003. Факультет інформатики
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Browsing 003. Факультет інформатики by Author "Avramenko, Olha"
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Item Benjamin-Feir Instability of Interfacial Gravity-Capillary Waves in a Two-Layer Fluid. Part I(2025) Avramenko, Olha; Naradovyi, VolodymyrThis study presents a detailed investigation of the modulational stability of interfacialwave packets in a two-layer inviscid incompressible fluid with finite layer thicknesses and interfacial surface tension. The stability analysis is carried out for a broad range of density ratios and geometric configurations, enabling the construction of stability diagrams in the (𝜌, 𝑘)-plane, where 𝜌 is the density ratio and 𝑘 is the carrier wavenumber. The Benjamin-–Feir index is used as the stability criterion, and its interplay with the curvature of the dispersion relation is examined to determine the onset of modulational instability. The topology of the stability diagrams reveals several characteristic structures: a localized loop of stability within an instability zone, a global upper stability domain, an elongated corridor bounded by resonance and dispersion curves, and a degenerate cut structure arising in strongly asymmetric configurations. Each of these structures is associated with a distinct physical mechanism involving the balance between focusing/defocusing nonlinearity and anomalous/normal dispersion. Systematic variation of layer thicknesses allows us to track the formation, deformation, and disappearance of these regions, as well as their merging or segmentation due to resonance effects. Limiting cases of semi-infinite layers are analyzed to connect the results with known configurations, including the "half-space–layer", "layer–half-space’" and "half-space–half-space" systems. The influence of symmetry and asymmetry in layer geometry is examined in detail, showing how it governs the arrangement and connectivity of stable and unstable regions in parameter space. The results provide a unified framework for interpreting modulational stability in layered fluids with interfacial tension, highlighting both global dispersion-controlled regimes and localized stability islands. This work constitutes Part I of the study; Part II will address the role of varying surface tension, which is expected to deform existing stability domains and modify the associated nonlinear–dispersive mechanisms.Item Benjamin-Feir Instability of Interfacial Gravity-Capillary Waves in a Two-Layer Fluid. Part II. Surface-Tension Effects(2026) Avramenko, Olha; Naradovyi, VolodymyrThis second part of the study develops a complete geometric and asymptotic description of how surface tension governs the modulationalstability of interfacial waves in a two-layer fluid. Extending the analytical framework of Part I, surface tension is treated as a freelyadjustable parameter, making it possible to trace the nonlinear and dispersive properties of the system across the full range of depthratios and density contrasts. Using the nonlinear Schr ̈odinger reduction together with long-wave asymptotics, the mechanisms thatshape the boundaries between stable and unstable regimes are identified and their dependence on surface tension is quantified. Thelong-wave structure is controlled by two special density values that mark the bases of the loop and the corridor on the stability diagrams.Their ordering switches at a threshold that exists only when the lower layer is deeper, and loop-type structures occur only in thisregime. A second organising parameter is the classical Bond threshold, at which the dispersive and nonlinear singularities coincide.When surface tension exceeds this value and the upper layer is sufficiently deep, the interaction between resonant and dispersive effectsproduces a capillary cut that replaces the corridor and characterises strongly capillary, upper-layer-dominated configurations. To unifythese observations, the full three-dimensional critical surfaces that separate different types of nonlinear and dispersive behaviour arecomputed. The familiar loop, corridor, and cut appear as planar sections of these surfaces, and their transitions follow directly fromthe deformation of the intersection between the resonant and dispersive sheets. Two depth ratios correspond to genuine geometricdegeneracies: equal layer depths, where the intersection reduces to a straight line, and the golden-ratio configuration, where the criticalsurface becomes horizontally tangent at the Bond threshold. Overall, Part II completes the geometric and physical classification ofmodulational stability in two-layer interfacial waves and provides a framework for future extensions incorporating shear, external forcing,flexible boundaries, or variable bathymetry.Item Benjamin–Feir Instability of Wave Packets at Interface of Liquid Half-Space and Layer(2026) Avramenko, Olha; Naradovyi, VolodymyrThe propagation of internal waves in a hydrodynamic system comprising a solid bottom and an upper half-space is investigated. The study is conducted within the framework of a nonlinear low-dimensional model incorporating surface tension on an interface using the method of multi-scale expansions. The evolution equation of the envelope of the wave packet takes the form of the Schr ̈odinger equation. Conditions for the Benjamin–Feir stability of the solution of the evolution equation are identified for various physical and geometrical characteristics of the system.Item Deviation of the interface between two liquid half-spaces with surface tension: multiscale approach(2024) Avramenko, OlhaThis paper investigates the deviation of the interface between two semi-infinite liquid media under the influence of surface tension and gravity using a multiscale analysis. The initial-boundary value problem is formulated based on key dimensionless parameters, such as the density ratio and the surface tension coefficient, to describe the generation and propagation of wave packets along the interface. A weakly nonlinear model is employed to examine initial deviations of the interface, enabling the derivation of integral solutions for both linear and nonlinear approximations. The linear approximation captures the fundamental structure of forward and backward waves, while nonlinear corrections account for higherorder effects derived through multiscale expansions. These corrections describe the evolution of the wave packet envelope, highlighting the interplay