The direct linearization framework is presented for the two-dimensional (2D) Toda equations associated with the infinite-dimensional Lie algebras , , and , as well as the Kac–Moody algebras , , , and for arbitrary integers , from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure

Elaborate regulatory feedback processes are thought to make biological development robust, that is, resistant to changes induced by sustained genetic or environmental perturbations. How this might be accomplished is still not completely understood, especially when models of some known feedback processes have been shown, some analytically and others by numerical simulations, to be ineffective in promoting

Ocean swell plays an important role in the transport of energy across the ocean, yet its evolution is not well understood. In the late 1960s, the nonlinear Schrödinger (NLS) equation was derived as a simplified model for the propagation of ocean swell over large distances. More recently, a number of dissipative generalizations of the NLS equation based on a simple dissipation assumption have been proposed

We construct a broad class of solutions of the Kadomtsev–Petviashvili (KP)-I equation by using a reduced version of the Grammian form of the -function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP-I analogues of the family of line-soliton

Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on , which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends

The problem of two-dimensional capillary-gravity waves on an inviscid fluid of finite depth interacting with a linear shear current is considered. The shear current breaks the symmetry of the irrotational problem and supports simultaneously counter-propagating waves of different types: Korteweg de-Vries (KdV)-type long solitary waves and wave-packet solitary waves whose envelopes are associated with

We develop a time and space-dependent predator—prey model. The predators' equation is a nonlocal hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators “hunt” for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey–parasites. The general well-posedness and stability results here obtained

The main object of this paper is to study the essential spectrum of a Hamiltonian system of arbitrary order with one singular endpoint under a class of perturbations. We first present a characterization of the essential spectrum in terms of singular sequences and then give the concept of perturbations small at singular endpoints of Hamiltonian systems. Based on the above characterization, the invariance

In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek–Jacobi type weight By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient and the sub-leading coefficient of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of can be expressed in terms

A rough nearly circular (and/or circular) plate submerged in some ice-covered ocean is undergoing a heaving motion. This radiation problem is reduced to a hypersingular integral equation over the surface of the rough plate. Perturbation method further simplifies the governing equation to get a system of hypersingular integral equations. The geometry of nearly circular plate has been conformally mapped

In this paper, we consider a problem inspired by the real-world need to identify the topographical features of ocean basins. Specifically, we consider the problem of estimating the bottom impermeable boundary to an inviscid, incompressible, irrotational fluid from measurements of the free-surface deviation alone, within the context of dispersive shallow-water wave models. The need to consider the shallow-water

This present work is based on developing a deterministic discrete formulation for the approximation of a multidimensional Smoluchowski (Coalescence) equation on a nonuniform grid. The mathematical formulation of the proposed method is simpler, easy to implement, and focuses on conserving the first-order moment. The new scheme resolved the issue of mass conservation along individual components in contrast

Resonant collisions among localized lumps and line solitons of the Kadomtsev–Petviashvili I (KP-I) equation are studied. The KP-I equation describes the evolution of weakly nonlinear, weakly dispersive waves with slow transverse variations. Lumps can only exist for the KP equation when the signs of the transverse derivative and the weak dispersion in the propagation direction are different, that is

Every underdetermined system of partial differential equations arising from a variational principle admits an infinite hierarchy of higher-order generalized symmetries. These symmetries are a consequence of the Noether dependencies among the Euler–Lagrange equations that follow from Noether's Second Theorem. This result is a consequence of a more general theorem on the existence of higher-order generalized

We consider a system of partial differential equations modeling tumors. The system under consideration describes the spatial dynamics of the tumor cells, extracellular matrix, and matrix degrading enzymes. We first carry out a complete group classification of the Lie point symmetries of this model. Next, we use symmetry techniques to construct invariant solutions for it. In addition, we consider a

In this paper, the -symmetric version of the Maccari system is introduced, which can be regarded as a two-dimensional generalization of the defocusing nonlocal nonlinear Schrödinger equation. Various exact solutions of the nonlocal Maccari system are obtained by means of the Hirota bilinear method, long-wave limit, and Kadomtsev–Petviashvili (KP) hierarchy method. Bilinear forms of the nonlocal Maccari

