http://repository.iiitd.edu.in/xmlui/handle/123456789/1713 Year-2025 Sat, 11 Apr 2026 17:42:02 GMT 2026-04-11T17:42:02Z http://repository.iiitd.edu.in/xmlui/handle/123456789/1771 Spectral instabilities in water wave models Bhavna; Pandey, Ashish Kumar (Advisor) This thesis investigates the spectral instabilities of various nonlinear water wave models through rigorous analytical techniques. Focusing on three fundamental types of instabilities, modulational instability, transverse instability, and high-frequency instability, the work provides a unified spectral framework to study how small perturbations evolve and potentially destabilize wave solutions in dispersive systems. We begin by analyzing modulational instability, wherein a periodic traveling wave becomes unstable to long-wavelength perturbations. Using perturbation theory and spectral analysis, we characterize conditions under which modulational instability arises in generalized Ostrovsky equations. The effect of dispersion, nonlinearity, and surface tension is examined in detail. The study then turns to transverse instability, where planar wave trains destabilize due to perturbations in the transverse direction. We consider rotation-modified and surface tension-influenced variants of the Kadomtsev–Petviashvili (KP) equation, the rotation-modified KP equation, and the KD equation, and identify parameter regimes leading to transverse spectral instabilities. Finally, we explore high-frequency instability, focusing on the behavior of the spectrum. We demonstrate how high-frequency perturbations can induce instabilities in small-amplitude periodic traveling waves. Altogether, the results contribute to a deeper understanding of how wave coherence is affected by perturbations of various scales and directions. The insights gained have potential implications for the stability of wave patterns in physical settings such as oceanography, fluid mechanics, and nonlinear optics. Fri, 01 Aug 2025 00:00:00 GMT http://repository.iiitd.edu.in/xmlui/handle/123456789/1771 2025-08-01T00:00:00Z http://repository.iiitd.edu.in/xmlui/handle/123456789/1768 Construction of some new diffeomorphisms using a combinatorial approach: the approximation by conjugation method Khurana, Divya; Banerjee, Shilpak (Advisor); Chaubey, Sneha (Advisor); Kunde, Philipp (Advisor) Ergodic theory often involves dealing with transformations of measure spaces that preserve specific measures, such as volume or probability measures. These transformations establish a framework for studying the behavior of dynamical systems. Combinatorial methods are pivotal in ergodic theory, providing valuable techniques for understanding the behavior of dynamical systems. They are often utilized in the study of measure-preserving transformations, aiding in characterizing invariant measures, exploring ergodic decomposition, and establishing connections between different dynamical systems. Moreover, numerous scenarios exist where we can effectively characterize a measure-preserving transformation and its long-term behaviour by approaching it as the limit of specific finite objects, such as periodic processes. This thesis discusses combinatorial constructions within Ergodic Theory, employing the “Approximation by Conjugation” method. This technique facilitates the construction of maps with specific topological and measure-theoretic characteristics on the manifolds that support a non-trivial circle action. We provide examples of volume-preserving diffeomorphisms with zero topological entropy. These examples exhibit intricate ergodic properties in both smooth and analytic categories. Specifically, we present an example of a smooth diffeomorphism with an invariant measure, which is a generic but non-ergodic volume measure. This diffeomorphism satisfies various other ergodic and topological dynamic properties on the 2-torus. In ergodic theory, generic points are essential for understanding the statistical and dynamical behavior of systems, as their orbits cover most of the phase space according to the invariant measure. We examine the distinctions between generic and non-generic points in smooth dynamical systems, focusing on establishing bounds for the sizes of their respective sets. We construct an explicit collection of sets that encompass all generic points within the system and focus on determining bounds for their Hausdorff dimension, leading to more insightful and compelling conclusions. Additionally, we explore the ergodic properties of diffeomorphisms and examine their differential maps with respect to a smooth measure in the projectivization of the tangent bundle. We investigate examples of diffeomorphisms with complex ergodic properties within the projectivized tangent bundle for the smooth and analytic case. We construct a smooth diffeomorphism whose differential is weakly mixing with respect to a smooth measure in the projectivization of the tangent bundle. In the case of the 2-torus, we also obtain the analytic counterpart of such a diffeomorphism by utilizing an analytic approximation technique. We present an example of an analytic diffeomorphism whose projectivized derivative extension exhibits weak mixing with respect to a smooth measure. All of these constructions are based on the original and quantitative version of the “Approximation by Conjugation” method. This approach utilizes highly explicit well-defined conjugation maps, explicit partial partitions of the space, specific approximation strategies, and geometric and combinatorial criteria to meet specific requirements. Fri, 01 Aug 2025 00:00:00 GMT http://repository.iiitd.edu.in/xmlui/handle/123456789/1768 2025-08-01T00:00:00Z http://repository.iiitd.edu.in/xmlui/handle/123456789/1763 Character analogues of cohen-type identities and related voronoi summation formulas Khyati; Banerjee, Debika (Advisor) Summation formulas play a vital role in analytic number theory. Several kinds of summation formulas exist, like the Poisson summation formula, Abel summation formula, Euler-Maclaurin formula, etc. In 1904, G. F. Voronoi proved that the error term in the Dirichlet divisor problem can be expressed in terms of infinite series involving Bessel functions. Additionally, he offered a broader version of the above summation formulas involving a test function f , where f (t) is a function of bounded variation. Consequently, he deduced a better bound for the error term in the Dirichlet divisor problem at that time. Following Voronoi’s astounding discovery, other number theorists like A. L. Dixon,W. L. Ferrar, J. R. Wilton, Koshliakov, M. Jutila etc looked into the formula and offered proofs under different conditions on the function f (x). Apart from its con-nection to different fields of mathematics, Voronoi-type summation formulas also have some applications in physics, especially in quantum graph theory. In 2014, B. C. Berndt and A. Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying Ramanujan’s type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, S. Kim extended the definition of twisted divisor sums to twisted sums of divisor functions . Here, we study identities associated with the aforementioned weighted divisor functions and the modified K-Bessel function in light of recent results obtained by D. Banerjee and B. Maji. Moreover, we provide a new expression for L(1, χ)from which the positivity of L(1, χ) for any real primitive character χ is established, which is important is the proof of Prime number theorems in arithmetic progression. In addition, we deduce Cohen-type identities and then exhibit the Voronoi-type summation formulas for them. Additionally, we discuss an equivalent version of the aforementioned results in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, we provide an identity for r6(n), which is analo-gous to Hardy’s famous result where r6(n) denotes the number of representations of natural number n as a sum of six squares. Fri, 04 Jul 2025 00:00:00 GMT http://repository.iiitd.edu.in/xmlui/handle/123456789/1763 2025-07-04T00:00:00Z http://repository.iiitd.edu.in/xmlui/handle/123456789/1716 Viscoelastic subdiffusive flows : modelling, analysis and computation Chauhan, Tanisha; Sircar, Sarthok (Advisor) This thesis explores the rheodynamics of viscoelastic subdiffusive fluids and high-lights the potential of fractional calculus in modelling these types of fluids. A novel fractional model is developed to investigate the regions of spatiotemporal instability. Direct numerical simulations are used to capture the macrostructures within the flow, utilizing a new structure tensor, which is physically realizable, contributing to a deeper understanding of this complex fluid dynamics phenomenon. The seven chapters are described as follows: Chapter 1 lays the foundation by introducing subdiffusive fluids and delving into the historical backdrop of fractional models applied to viscoelastic flows. This chapter surveys past analytical outcomes, numerical strategies, and experimental findings, exhibiting a wide range of applications in various fields. In Chapter 2, the fundamental groundwork is established, including a thorough explanation of several types of instability: convective, absolute, and evanescent modes. This chapter elucidates numerical methods, especially finite difference schemes, designed to handle various equation components such as advection and diffusion terms and also discusses the different fractional derivatives and their corresponding numerical approximations. Chapter 3 embarks on a comprehensive investigation of temporal and spatiotemporal linear stability analyses within the viscoelastic subdiffusive plane Poiseuille flow. The Fractional Upper Convected Maxwell model is derived and explored under low to moderate Reynolds numbers (Re) and Weissenberg numbers (We) to uncover regions of topological transition in advancing flow interfaces. The relation between the exponent in the time scale of the microscale models (tα ) and fractional order (α) in stress constitutive equations is derived. Using Brigg’s method of analytic continuation, insights into instability modes as fractional derivative order changes are discussed. The stability studies are limited to two exponents: monomer diffusion in Rouse chain melts, α = 1/2, and in Zimm chain solutions, α = 2/3. The presence of a non-homogeneous environment with hindered flow is revealed by the discovery of an abnormal region of temporal stability at high fluid inertia, highlighting the potential of the model to accurately capture some experimentally observed flow-instability transitions in subdiffusive flows. Chapter 4 presents a theory to quantify the development of the spatiotemporal macrostructures for viscoelastic sub-diffusive flows by decomposing the polymer conformation tensor into the so-called structure tensor. Our method bypasses the traditional arithmetic decomposition’s fundamental flaw, which is that the fluctuating conformation tensor fields might not be positive definite and, as a result, lose their physical significance. By defining and building a geodesic via the inner product on its tangent space, the space of positive definite matrices is converted into a Riemannian manifold using some well-proven results in matrix analysis. Three scalar invariants of the structure tensor are defined by means of this geodesic. The maximum amount of time that the perturbative solution’s evolution may be accurately predicted by linear theory along the Euclidean manifold is found. Chapter 5 introduces a novel family of time-asymptotically stable, implicit-explicit, adaptive, time integration methods (denoted with the θ -method) for the solution of the fractional advection-diffusion-reaction equations. The computationally explicit L1 method is generalised by this class of temporal integration techniques. For a specific range of Peclet numbers, the dispersion relation analysis of the method which takes into account the group velocity and the phase speed, indicates a favorably large region. The one-dimensional fractional diffusion equation is used to validate the method’s correctness and effectiveness. Chapter 6 includes the direct numerical simulations of viscoelastic, subdiffusive, plane Poiseuille flow in the regime of low to moderate Reynolds number and low Weissenberg number. These simulations successfully capture the flow structures by offering (i) a better resolution of the instantaneous regions of elastic shocks (which are the alternating regions of expanded and compressed polymer volume, in comparison with the volume of the mean conformation tensor), and (ii) a higher resolution to identify areas where the mean conformation tensor tends to diverge significantly from the instantaneous conformation tensor, supporting the experimentally observed transition of subdiffusive flows into flow instability. Chapter 7 summarizes the real-life applications involving viscoelastic subdiffusive fluids, focusing on their rheological properties. Fractional models, although instrumental in comprehending intricate phenomena with non-local and memory-dependent behaviour, do present certain limitations. To wrap up, this chapter points out the main challenges in understanding these flows and emphasizes potential future problems that need serious attention in this new category of complex fluids Mon, 27 Jan 2025 00:00:00 GMT http://repository.iiitd.edu.in/xmlui/handle/123456789/1716 2025-01-27T00:00:00Z