between dispersion, nonlinearity, and surface tension. Integral expressions are provided for both linear and nonlinear solutions, including those illustrating the role of even and odd initial deviations of the interface. Comparisons between linear and nonlinear approximations emphasize their interconnectedness. The linear model defines the primary wave dynamics, while the nonlinear terms contribute higher harmonics, refining the solutions and facilitating stability analysis. The results reveal significant contributions from higher-order harmonics in determining the dynamics of the interface. Furthermore, the study explores the conditions under which the nonlinear envelope remains stable, including constraints on initial amplitudes to prevent instability. This research opens new perspectives for further analysis of stability and wave dynamics at fluid interfaces using symbolic computations. Potential applications include the study of wave behavior under various geometric configurations and fluid properties. The findings contribute to advancing hydrodynamic wave modeling and establish a foundation for future research in this field.Item Modulational stability of wave packets at fluid interface of layer and half-space(2025) Avramenko, Olha; Naradovyi, VolodymyrThe modulational stability of internal wave packets propagated along the surface of a hydrodynamic system consisting of a lower half-space and an upper layer covered with a rigid lid is investigated. The study is conducted within the framework of a nonlinear low-dimensional model incorporating surface tension on an interface using the method of multi-scale expansions implemented via symbolic computation. The evolution equation of the envelope of the wave packet takes the form of the Schrodinger equation. Conditions ¨ for the modulational stability of the solution of the evolution equation are identified for various physical and geometrical characteristics of the system. Significant influence on the modulational stability of the system’s geometrical characteristics and surface tension is observed for relatively small liquid layer thicknesses. For large layer thicknesses, the stability diagram degenerates to that of a system composed of two half-spaces.Item Peculiarities of initial condition specification in a problem of wave packet propagation in layered fluid(Дніпровський національний університет імені Олеся Гончара, 2024) Avramenko, OlhaThe problem of wave packet propagation along the interface of two semiinfinite fluids with different densities is considered within the framework of a weakly nonlinear model, taking surface tension into account. The method of multiple scales expansions is applied. The analytical analysis of admissible initial conditions is carried out in two stages. In the first stage, the initial perturbation of the free surface is specified as a smooth function symmetric about the central point. This function is expanded into a series of the first harmonics, taking into account the dispersion relation. In the second stage, a sequence of second harmonics is constructed that satisfies the evolution equation, namely, the nonlinear Schrödinger equation.Item PINN Modeling of Interfacial Gravity-Capillary Waves(Національний університет "Києво-Могилянська академія", 2025) Avramenko, Olha; Sontikov, MaksymThis paper presents an automated computational framework for modeling hydrodynamic processes using physics-informed neural networks (PINNs). The modular system integrates all stages of numerical experimentation — from data generation and model training to validation and accuracy evaluation — ensuring reproducibility, flexibility, and scalability. The framework was verified on the classical problem of interfacial gravity–capillary waves between two incompressible fluids, using the analytical solution as a benchmark for numerical assessment. Computational experiments showed that increasing the number of training points from 400 to 1000 improved accuracy and convergence, with the Extended configuration achieving 98.86% accuracy and a MAPE of 1.14%, while Adaptive_LR remained stable. The results confirm the reliability and efficiency of the proposed PINN-based framework for solving complex hydrodynamic problems governed by nonlinear partial differential equations.Item PINN-based machine learning for modeling internal waves insemi-infinite fluids(2025) Avramenko, Olha; Kompan, Serhii; Sarana, MaksymThis study investigates the application of Physics-Informed Neural Networks (PINNs) for modelingwave processes at the interface between two incompressible fluids of differing densities. As a first step,the linear formulation of the problem is considered, which admits an analytical solution based on aspectral method involving Fourier decomposition of the initial perturbation. This solution serves as abenchmark for testing and validating the accuracy of the PINN predictions.The implementation is carried out in Python using specialized libraries such as TensorFlow, NumPy,SciPy, and Matplotlib, which provide both efficient deep learning frameworks and tools for solving mathe-matical physics problems numerically. The approach integrates artificial intelligence with domain-specificknowledge in hydrodynamics, enabling the construction of interpretable and physically consistent mod-els. Particular attention is given to the organization of the computational experiment, automation ofvisualizations, and storage of intermediate results for further analysis. The PINN model includes a lossfunction that encodes the governing equations and boundary conditions, and the training is conductedon randomly sampled points across the spatio-temporal domain. The influence of network architectureand training parameters on solution accuracy is examined. Visualization of loss function evolutionand predicted wave profiles provides insight into convergence behavior and physical plausibility of thesolutions.A comparative analysis between the PINN-based and analytical solutions across different time in-stances is presented, revealing phase shifts and amplitude deviations. The model demonstrates goodagreement at early times and a gradual accumulation of errors as time progresses—an expected featureof this class of methods. The results confirm the feasibility of applying the PINN framework to linearhydrodynamic problems, laying the groundwork for future extensions to weakly and strongly nonlinearregimes, including studies of wave stability and nonlinear wave dynamics.