Parabolic cylinder functions are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratio and the double ratio in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for

In this paper, we study stationarity of an age-structured human immunodeficiency virus (HIV) model with a delay and driven by the Ornstein–Uhlenbeck process. The HIV model involves cell-to-virus infection and cell-to-cell transmission, and the two transmission modes have a short intracellular latency described by the delay. As the death rate and virus production rate of infected cells have an explicit

We present and analyze a novel manifestation of the revival phenomenon for linear spatially periodic evolution equations , in the concrete case of three nonlocal equations that arise in water wave theory and are defined by convolution kernels. Revival in these cases is manifested in the form of dispersively quantized cusped solutions at rational times. We give an analytic description of this phenomenon

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation , with a step-like boundary values: as and as for all , where is a constant. In a recent paper, we presented the long-time asymptotics of the solution of this problem along the rays , where is a constant. In the present paper, we extend the asymptotics into a region that is asymptotically closer to the ray than

We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex-valued weight function, , over the interval , where is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics

This paper investigates the infinite time blow-up of solutions with arbitrary high initial energy to wave equations with weak damping term, strong damping term, and logarithmic nonlinearity. This problem has been studied previously with the assumptions that there is no strong damping term and the initial displacement and initial velocity have the same sign. However, from the physical point of view

In this work, we investigate the existence of nonmonotone traveling wave solutions to a reaction-diffusion system modeling social outbursts, such as rioting activity, originally proposed in Berestycki et al (Netw Heterog Media. 2015;10(3):443–475). The model consists of two scalar values, the level of unrest and a tension field . A key component of the model is a bandwagon effect in the unrest, provided

At the time when this paper was written, quarantine-related strategies (from full lockdown to some relaxed preventive measures) were the only available measure to control coronavirus disease 2019 (COVID-19) epidemic. However, long-term quarantine and especially full lockdown is an extremely expensive measure. To explore the possibility of controlling and suppressing the COVID-19 epidemic at the lowest

In this paper, squared eigenfunction symmetry of the differential-difference modified Kadomtsev–Petviashvili (DmKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the DmKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R-Toda) hierarchy. An invertible transformation is given to connect the positive and negative R-Toda

The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity of the Korteweg–de Vries equation and the full linear dispersion relation of unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear

The transonic flow problem is an important one in fluid dynamics or gas dynamics. Local solvability of the mixed problem of Chaplygin's hodograph equation in hyperbolic bounded domains near the degenerate line is considered by using abc method.

We study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi‐linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken

Two‐dimensional materials such as graphene have become crucial components of most state‐of‐the‐art plasmonic devices. The possibility of not only generating plasmons in the terahertz regime, but also tuning them in real time via chemical doping or electrical gating make them compelling materials for engineers seeking to build accurate sensors. Thus, the faithful modeling of the propagation of linear

In this paper, a fractional‐order nonlinear reaction diffusion system is proposed to remove the multiplicative Gamma noise. The new reaction diffusion system consists of three equations: the regularized Perona and Malik (PM) equation , which is used for presmoothing the image that is contaminated by noise; the time‐delay regularization equation, which is used for incorporating the past information

A hybrid asymptotic‐numerical theory is developed to analyze the effect of different types of localized heterogeneities on the existence, linear stability, and slow dynamics of localized spot patterns for the two‐component Schnakenberg reaction‐diffusion model in a 2‐D domain. Two distinct types of localized heterogeneities are considered: a strong localized perturbation of a spatially uniform feed

In this paper, we consider the monodromy and, in particular, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. The isomonodromy sets of accessory parameters

We show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators on have a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog

We propose a random batch method (RBM) for a contractive interacting particle system on a network, which can be formulated as a first‐order consensus model with heterogeneous intrinsic dynamics and convolution‐type consensus interactions. The RBM was proposed and analyzed recently in a series of work by the third author and his collaborators for a general interacting particle system with a conservative

We consider a spatial (line) model for invasion of a population by a single mutant with a stochastically selectively neutral fitness landscape, independent from the fitness landscape for nonmutants. This model is similar to those considered earlier. We show that the probability of mutant fixation in a population of size , starting from a single mutant, is greater than , which would be the case if there

The article mainly studies the isentropic approximation of the compressible Euler equations in Besov space , provided the initial entropy changes closing a constant in the Besov spaces, which extends and improves the result in Sobolev space by Jia and Pan.

