Studies on integrability of some perturbed nonlinear evolution equations

Usually typical dynamical systems are non integrable. But few systems of practical interest are integrable. The soliton concept is a sophisticated mathematical construct based on the integrability of a class ol' nonlinear differential equations. An important feature in the clevelopment. of the theory of solitons and of complete integrability has been the interplay between mathematics and physics. Every integrable system has a lo11g list of special properties that hold for integrable equations and only for them. Actually there is no specific definition for integrability that is suitable for all cases. .There exist several integrable partial clillerential equations( pdes) which can be derived using physically meaningful asymptotic teclmiques from a very large class of pdes. It has been established that many 110nlinear wa.ve equations have solutions of the soliton type and the theory of solitons has found applications in many areas of science. Among these, well-known equations are Korteweg de-Vries(KdV), modified KclV, Nonlinear Schr6dinger(NLS), sine Gordon(SG) etc..These are completely integrable systems. Since a small change in the governing nonlinear prle may cause the destruction of the integrability of the system, it is interesting to study the effect of small perturbations in these equations. This is the motivation of the present work.

Fluid mechanics-non-linear waves studies on korteweg-de vries equations

In this thesis the author has presented qualitative studies of certain Kdv equations with variable coefficients. The well-known KdV equation is a model for waves propagating on the surface of shallow water of constant depth. This model is considered as fitting into waves reaching the shore. Renewed attempts have led to the derivation of KdV type equations in which the coefficients are not constants. Johnson's equation is one such equation. The researcher has used this model to study the interaction of waves. It has been found that three-wave interaction is possible, there is transfer of energy between the waves and the energy is not conserved during interaction.

Swift heavy ion induced surface and microstructural evolution in metallic glass thin films

Swift heavy ion induced changes in microstructure and surface morphology of vapor deposited Fe–Ni based metallic glass thin films have been investigated by using atomic force microscopy, X-ray diffraction and transmission electron microscopy. Ion beam irradiation was carried out at room temperature with 103 MeV Au9+ beam with fluences ranging from 3 1011 to 3 1013 ions/cm2. The atomic force microscopy images were subjected to power spectral density analysis and roughness analysis using an image analysis software. Clusters were found in the image of as-deposited samples, which indicates that the film growth is dominated by the island growth mode. As-deposited films were amorphous as evidenced from X-ray diffraction; however, high resolution transmission electron microscopy measurements revealed a short range atomic order in the samples with crystallites of size around 3 nm embedded in an amorphous matrix. X-ray diffraction pattern of the as-deposited films after irradiation does not show any appreciable changes, indicating that the passage of swift heavy ions stabilizes the short range atomic ordering, or even creates further amorphization. The crystallinity of the as-deposited Fe–Ni based films was improved by thermal annealing, and diffraction results indicated that ion beam irradiation on annealed samples results in grain fragmentation. On bombarding annealed films, the surface roughness of the films decreased initially, then, at higher fluences it increased. The observed change in surface morphology of the irradiated films is attributed to the interplay between ion induced sputtering, volume diffusion and surface diffusion

Queues with inter- ruption in Random/ Markovian Environment

In many situations probability models are more realistic than deterministic models. Several phenomena occurring in physics are studied as random phenomena changing with time and space. Stochastic processes originated from the needs of physicists.Let X(t) be a random variable where t is a parameter assuming values from the set T. Then the collection of random variables {X(t), t ∈ T} is called a stochastic process. We denote the state of the process at time t by X(t) and the collection of all possible values X(t) can assume, is called state space

Possibility of waveguide formation on organic nonlinear crystal methyl para-hydroxy benzoate using high energy ion irradiation

Organic nonlinear optical single crystals of Methyl para-Hydroxy Benzoate (MHB) have been grown using gel-solution technique. These crystals are cut along z-axis and are bombarded with Ag14+ ions of energy 100 MeV. The results show an increase in refractive index at the ion irradiated region. The dielectric constant of the irradiated crystal is increased more than 15 times compared to that of a nonirradiated crystal. The result of these changes and comparative study of second harmonic generation (SHG) efficiency before and after irradiation is discussed.

