Search Results (44)

Research on surfaces generated by curves plays a central role in linking differential geometry with physical applications, especially following Hasimoto’s transformation and the development of Hasimoto-inspired surface models. In this work, we introduce a new class of such surfaces, referred to as RM Hasimoto surfaces, constructed by employing the rotation-minimizing (RM) Darboux frame along both timelike and spacelike curves in Minkowski 3-space ${\mathbb{E}}_1^3$. In contrast to the classical Hasimoto surfaces defined via the Frenet or standard Darboux frames, the RM approach eliminates torsional difficulties and reduces redundant rotational effects. This leads to more straightforward expressions for the first and second fundamental forms, as well as for the Gaussian and mean curvatures, and facilitates a clear classification of key parameter curves. Furthermore, we establish the associated evolution equations, analyze the resulting geometric invariants, and present explicit examples based on timelike and spacelike generating curves. The findings show that adopting the RM Darboux frame provides greater transparency in Lorentzian surface geometry, yielding sharper characterizations and offering new perspectives on relativistic vortex filaments, magnetic field structures, and soliton behavior. Thus, the RM framework opens a promising direction for both theoretical studies and practical applications of surface geometry in Minkowski space. Full article

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 applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. Full article

The paper advances a new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the indexes, we generate the Darboux–Crum nets of the rational canonical Sturm–Liouville equations (RCSLEs). It is shown that each RCSLE in question has four infinite sequences of quasi-rational solutions (q-RSs) such that their polynomial components from each sequence form a X-Jacobi DPS composed of simple pseudo-Wronskian polynomials (p-WPs). For each p-th order rational Darboux Crum transform of the Jacobi-reference (JRef) CSLE, used as the starting point, we formulate two rational Sturm–Liouville problems (RSLPs) by imposing the Dirichlet boundary conditions on the solutions of the so-called ‘prime’ SLE (p-SLE) at the ends of the intervals (−1, +1) or (+1, ∞). Finally, we demonstrate that the polynomial components of the q-RSs representing the eigenfunctions of these two problems have the form of simple p-WPs composed of p Romanovski–Jacobi (R-Jacobi) polynomials with the same pair of indexes and a single classical Jacobi polynomial, or, accordingly, p classical Jacobi polynomials with the same pair of positive indexes and a single R-Jacobi polynomial. The common, fundamentally important feature of all the simple p-WPs involved is that they do not vanish at the finite singular endpoints—the main reason why they were selected for the current analysis in the first place. The discussion is accompanied by a sketch of the one-dimensional quantum-mechanical problems exactly solvable by the aforementioned infinite and finite EOP sequences. Full article

This paper investigates the (2 + 1)-dimensional nonlocal Hirota–Maxwell–Bloch (NH-MB) system under various types of nonlocality. The mathematical consistency of possible nonlocal structures is analyzed, and three types that lead to a well-posed system are identified. The integrability of the system is established through its Lax pair representation, and a Darboux transformation is constructed. Exact soliton solutions are obtained for both the defocusing and focusing cases. The results obtained may find applications in nonlinear optics, quantum theory, and the theory of integrable systems. Full article

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (−1, +1) and the finite EOP sequences on the positive interval (1, ∞). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1, ∞). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a′) exactly quantized by the EOPs in question. Full article

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 pairs of the equation, we successfully derived the corresponding Darboux transformation and Bäcklund transformation for this equation, which are fundamental to our solving process. Subsequently, we constructed and calculated the recursive operator for this equation, providing an effective approach to tackling complex problems within this domain. These results are crucial for advancing our understanding of the underlying principles of soliton theory and their implications on related natural phenomena. Our findings not only enrich the theoretical framework but also offer practical tools for further research in nonlinear wave dynamics. Full article

