precisely sharp energy thresholds for blow‐up and global existence are obtained by analyzing potential well structures for associated functionals.

In this paper, we obtain some basic results of quaternion algorithms and quaternion calculus on time scales. Based on this, a Liouville formula and some related properties are derived for quaternion dynamic equations on time scales through conjugate transposed matrix algorithms. Moreover, we introduce the quaternion matrix exponential function by homogeneous quaternion matrix dynamic equations. Also

In this work, we consider a reaction–diffusion system, modeling the interaction between nutrients, phytoplankton, and zooplankton. Using a semigroup approach in , we prove global existence, uniqueness, and positivity of the solutions. The nonlinearity is handled by providing estimates in , allowing to deal with most of the functional responses that describe predator/prey interactions (Holling I, II

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As

We are concerned with the quantitative study of the electric field perturbation due to the presence of an inhomogeneous conductive rod embedded in a homogenous conductivity. We sharply quantify the dependence of the perturbed electric field on the geometry of the conductive rod. In particular, we accurately characterize the localization of the gradient field (i.e., the electric current) near the boundary

In Part II of this series of papers, we consider an initial‐boundary value problem for the Kolmogorov–Petrovskii–Piscounov (KPP)‐type equation with a discontinuous cut‐off in the reaction function at concentration . For fixed cut‐off value , we apply the method of matched asymptotic coordinate expansions to obtain the complete large‐time asymptotic form of the solution, which exhibits the formation

Wolbachia‐based biocontrol has recently emerged as a potential method for the prevention and control of dengue and other vector‐borne diseases. Major vector species, such as Aedes aegypti females, when deliberately infected with Wolbachia become far less capable of getting infected and transmitting the virus to human individuals. In this paper, we propose and qualitatively analyze a simplified model

A novel modified nonlinear Schrödinger equation is presented. Through a traveling wave ansatz, the equation is solved exactly and analytically. The soliton solution is characterized in terms of waveform and wave speed, and the dependence of these properties upon parameters in the equation is detailed. It is discovered that some parameter settings yield unique waveforms, while others yield degeneracy

The inverse scattering transform for the focusing nonlinear Schrödinger equation is presented for a general class of initial conditions whose asymptotic behavior at infinity consists of counterpropagating waves. The formulation takes into account the branched nature of the two asymptotic eigenvalues of the associated scattering problem. The Jost eigenfunctions and scattering coefficients are defined

A novel complex varying‐parameter finite‐time zeroing neural network (VPFTZNN) for finding a solution to the time‐dependent division problem is introduced. A comparative study in relation to the zeroing neural network (ZNN) and finite‐time zeroing neural network (FTZNN) is established in terms of the error function and the convergence speed. The error graphs of the VPFTZNN design show promising results

We investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed

Using the theory introduced by Casper and Yakimov, we investigate the structure of algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) on , and we derive algebraic and differential relations for these MVOPs. A particular case of importance is that of MVOPs with respect to a matrix weight of the form on the real line, where is a scalar polynomial

Systems exhibiting degeneracies known as exceptional points have remarkable properties with powerful applications, particularly in sensor design. These degeneracies are formed when eigenstates coincide, and the remarkable effects are exaggerated by increasing the order of the exceptional point (i.e., the number of coincident eigenstates). In this work, we use asymptotic techniques to study ‐symmetric

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

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

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

We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of Min and Chen [Math. Methods Appl Sci. 2019;42:301‐321] where a second‐order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system

We consider Kolmogorov‐Petrovskii‐Piscounov (KPP) type models in the presence of a discontinuous cut‐off in reaction rate at concentration . In Part I, we examine permanent form traveling wave solutions (a companion paper, Part II, is devoted to their evolution in the large time limit). For each fixed cut‐off value , we prove the existence of a unique permanent form traveling wave with a continuous

We consider nonclassical symmetries of partial differential equations (PDEs) in dimensions. Given a th‐order ordinary differential equation in the unknown we are able to find the most general scalar PDE of a given order which can be reduced via a nonclassical symmetry to .

We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period 2 in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel is expressed in terms of non‐Hermitian matrix valued orthogonal polynomials (OPs). This model belongs to a class of models for which the existing techniques for studying

We consider the mass‐in‐mass (MiM) lattice when the internal resonators are very small. When there are no internal resonators the lattice reduces to a standard Fermi‐Pasta‐Ulam‐Tsingou (FPUT) system. We show that the solution of the MiM system, with suitable initial data, shadows the FPUT system for long periods of time. Using some classical oscillatory integral estimates we can conclude that the error

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

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