A high-order single-step implicit method, the Legendre–Darboux Method of order six (LDM6), is introduced for solving both linear and nonlinear initial value problems. Unlike classical Taylor expansions, LDM6 systematically constructs higher-ord...
In this paper, we introduce a modification of the Singular Manifold Method in order to derive the associated spectral problem for a generalization of the complex version of the modified Korteweg–de Vries equation. This modification yields the r...
This paper focuses on investigating the bifurcation of limit cycles and centers within a specific class of three-dimensional cubic systems possessing
The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quan...
A vector modified Yajima–Oikawa long-wave–short-wave equation is proposed using the zero-curvature presentation. On the basis of the Riccati equations associated with the Lax pair, a method is developed to construct multi-fold classical a...
This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie alge...
Willmore defined embedded surfaces on
With a specific Darboux transformation, we construct solutions to the sine-Gordon equation. We use both the simple Darboux transformation as well as the multiple Darboux transformation, which enables the obtainment of compact solutions of this equati...
Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving...
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...
In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear Schrödinger e...
In this paper, the linearizability of a 2:−3 resonant system with quadratic nonlinearities is studied. We provide a list of the conditions for this family of systems having a linearizable center. The conditions for linearizablity are obtained by comp...
In this paper, we show that the standard techniques that are utilized to study the classical-like properties of the pure states for Hermitian systems can be adjusted to investigate the classicality of pure states for non-Hermitian systems. The method...
We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In eac...
Solutions to the Davey–Stewartson equations (DS) in terms of the determinants of order
This paper conducts an in-depth study on the self-similar transformation, Darboux transformation, and the excitation and propagation characteristics of high-order bright–dark rogue wave solutions in the (2+1)-dimensional variable-coefficient Za...
This review begins with the standard Lie symmetry theory for nonlinear PDEs and explores extensions of symmetry analysis. First, it introduces three key symmetry reduction methods: the classical symmetry method, conditional symmetries, and the CK dir...
This paper explores the third-order nonlinear coupled KdV equation utilizing prolongation structure theory and gauge transformation. By applying the prolongation structure method, we obtained an extended version of the equation. Starting from the Lax...
This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method...
The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of...
Nonlinear evolution equations play a key role in modeling various physical processes, such as wave propagation in nonlinear optical and hydrodynamic media, as well as in the dynamics of plasma and quantum systems. In this paper, we study an integrabl...
Three-dimensional (3D) reconstruction is an essential technique to visualize and monitor the growth of agricultural and forestry plants. However, inspecting tall plants (trees) remains a challenging task for single-camera systems. A combination of lo...
Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili
This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form
Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie...
Accurate delineation of stream networks in low-gradient wetlands remains challenging due to subtle topographic variation and dense vegetation cover. This study systematically evaluated 48 Unmanned Aerial Vehicle Light Detection and Ranging (UAV-LiDAR...
The formation of rogue oceanic waves may be the result of different causes. Various factors (winds, currents, dispersive focussing, depth, nonlinear focussing and instability) make this subject intriguing, and yet its understanding is quite relevant...