This paper investigates a four-dimensional predator-prey model with cross-species infection and Holling type II functional response. The model incorporates logistic growth for susceptible prey, susceptible-infected (SI)-type disease transmission with mass-action incidence, and a biologically realistic mechanism by which predators become infected through consumption of infected prey. We derive five key ecological and epidemiological threshold parameters governing predator persistence and disease invasion in different ecological scenarios. Analytical results establish positivity, boundedness, and conditions for the existence, feasibility, and stability of all equilibria. These include disease-free coexistence, endemic prey-only, and full endemic coexistence states. Global stability of the disease-free coexistence equilibrium is obtained using the Lyapunov method. Bifurcation analyses reveal transcritical, Hopf, and saddle-node bifurcations, explaining the transitions between extinction, stable coexistence, oscillatory dynamics, and bistability. Co-dimension-two analysis identifies organizing centers that structure the parameter space and clarifies mechanisms underlying regime shifts. Numerical simulations using MATLAB and MatCont confirm theoretical findings and illustrate diverse dynamical behaviors. These behaviors include predator extinction driven by highly infectious diseases, predation-mediated disease control, sustained oscillations, and multiplicity. Spatial extension via a reaction-diffusion framework demonstrates diffusion-driven Turing instability and pattern formation. The results provide integrated ecological and epidemiological insights into cross-infection dynamics and predator-prey coexistence.

We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space E13. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior characteristic of RM frames, yielding a natural geometric description of curves in a Lorentzian environment. For unit speed non-null curves, the governing equations of the RM Darboux frame are derived, and precise connections between the RM curvature functions and the classical Frenet and Darboux invariants are obtained, thereby elucidating the geometric significance of RM curvatures in Lorentzian geometry. Within this setting, multiple classes of ruled surfaces are generated using RM Darboux frame vector fields. Necessary and sufficient conditions for developability, minimality, and flatness are formulated exclusively in terms of RM curvature quantities. The role of the causal character of the generating curve is analyzed in detail, revealing distinct geometric behaviors for space-like and time-like cases. These findings indicate that the RM Darboux framework constitutes a flexible and effective approach for modeling curve-induced surface geometries in Minkowski space, with potential relevance to relativistic kinematics, world sheet constructions, and geometric problems arising in mathematical physics.

This paper introduces a new class of surface constrained curves, termed generalized Bhelices, in the Euclidean three-space E3. The analysis is developed through a rotational modification of the classical Darboux frame, called the B-Darboux frame {T, chi 1, chi 2}. This frame combines the effects of geodesic and normal curvatures into two coupled invariants, referred to as the B-Darboux curvatures, providing a unified description of curve geometry on regular surfaces. Within this framework, three types of generalized B-helices are defined, and their characterization conditions are expressed in terms of these curvature functions. The spherical representations corresponding to the B-Darboux frame vectors are also examined, and explicit relations for their curvature and torsion are derived. It is shown that each type of generalized B-helix generates a circular indicatrix on the unit sphere, establishing a direct geometric link between spatial and spherical curves. To validate the theoretical developments, illustrative examples and graphical simulations of ruled surfaces associated with the frame vectors are presented, confirming the applicability and effectiveness of the proposed model.

This study presents a new geometric formulation for constructing embankment-type ruled surfaces in the Lorentzian framework of Minkowski 3-space E31, by means of the orthogonal modified frame (OMF). The OMF serves as an orthogonal moving frame that is fully compatible with the Minkowski metric, providing a consistent representation of the causal behavior of both spacelike and timelike curves. Within this framework, three distinct surface families are established namely, the OMF embankment surface, the OMF embankment-like surface, and the OMF tubembankment- like surface. Each class is developed together with its explicit parametric expression and associated geometric invariants. The corresponding first and second fundamental forms are derived, from which closed analytical expressions for the Gaussian curvature and mean curvature are obtained. These computations lead to precise differential conditions governing the developability and minimality of the generated surfaces. Representative examples illustrate the smooth curvature distribution and the Lorentzian metric consistency of the OMF-based models, demonstrating clear advantages over traditional Euclidean constructions. Overall, the proposed formulation offers an efficient and unified framework for analyzing and modeling ruled surfaces in Lorentzian geometry, with prospective applications in relativistic motion theory, geometric design, and computer-aided kinematic simulation.

