Кафедра математики

Permanent URI for this collection

https://ekmair.ukma.edu.ua/handle/123456789/12312

Browse

Browsing Кафедра математики by Author "Avramenko, Olha"

Now showing 1 - 5 of 5
  • Thumbnail Image
    Item
    Benjamin-Feir Instability of Interfacial Gravity–Capillary Waves in a Two-Layer Fluid. Part I
    (2025) Avramenko, Olha; Naradovyi, Volodymyr
    This 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.
  • Thumbnail Image
    Item
    Deviation of the interface between two liquid half-spaces with surface tension: multiscale approach
    (2024) Avramenko, Olha
    This 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.
  • Thumbnail Image
    Item
    Modulational stability of wave packets at fluid interface of layer and half-space
    (2025) Avramenko, Olha; Naradovyi, Volodymyr
    The 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.
  • Thumbnail Image
    Item
    Peculiarities of initial condition specification in a problem of wave packet propagation in layered fluid
    (Дніпровський національний університет імені Олеся Гончара, 2024) Avramenko, Olha
    The 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.
  • Thumbnail Image
    Item
    PINN-based machine learning for modeling internal waves insemi-infinite fluids
    (2025) Avramenko, Olha; Kompan, Serhii; Sarana, Maksym
    This 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.