LEE- Matematik Mühendisliği-Yüksek Lisans
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ÖgeExact soliton solutions of cubic nonlinear Schrödinger equation with a momentum term(Graduate School, 2024-12-26)There are various interconnections between the positive sciences, with differential equations serving as a fundamental bridge linking mathematics to other scientific disciplines. Nonlinear wave phenomena have recently gained considerable attention due to their theoretical significance and applied relevance. Nonlinear optical wave equations not only facilitate the development of advanced techniques but also play a crucial role in elucidating natural phenomena across diverse fields, including biology, nonlinear optics, and quantum physics. Among these, solitons—localized nonlinear waves—stand out as valuable tools for understanding complex nonlinear systems. Solitons are widely studied in areas such as plasma physics, nonlinear optics, and quantum mechanics. Optical solitons, in particular, have drawn significant interest due to the inherently interdisciplinary nature of soliton theory, making it a pivotal topic for advancing technologies like high-speed data transmission. The external potential strongly influences the shape and stability of optical pulses. In quantum mechanics and nonlinear optics, potentials with parity-time symmetry (PT -symmetry) are frequently utilized. Numerous studies in the literature examine the stability of nonlinear Schrödinger (NLS) equations with PT -symmetry. These equations admit various nonlinear wave solutions, including solitons, which are localized waves that propagate without distortion. Solitons demonstrate remarkable resilience during collisions, retaining their properties even after interacting with other waves. This work investigates the soliton solutions and their stability in an NLS equation incorporating a momentum term and cubic nonlinearity under an external PT -symmetric potential. The governing equation is expressed as: iu_(z) +αu_(xx) −iΓu_(x) +φ|u|^2u+V_(PT) u = 0. Here, z denotes the scaled propagation distance, u is the differentiable complex-valued slowly varying amplitude, u_(xx) represents diffraction, Γ is the momentum term taken as a constant, and V_(PT) denotes the external potential. The PT -symmetric potential is defined as: VPT = V(x) +iW(x)=V0 +V1sech(x) +V2sech^2(x)+i[W0sech(x)tanh(x) +W1tanh(x)]. Here, V(x) and W(x) represent the real and imaginary components of the potential, where V(x) is an even function and W(x) is an odd function. A detailed introduction to solitons and their interdisciplinary significance is provided in Chapter 1. The NLS equation is introduced, along with its recent developments, including the momentum term and PT -symmetry. The chapter also outlines the research objectives and the thesis hypothesis, emphasizing the importance of the momentum term in the NLS equation. Chapter 2 describes the Spectral Renormalization (SR) Method, an iterative Fourier technique used to numerically solve the NLS equation with a momentum term and a PT -symmetric potential. The method is adapted to the problem at hand, and numerical solutions are obtained. In Chapter 3, the structure of the NLS equation without potential is analyzed to investigate the effect of the momentum term. Variations in soliton structures are examined in relation to changes in the momentum term coefficient, Γ, and the propagation constant, µ. Chapter 4 explores exact solutions of the NLS equation with a momentum term and PT -symmetric potential. Using the ansatz u(x,z) = f(x)e^i(µz+g(x)), where f(x) and g(x) are real-valued functions, analytical solutions are derived. These solutions are compared with the numerical results, which shows excellent agreement. The chapter also verifies the parity-time symmetry properties of the potential, confirming that its imaginary part is odd and its real part is even. Chapter 5 focuses on the stability analysis of soliton solutions. The Split-Step Fourier method is employed to investigate nonlinear stability, while linear stability is examined through the linear spectrum. The results indicate that the solitons become unstable with even slight increases in the momentum term coefficient, Γ. Additionally, enhancing the complex component of the potential increases instability, whereas increasing the real component improves stability. The acquired results are summed up in Chapter 6. Moreover, a brief discussion on potential future research is included. All numerical results were obtained using MATLAB2023®.