Conformal maps preserve angles and maintain the local shape of geometric structures. The osculating curve plays an important role in analyzing the variations in curvature, providing a detailed understanding of the local geometric properties and the impact of conformal transformations on curves and surfaces. In this paper, we study osculating curves on regular surfaces under conformal transformations. We obtained the conditions required for osculating curves on regular surfaces R and $\tilde{R}$ to remain invariant when subjected to a conformal transformation $\psi :R\to \tilde{R}$. The results presented in this paper reveal the specific conditions under which the transformed curve $\tilde{\sigma }=\psi \circ \sigma$ preserves its osculating properties, depending on whether $\tilde{\sigma }$ is a geodesic, asymptotic, or neither. Furthermore, we analyze these conditions separately for cases with zero and non-zero normal curvatures. We also explore the behavior of these curves along the tangent vector $T_{\sigma }$ and the unit normal vector $P_{\sigma }$. Full article

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way. Full article

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 direct method. Next, it presents two finite symmetry transformation group methods—one related to Lax pairs and one independent of them. The fourth section reviews four nonlocal symmetry methods based on conserved forms, conformal invariants, Darboux transformations, and Lax pairs. The final section covers supersymmetry theory and supersymmetric dark equations. Each method is illustrated with examples and references. Full article

Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs of matrix eigenproblems as a foundation, binary Darboux transformations are constructed. These transformations, initiated with a zero seed solution, facilitate the generation of soliton solutions for the SS-type integrable matrix mKdV equations presented. Full article

In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When $x\in \mathbb{R}$ and $t\in \mathbb{R}$, the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and N-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales ($\mathbb{R}$, $\mathbb{Z}$, q-discrete, $\mathbb{C}$). Full article

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 Zakharov equation. The Zakharov equation is instrumental for studying complex nonlinear interactions in these areas, with specific implications for energy transfer processes in plasma and nonlinear wave propagation systems. By analyzing bright–dark rogue wave solutions—phenomena that are critical in understanding high-energy events in optical and fluid environments—this research elucidates the intricate dynamics of energy concentration and dissipation. Using the self-similar transformation method, we map the (2+1)-dimensional equation to a more tractable (1+1)-dimensional nonlinear Schrödinger equation form. Through the Lax pair and Darboux transformation, we successfully construct high-order solutions that reveal how variable coefficients influence rogue wave features, such as shape, amplitude, and dynamics. Numerical simulations demonstrate the evolution of these rogue waves, offering novel perspectives for predicting and mitigating extreme wave events in engineering applications.This paper crucially advances the practical understanding and manipulation of nonlinear wave phenomena in variable environments, providing significant insights for applications in optical fibers, atmospheric physics, and marine engineering. Full article

The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two second-order Darboux–Crum transformations (DCTs) with the seed functions represented by specially chosen pairs of ‘basic’ quasi-rational solutions (q-RSs), i.e., such that their analytical continuations do not have zeros in the complex plane. It is proven that both transformations generally either increase or decrease by 2 the exponent difference (ExpDiff) for the mentioned pole while keeping two other parameters unchanged. The change is more complicated in the latter case if the ExpDiff for the pole of the original CSLE at the origin is smaller than 2. It was observed that the DCTs in question do not preserve bound energy levels according to the conventional supersymmetry (SUSY) rules. To understand this anomaly, we split the DCT in question into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs of our interest. We found that the first Darboux transformation turns the initial CSLE into the Heun equation written in the canonical form while the second transformation brings us back to the canonical form of the hypergeometric equation. It is shown that the first of these transformations necessarily places the mentioned ExpDiff into the limit-circle (LC) range and then the second transformation keeps the pole within the LC region, violating the conventional prescriptions of SUSY quantum mechanics. Full article

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials. Full article

The $4\times 4$ trace-free complex matrix set is introduced in this study. By using it, we are able to create a novel soliton hierarchy that is reduced to demonstrate its bi-Hamiltonian structure. Additionally, we give the Darboux matrix T, whose elements are connected to the spectral parameter in accordance with the various positions and numbers of the spectral parameter $\lambda$. The Darboux transformation approach has also been successfully applicated to superintegrable systems. Full article