60 Results Found

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé tru...

In this paper, we describe the global dynamics of the Painlevé–Gambier equations numbered XVIII: $x^{″}-{\left(x^{\prime }\right)}^2/\left(2x\right)-4x^2=0$, XXI: $x^{″}-3{\left(x^{\prime }\right)}^2/\left(4x\right)-3x^2$, and XXII: $x^{″}-3{\left(x^{\prime }\right)}^2/\left(4x\right)+1=0$...

Broer and Kaup (BK) equations that include both spatial and temporal variables were independently introduced in the early 1970s. Renowned for describing the propagation of bidirectional long waves in shallow water, the BK equations are an important c...

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group ${\widehat{A}}_{N-1}^{\left(1\right)}$ contains the conventional Bäcklund tran...

This article investigates the widespread occurrence of wave-like behavior in biological systems and reflects on the importance of reaction–diffusion frameworks to explain fast information transfer. Focusing on the classical Fisher–Kolmogo...

Painlevé transcendents are usually considered as complex functions of a complex variable, but, in applications, it is often the real cases that are of interest. We propose a classification of the real solutions of Painlevé IV (or, more...

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of appl...

The (2+1)-D generalized B-type Kadomtsev–Petviashvili (BKP) equation is studied in this work utilizing Painlevé property, Lie-symmetry method and generalized Kudryashov method (GKM). This study aims to pass the Painlevé test and o...

The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of...

The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper, we argue against the commonly held belief that the symmetry group of a given equation is solely determined by i...

Asymptotic analysis is a branch of mathematical analysis that describes the limiting behavior of the function. This behavior appears when we study the solution of differential equations analytically. The recent work deals with a special class of thir...

We demonstrate the way to derive the second Painlevé equation $P_2$ and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transf...

This research focuses on the following equation: $(\overline{g}g-1)(g\underline{g}-1)+B\left(z\right){\displaystyle \frac{g^{\prime }\left(z\right)}{g\left(z\right)}}=R(z,g)={\displaystyle \frac{P(z,g)}{Q(z,g)}},$ where $P(z,g)$ and $Q(z,g)$ represent polynomials in g, possessing rational coefficients in z and lacking shared roots, $B\left(z\right)$ is...

In this paper, we investigate the long-term asymptotic behavior of solutions to the Cauchy problem for the n-component coupled higher-order nonlinear Schrödinger (nC-HNLS) equation on a line with decaying initial data. Through the application of...

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable...

We apply a singularity analysis to investigate the integrability properties of the gravitational field equations in Weyl Integrable Spacetime for a spatially flat Friedmann–Lemaître–Robertson–Walker background spacetime induce...

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and sig...

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé an...

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under $A_{N-1}^{\left(1\right)}$ symmetry. This formalism...

The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equat...

We consider a generalization of the modified Korteweg–de Vries–Burgers equation with a nonlinear source. Using the Painlevé test for partial differential equation with the Kruskal variable, we show that the corresponding Cauchy pro...

The $(2+1)$-dimensional Boussinesq equation plays an important role in mathematical physics. In this paper, we investigate some exact solutions of the $(2+1)$-dimensional variable-coefficient Boussinesq equation. Firstly, the Painlevé analysis is...

In this paper, the symmetries and conservation laws of a variable-coefficient generalized Calogero–Bogoyavlenskii–Schiff (vcGCBS) equation are investigated by modeling the propagation of long waves in nonlinear optics, fluid dynamics, and...

This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient an...

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of t...

A geometric-analytic approach to studying invariants of solvable nonlinear ordinary differential equations is developed. In particular, there is described in detail a general scheme of constructing solvable nonlinear ordinary differential equations,...

In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Lagu...

Using the coefficients of a system semilinear cubic in the first derivative second order differential equations one defines a connection in the space of the independent and dependent variables, which is specified modulo two free parameters. In this w...

In this investigation, a novel (3+1)-dimensional Lax integrable Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation is constructed and analyzed analytically. The Painlevé integrability for the mentioned model is examin...

In this paper, the (2+1)-dimensional q-deformed Sinh-Gordon model has been investigated via $(\frac{G^{{}^{\prime }}}{G},\frac{1}{G})$-expansion and Sine-Gordon-expansion methods. These techniques successfully retrieve trigonometric as well as hyperbolic solutions, along necess...

Recent work on the evaluation of large language models emphasizes that the relevant unit of intelligence is not the artificial system alone but the human–AI hybrid. In parallel, topological and dynamical models of cognition based on Painlev&eac...

This paper proposes that the origin of life may be understood as a cascade of dynamical integration events governed by Painlevé transcendental equations—a class of nonlinear differential equations arising widely in physics, from quantum...

In this paper, the Gilson–Pickering (GP) equation with applications for wave propagation in plasma physics and crystal lattice theory is studied. The model with wave propagation in plasma physics and crystal lattice theory is explained. A colle...

We propose a novel mathematical framework for understanding consciousness as a dynamical phenomenon governed by nonlinear integrable equations. The central hypothesis identifies conscious state dynamics with the Painlevé VI equation and its co...

This paper presents a detailed study of the $(3+1)$-dimensional Zakharov–Kuznetsov–Burgers equation to investigate shock-wave phenomena in dusty plasmas with quantum effects. The model provides significant physical insight into nonlinear di...

In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix e...

We investigate the onset of hyperbolicity in Jensen polynomials $J_{d,n}$ associated with the Riemann $\mathsf{\Xi }$-function and identify a robust parity-driven bifurcation with a natural geometric interpretation. Numerical analysis for degrees $5\le d\le 16$ reveal...

Symmetry governs complex systems from particle physics to biology. We demonstrate that consciousness dynamics follow symmetry-breaking cascades described by Painlevé confluence topology, bridging quantum topology, neuroscience, and consciousne...

In this paper, a variable-coefficient Davey–Stewartson (vcDS) system is investigated for modeling the evolution of a two-dimensional wave-packet on water of finite depth in inhomogeneous media or nonuniform boundaries, which is where its novelt...

This paper focuses on the variable coefficients geophysical KdV (VCGKdV) equation, which involves time-dependent perturbation, nonlinearity and dispersion parameters. It is a more realistic model than its constant coefficient counterpart and can be u...

The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary...

This study investigates the solitary wave solutions of the nonlinear fractional Jimbo–Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interes...

We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this...

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predat...

The present work strives to explore some qualitative analysis for the governing equation describing the dynamic response of a piezoelectro-magnetic circular rod. As a result of the integrability study of the governed equation, which furnishes valuabl...

This work investigates the dynamical properties of the Kolmogorov–Petrovskii–Piskunov (KPP) equation. We begin by establishing its non-integrability through the Painlevé test. Using a traveling wave transformation, we reduce the eq...

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sens...

Via constructing an explicit Lagrangian for which the perturbation equations are analogs of a scalar field propagating in a planar black-hole space–time, it is found that all planar black holes conformal to a Painlevé–Gullstrand-ty...

The classical numerical methods for differential equations are a well-studied field. Nevertheless, these numerical methods are limited in their scope to certain classes of equations. Modern machine learning applications, such as equation discovery, m...

We study the Hankel determinant generated by a singularly perturbed Jacobi weight $w(x,s):={(1-x)}^{\alpha }{(1+x)}^{\beta }{\mathrm{e}}^{-\frac{s}{1-x}},x\in [-1,1],\alpha >0,\beta >0s\ge 0.$ If $s=0$, it is reduced to the classical Jacobi weight. For $s\&$...