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ÖgeOn the hypersurfaces with non-diagonalizable shape operator in Minkowski spaces(Graduate School, 2022)Let $M$ be an oriented hypersurface of the Minkowski space $\mathbb E^{n+1}_1$. One of the most important extrinsic object of $M$ is its shape operator $S$ defined by the Wiengarten formula $$SX=-\tilde \nabla_X N,$$ where $N$ is the unit normal vector field to $M$ whenever $X \in TM$. The shape operator can be used to determine how the tangent plane and its normal move in all directions. Note that $S$ is a self adjoint endomorphism in $TM$. Therefore, it is diagonalizable when $M$ is Riemannian. However, if $M$ is Lorentzian, then its shape operator can be non-diagonalizable. In this case, the shape operator $S$ has four canonical forms. These canonical forms are written with respect to either an orthonormal basis or a pseudo-orthonormal basis. If the basis is orthonormal, then it is called a orthonormal frame field. An orthonormal frame of vector fields in a neighborhood of any point in $M$ is a basis $\{ E_1, \hdots, E_n \}$ such that $$(E_1,E_1)=-1, \quad (E_1,E_i)=0, \quad (E_i,E_j)=\delta_{ij}$$ for $2 \leq i, \ j \leq n$. On the other hand if the basis is pseudo-orthonormal, then it is called a pseudo-orthonormal frame field. A pseudo-orthonormal frame of vector fields in a neighborhood of any point in $M$ is a basis { X, Y, E_1, \hdots, E_{n-2} } such that $$(X,X)= (Y,Y)=0, \quad (X,E_i)=(Y,E_i)=0, \quad (X,Y)=-1$$ and $$(E_i, E_j)=\delta_{ij}$$ for $1 \leq i, \ j \leq n-2$. The eigenvalues and eigenvectors of $S$ are called the principal curvatures and principal directions of $M$, respectively. If the shape operator $S$ is diagonalizable and $M$ has constant principal curvatures, then the hypersurface $M$ is said to be isoparametric. If $S$ is non-diagonalizable and its minimal polynomial is constant, then $M$ is said to be isoparametric. In this thesis, we study isoparametric hypersurfaces with non-diagonalizable shape operator in Minkowski space $\mathbb E^{4}_1$. This thesis consists of five sections. First section is introduction. In the second section, we give some basic concepts on Lorentzian inner product and also some basic facts on hypersurfaces of $\mathbb E^{n+1}_1$. In the third section, a theorem about isoparametric hypersurfaces is given. We note that these theorems are proved by Magid in 1985. We prove these theorems by using another method. In fact, this theorem implies that there is only four classes of isoparametric hypersurface using the Codazzi and Gauss equations in $\mathbb E^{4}_1$. Then, we give another theorem which proves that there is no isoparametric hypersurface in $\mathbb E^{4}_1$ with complex principal curvatures. In the fourth section, we construct a family of hypersurfaces with non-diagonalizable shape operator in $\mathbb E^{5}_1$. We obtain the shape operator, the mean curvature, Gauss-Kronecker curvature and Levi-Civita connection of this hypersurface. Then, we give the necessary and sufficient condition for this hypersurface to be minimal with a theorem.