Green's functions to linear second-order ordinary differential equations with general separated boundary conditions (BCs) are derived, where the parameters used in the BCs are allowed to take negative values. Previous results only considered the nonnegative parameters and such separated BCs arise in the usual Thomas–Fermi BCs. Properties of the Green's functions with possibly negative parameters are

In this work, the dispersive shock wave (DSW) solution of a Boussinesq Benjamin–Ono (BBO) equation, the standard Boussinesq equation with dispersion replaced by nonlocal Benjamin–Ono dispersion, is derived. This DSW solution is derived using two methods, DSW fitting and from a simple wave solution of the Whitham modulation equations for the BBO equation. The first of these yields the two edges of the

We consider a model for the flow of two immiscible fluids in a two-dimensional thin strip of varying width. This represents an idealization of a pore in a porous medium. The interface separating the fluids forms a freely moving interface in contact with the wall and is driven by the fluid flow and surface tension. The contact-line model incorporates Navier-slip boundary conditions and a dynamic and

This study is motivated by several engineering papers that describe the so-called acoustic black hole; for example, a beam with a monotonically decreasing thickness toward one of the endpoints. It appears that the time of propagation of a signal toward such an endpoint is infinite so that the signal is “trapped.” Also, the amplitude of a signal increases with no bound near this point. The main objective

The existence of soliton families in nonparity-time-symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one- and two-dimensional nonlinear Schrödinger equations with localized Wadati-type nonparity-time-symmetric complex potentials. By utilizing the conservation

Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified KdV (Korteweg–de Vries) equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of the -method

For the predator–prey model with the Sigmoid functional response, the known result is on the global stability of its positive equilibrium when it is locally stable. Here, we characterize existence of particular type of limit cycles using qualitative theory and geometric singular perturbation methods. The main results are as follows. If the positive equilibrium exists and is a weak focus, it is a stable

We consider the mapping properties of the integral operator arising in nonlocal slender body theory (SBT) for the model geometry of a straight, periodic filament. It is well known that the classical singular SBT integral operator suffers from high wavenumber instabilities, making it unsuitable for approximating the slender body inverse problem, where the fiber velocity is prescribed and the integral

In this paper, based on the Hirota and Maxwell–Bloch (H-MB) system and its application in the theory of the femtosecond pulse propagation through an erbium doped fiber, we define two kinds of nonlocal Hirota and Maxwell–Bloch (NH-MB) systems, namely, -symmetric NH-MB system and reverse space-time NH-MB system. Then, we construct the Darboux transformations of these NH-MB systems. Meanwhile, we derive

First, we show that the system consisting of integer-order partial differential equations (PDEs) and time-fractional PDEs with the Riemann–Liouville fractional derivative has an elegant local symmetry structure. Then with the symmetry structure we consider two particular cases where one is the pure time-fractional PDEs whose symmetry invariant condition is divided into two parts of integer-order and

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

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

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

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.

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 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

We study a class of nonlinear fractional difference equations with nonlocal boundary conditions at resonance. The system is inspired by the three‐point boundary value problem for differential equations that have been extensively studied. It is also an extension to a fractional difference equation arising from real‐world applications. Converting the problem to an equivalent system corresponding to the

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

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 consider the operator subject to the Dirichlet or Robin condition, where a domain is bounded or unbounded. The symbol stands for a second‐order self‐adjoint differential operator on such that the spectrum of the operator contains several discrete eigenvalues , . These eigenvalues are thresholds in the essential spectrum of the operator . We study how these thresholds bifurcate once we add a small

We equip a knot $K$ with a set of colored bonds, that is, colored intervals properly embedded into $\mathbb{R}^3 \setminus K$. 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

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