This study investigates the geometry of osculating type-2 ruled surfaces in Minkowski 3-space E13, formulated through the Type-2 Bishop frame associated with a spacelike curve whose principal normal is timelike and binormal is spacelike. Using the hyperbolic transformation linking the Frenet–Serret and Bishop frames, we analyze how the Bishop curvatures ζ1 and ζ2 affect the geometric behavior and formation of such surfaces. Explicit criteria are derived for cylindrical, developable, and minimal configurations, together with analytical expressions for Gaussian and mean curvatures. We also determine the conditions under which the base curve behaves as a geodesic, asymptotic line, or line of curvature. Several illustrative examples in Minkowski 3-space are provided to visualize the geometric influence of ζ1 and ζ2 on flatness, minimality, and developability. Overall, the Type-2 Bishop frame offers a smooth and effective framework for characterizing Lorentzian geometry and symmetry of osculating ruled surfaces, extending classical Euclidean results to the Minkowski setting.

We study the perturbed and modified nonlinear Schrödinger equation (PNLSE with Kerr nonlinearity and MNLSE), establish their integrability under explicit parameter conditions via the Painlevé test, and construct analytical traveling-wave families. Methodologically, we (i) prove integrability through a complete dominant-balance and resonance analysis, (ii) derive closed-form polynomial/elliptic solutions with the Unified Method (UM) for both models, and (iii) obtain periodic solutions using the Energy Balance Method (EBM). The novelty is the combined, comparative use of UM and EBM on these two optical models, with parameter maps linking coefficients to solution classes (solitary, soliton/dromion, and elliptic/periodic profiles). We also interpret the physical relevance for nonlinear optics (optical pulse propagation with Kerr response, self-steepening) and compare our exact profiles with prior solution families. These results provide verifiable benchmarks for NLSE-type dynamics and broaden the toolbox for modeling optical solitons and periodic waves in Kerr media.

We study stochastic variants of the Kairat-II and Kairat-X equations in (3 + 1) dimensions, two canonical models in soliton theory. Random fluctuations are incorporated through a Wiener process, yielding a multiplicative stochastic embedding of the wave fields. By combining the enhanced direct algebraic technique with the new projective Riccati equation approach, we obtain closed-form stochastic soliton solutions and analyze how noise modulates their amplitude and localization. The solutions are illustrated with consistent 3D surface plots (mean field vs. sample paths) and 2D time traces to highlight wave geometry and variability. In addition, we employ the energy balance approach to separate kinetic and potential contributions and to verify an energy balance relation for the derived solutions, thereby clarifying their physical plausibility and stability under noise. The results provide exact, easily verifiable benchmarks for stochastic nonlinear wave models and a practical template for incorporating randomness into nonlinear dispersive systems.

In this study, We explore for Minkowski 3-space E13 harmonic surfaces’ geometric features by employing a common tangent vector field along a curve situated on the surface. Our analysis is grounded in the rotation minimizing (RM) Darboux frame, which offers a robust alternative to the classical Frenet frame particularly valuable in the Lorentzian setting, where singularities frequently arise. The RM Darboux frame, tailored to curves lying on surfaces, enables the expression of fundamental invariants such as geodesic curvature, normal curvature, and geodesic torsion. We derive specific conditions that characterize harmonic surfaces based on these invariants. We also clarify the connection between the components of the RM Darboux frame and thesurface’s mean curvature vector. This formulation provides fresh perspectives on the classification and intrinsic structure of harmonic surfaces within Minkowski geometry. To support our findings, we present several illustrative examples that demonstrate the applicability and strength of the RM Darboux approach in Lorentzian differential geometry.

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 E13. 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.

This paper gives explicit positive solutions for several higher-order and higher-dimensional systems of rational difference equations (RDEs). We study how solutions behave over time for a wide range of initial data in (0,∞ ) , and we prove when solutions stay positive and bounded. We also give clear conditions under which solutions approach fixed points or settle into periodic cycles. To support the theory, we include numerical examples that show how the system order and the initial values shape the long-term dynamics. RDEs form one of the most important classes of nonlinear discrete systems because they naturally appear in population dynamics, epidemiology, economic growth models, and control processes. They are also known to generate rich dynamical behavior, including oscillations and chaos, which makes their qualitative study both challenging and useful. The novelty of our work is that we move beyond the mostly studied low-order cases and provide explicit closed-form solutions and stability results for higher-order, multi-dimensional systems, where such results are rare. The significance is that these formulas make analysis and simulation straightforward, and they offer practical tools for models in areas such as mathematical biology, economics, and control. In addition, our results highlight how higher-order interactions can create new stability patterns and periodic behaviors that do not appear in simpler systems. These findings not only strengthen the theoretical framework of nonlinear discrete-time systems but also extend the range of applications where rational difference equations can be effectively used in practice.