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ÖgeSuppression of symmetry-breaking bifurcations of optical solitons in parity-time symmetric potentials(Graduate School, 2022)Optical soliton refers to any optical field that maintains its special structure during propagation because of the balance between diffraction and self-phase modulation of the medium. The dynamics of optical solitons are investigated comprehensively due to their fundamental structures and potential applications. In particular, optical solitons play an important role in fiber optic communication system that uses pulses of infrared light to transmit information from one place to another over a long distance. The propagation of the electromagnetic wave in optical fibers is modeled by the cubic-quintic nonlinear Schrödinger (CQNLS) equation iΨ_z+Ψ_{xx}+α|Ψ|^2Ψ+β|Ψ|^4Ψ=0, where Ψ(x,z) is normalized complex-valued slowly varying pulse envelope of the electric field, z is the scaled propagation distance, x is the transverse coordinate, Ψ_{xx} corresponds to diffraction, α and β are the coefficients of cubic and quintic nonlinearities, respectively. A higher-order dispersion needs to be considered for performance enhancement along trans-oceanic and trans-continental distances. Fourth order dispersion needs to be taken into account for short pulse widths where the group velocity dispersion changes within the spectral bandwidth of the signal. In addition, it is known from many studies in the literature that an external potential added to the system can be also beneficial for performance improvement. In this thesis, we consider the nonlinear paraxial beam propagation in cubic-quintic nonlinearity with a complex parity-time (PT) symmetric potential and fourth order dispersion. This propagation is modeled by the following CQNLS equation iΨ_z+Ψ_{xx}−γΨ_{xxxx}+V(x)Ψ+α|Ψ|^2Ψ+β|Ψ|^4Ψ=0, where γ>0 is the coupling constant of the fourth order dispersion, V(x) represents a complex PT-symmetric potential. In this thesis, we consider PT-symmetric potentials that are of the form V(x)=g^2(x)+c0*g(x)+ig′(x) where g(x) is an arbitrary real and even function, c0 is an arbitrary real constant and PT-symmetric solitons undergo symmetry breaking. We take a localized double-hump function g(x) in the form of g(x)=A*[exp(−(x+x0)^2)+exp(−(x−x0)^2)] where A and x0 are related to the modulation strength and separation of PT-symmetric potential, respectively. The soliton solutions of CQNLS equation with fourth order dispersion and a complex PT-symmetric potential are numerically obtained by means of the squared-operator method since the equation is nonintegrable. The linear stability analysis of the numerically obtained solitons is examined by linear spectrum analysis and the nonlinear stability analysis is examined by nonlinear evolution with split-step Fourier method. The existence of symmetry breaking of solitons and suppression of symmetry-breaking bifurcations have been investigated. To examine the effect of fourth order dispersion on this symmetry breaking, the coefficient of fourth order dispersion γ is incremented gradually. Consequently, we have demonstrated that the symmetry-breaking bifurcation of the solitons in this problem is completely suppressed as the strength of the fourth order dispersion increases. Moreover, increasing the strength of fourth order dispersion positively influences the linear and nonlinear stability behaviors of solitons.
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ÖgeBiconservative and biharmonic surfaces in Euclid and Minkowski spaces(Graduate School, 2024-07-02)In 1964, Sampson and Eells formulated the concept of biharmonic maps as an extension of harmonic maps while investigating the energy functional $E$ between Riemannian manifolds, a subject of both geometric and physical significance. Subsequently, numerous mathematicians have shown interest in the study of biharmonic mappings.\\ By the definition, the bienergy functional between semi-Riemannian manifolds $(M^m,g)$ and $(N^n,\tilde{g})$ is defined by $$E_2(\varphi)=\frac {1}{2} \int_M \|\tau(\varphi)\|^2 v_g$$ for a smooth map $\varphi:M \to N$, where $\tau(\varphi)$ represents the tension field of $\varphi$. $\varphi:M\to N$ is said to be biconservative if it is a critical point of $E_2$. This condition is equivalent to satisfying the Euler-Lagrange equation associated with the bienergy functional $$\tau_2(\varphi)=0,$$ where $\tau_2$ is the bitension field defined by $$\tau_2(\varphi):=\Delta\tau(\varphi)-\mathrm{tr\,} \tilde{R}(d\varphi,\tau(\varphi))d\varphi.$$\\ In the 1980s, B. Y. Chen conducted research on biharmonic submanifolds within Euclidean spaces as a component of B. Y. Chen's initiative to comprehend submanifolds of finite type in semi-Euclidean spaces. B. Y. Chen proposed an other characterization of biharmonic submanifolds in these spaces. Let $x:M\to \mathbb E^n_r$ be an isometric immersion. By examining normal and tangential parts of $\tau_2(x)$, the following results can be obtained.\\ Let $x: M^m \rightarrow \mathbb E^n_r$ be an isometric immersion of an $n$-dimensional semi-Riemannian submanifold $M^m$ into the semi-Euclidean space $\mathbb E^n_r$. If $x$ satisfies the fourth-order semi-linear PDE system given by the equations $$\Delta^\perp H+\mathrm{trace}h(A_H(\cdot),\cdot)=0$$ and $$m\mathrm{grad} \Vert H \Vert^2 +4\mathrm{trace} A_{\nabla^\perp_\cdot H}(\cdot)=0,$$ then $M^m$ is biharmonic.\\ On the other hand, if a mapping $\varphi: M \to N$ satisfies the weaker condition $$\langle \tau_2(\varphi), d\varphi \rangle = 0,$$ then it is said to be biconservative. Mainly, if $x: M \to N$ is an isometric immersion, then the previous equation is equivalent to $$\tau_2(x)^\top = 0,$$ where $\tau_2(x)^\top$ represents the tangential part of $\tau_2(x)$. In this case, $M$ is said to be a biconservative submanifold of $N$.\\ In this thesis, we mainly focus on biharmonic and biconservative surfaces in four dimensional Euclidean and Minkowski spaces. The first section provides a concise overview of the historical background and underlying principles concerning biharmonic and biconservative submanifolds, as well as an overview of the research conducted far in this field. In the second section, we give some basic notations and basic facts about submanifolds of semi- Euclidean spaces, the definition of biconservative submanifolds and we introduce the rotational surfaces. In the third section, we give biconservative PNMCV surfaces in $\mathbb E^4$. We obtain local parameterizations of these surfaces and demonstrate that they are not biharmonic. In the fourth section, we give biconservative rotational surfaces in $\mathbb E_1^4$. We study with three different class of rotational surfaces and obtain the condition for each of them to be biconservative. In the concluding section, the derived conclusions are presented, along with recommendations regarding possible future researches.
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ÖgeInnovative computational techniques for accurate internal defect detection in trees: A stress wave tomography approach enhanced by machine learning(Graduate School, 2024-06-10)The detection of internal defects in trees holds critical importance given the health of forest ecosystems and the industrial significance of wood products. The identification of these internal defects without damaging the wood is a significant factor in the forestry industry and in the production of wood products. While traditional methods often require cutting or processing the wood, non-invasive techniques such as stress wave tomography offer the possibility of identifying internal defects without disrupting the wood's structure. This contributes both to the sustainable management of forest resources and to the improvement of wood product quality. A branch of artificial intelligence, machine learning algorithms allow computer systems to analyze data, recognize patterns, make decisions, and solve problems. These algorithms are critical tools in analyzing large datasets obtained from non-invasive techniques like stress wave tomography, and in accurately detecting and classifying internal defects. In this thesis, an algorithm design capable of generating stress wave tomography based on ray segmentation and machine learning has been developed for the purpose of detecting internal defects in trees. A two-stage algorithm has been proposed based on data obtained from stress waves produced by sensors mounted on trees and on the segmented propagation rays generated from these data. In the first step, a ray segmentation method maps the velocity of stress waves to create segmented sensors. In the second step, data obtained from these segmented rays are processed using K-Nearest Neighbors (KNN) and Gaussian Process Classifier (GPC) algorithms to create a tomographic image of defects within the tree. The algorithm carries the potential to detect internal defects in wood without causing damage and provides more precise results compared to traditional methods. Implemented using the Python programming language, the algorithm equips researchers with the ability to understand and analyze the internal structure of trees. This method stands out as a practical tool for contributing to forest health assessment and conservation through stress wave tomography. During experiments, data from four real trees were collected via sensors, and an algorithm was developed to generate four sets of synthetic defective tree data in the sensor's data format. Real tree data was provided by Istanbul University Cerrahpaşa Faculty of Forestry. All tree data were individually used to feed the proposed defect detection algorithm, and the outputs were transformed into tomographic images. Success rates above 90% were achieved for all evaluation metrics. Compared to related studies, the results showed improvements ranging from 7% to 22% relative to the literature. This thesis aims to contribute to the development of the sustainable wood industry by offering a new approach to detecting internal tree defects. Although the results obtained are quite good compared to the results in the scientific literature, it is thought that even better results will be obtained by optimizing the parameters of the algorithm or by differentiating the machine learning algorithms integrated into the method.