Using a common tangent vector field to a surface along a curve, in this study we discussed a new Darboux frame that we referred to as the rotation-minimizing Darboux frame (RMDF) in Minkowski 3-space. The parametric equation resulting from the RMDF frame for an imbricate-ruled surface was then provided. As a result, minimal (or maximal for timelike surfaces) ruled surfaces were derived, along with the necessary and sufficient criteria for imbricate-ruled surfaces to be developable. The surfaces also described the parameter curves of these surfaces' asymptotic, geodesic, and curvature lines. We also gave an example to emphasize the most significant results.

Using a common tangent vector field to a surface along a curve, in this study we discussed a new Darboux frame that we referred to as the rotation -minimizing Darboux frame (RMDF) in Minkowski 3 -space. The parametric equation resulting from the RMDF frame for an imbricate -ruled surface was then provided. As a result, minimal (or maximal for timelike surfaces) ruled surfaces were derived, along with the necessary and sufficient criteria for imbricate -ruled surfaces to be developable. The surfaces also described the parameter curves of these surfaces' asymptotic, geodesic, and curvature lines. We also gave an example to emphasize the most significant results.

We investigate the equiform-Bishop Hasimoto surfaces ℘(\sigma, t) in Minkowski 3-space in this paper. In E^3_1, in different three cases, the geometric properties of equiform-Bishop Hasimoto surfaces are discussed. For each case, the equiformBishop Gaussian and mean curvatures of the equiform-Bishop Hasimoto surface are determined. Then, in E^3_1, we characterize the parameter equiform Bishop curves of equiform-Bishop Hasimoto surfaces.

In this paper, using the Hirota bilinear method, we investigate the perturbed Gerdjikov–Ivanov equation (PGIE) for lump soliton, rogue wave, periodic and kink cross rational solutions, homoclinic breathers, M-shaped and many other interactions. All these solutions have so many real life applications for example lump wave help to study the behavior of vortices and eddies in the ocean, which are used for climate modeling and fisheries management. Rogue waves are used in maritime safety, M-shaped solitons are used in optical fiber communications to transmit data over long distances without distortion, improving the efficiency and reliability of high-speed data transmission. Breathers can be used in fiber optic systems to generate ultrafast optical pulses, which have applications in telecommunications and laser technology. We also discuss the dynamical behaviour of our solutions in various dimensions.

In this paper, we introduce a normal spacelike developable surface that is normal to a surface Ω along a spacelike curve α in Minkowski 3-space 31 .We study the existence and singularities of normal spacelike developable surface through two invariants of the spacelike curves on a surface.Furthermore, we will be interested in the case when the spacelike curve is a geodesic curve and when it lies on a surface of revolution.

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.

The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces.

In this paper, some geometric properties of equiform Smarandache ruled surfaces in Minkowski space E13 using an equiform frame are investigated. Also, we give the sufficient conditions that make these surfaces are equiform developable and equiform minimal related to the equiform curvatures and when the equiform base curve contained in a plane or general helix. Finally, we provide an example, such as these surfaces.

This paper seeks to introduce the wave structures and dynamics properties of the perturbed Chen–Lee–Liu equation (PCLLE). Using the traveling wave transformation, we derive the corresponding traveling wave system from the original equation and construct a conserved quantity named as Hamiltonian. Subsequently, we establish periodic solutions and the existence of soliton using the bifurcation method. The bifurcation method is a mathematical technique used to study how the qualitative behavior of a system changes as one or more parameters of the system are varied. It involves analyzing the system’s equilibrium points and studying how they behave as the parameters are changed. Finally, we construct the exact traveling wave solutions using the complete discriminant system (CDS) of polynomial method (CDSPM) to explicitly validate our findings.