Theory of differential inequalities with appliciations to singular perturbiation problems and method of lines

During recent years, the theory of differential inequalities has been extensively used to discuss singular perturbation problems and method of lines to partial differential equations. The present thesis deals with some differential inequality theorems and their applications to singularly perturbed initial value problems, boundary value problems for ordinary differential equations in Banach space and initial boundary value problems for parabolic differential equations. The method of lines to parabolic and elliptic differential equations are also dealt The thesis is organised into nine chapters

Discrete commutative difference operator theory

The object of this thesis is to formulate a basic commutative difference operator theory for functions defined on a basic sequence, and a bibasic commutative difference operator theory for functions defined on a bibasic sequence of points, which can be applied to the solution of basic and bibasic difference equations. in this thesis a brief survey of the work done in this field in the classical case, as well as a review of the development of q~difference equations, q—analytic function theory, bibasic analytic function theory, bianalytic function theory, discrete pseudoanalytic function theory and finally a summary of results of this thesis

Investigations On A Microstrip Excited Rectangular Dielectric Resonator Antenna

The thesis relates to the investigations carried out on Rectangular Dielectric Resonator Antenna configurations suitable for Mobile Communication applications. The main objectives of the research are to: - numerically compute the radiation characteristics of a Rectangular DRA - identify the resonant modes - validate the numerically predicted data through simulation and experiment 0 ascertain the influence of the geometrical and material parameters upon the radiation behaviour of the antenna ° develop compact Rectangular DRA configurations suitable for Mobile Communication applications Although approximate methods exist to compute the resonant frequency of Rectangular DRA’s, no rigorous analysis techniques have been developed so far to evaluate the resonant modes. In this thesis a 3D-FDTD (Finite Difference Time Domain) Modeller is developed using MATLAB® for the numerical computation of the radiation characteristics of the Rectangular DRA. The F DTD method is a powerful yet simple algorithm that involves the discretimtion and solution of the derivative form of Maxwell’s curl equations in the time domain.

Magnetic and topographical modifications of amorphous Co–Fe thin films induced by high energy Ag7+ ion irradiation

We have investigated the effects of swift heavy ion irradiation on thermally evaporated 44 nm thick, amorphous Co77Fe23 thin films on silicon substrates using 100 MeV Ag7+ ions fluences of 1 1011 ions/ cm2, 1 1012 ions/cm2, 1 1013 ions/cm2, and 3 1013 ions/cm2. The structural modifications upon swift heavy irradiation were investigated using glancing angle X-ray diffraction. The surface morphological evolution of thin film with irradiation was studied using Atomic Force Microscopy. Power spectral density analysis was used to correlate the roughness variation with structural modifications investigated using X-ray diffraction. Magnetic measurements were carried out using vibrating sample magnetometry and the observed variation in coercivity of the irradiated films is explained on the basis of stress relaxation. Magnetic force microscopy images are subjected to analysis using the scanning probe image processor software. These results are in agreement with the results obtained using vibrating sample magnetometry. The magnetic and structural properties are correlated

Studies on the interaction of two level atoms with quantized electromagnetic field

The present study described about the interaction of a two level atom and squeezed field with time varying frequency. By applying a sinusoidal variation in the frequency of the field, the randomness in population inversion is reduced and the collapses and periodic revivals are regained. Quantum optics is an emerging field in physics which mainly deals with the interaction of atoms with quantised electromagnetic fields. Jaynes-Cummings Model (JCM) is a key model among them, which describes the interaction between a two level atom and a single mode radiation field. Here the study begins with a brief history of light, atom and their interactions. Also discussed the interaction between atoms and electromagnetic fields. The study suggest a method to manipulate the population inversion due to interaction and control the randomness in it, by applying a time dependence on the frequency of the interacting squeezed field.The change in behaviour of the population inversion due to the presence of a phase factor in the applied frequency variation is explained here.This study also describes the interaction between two level atom and electromagnetic field in nonlinear Kerr medium. It deals with atomic and field state evolution in a coupled cavity system. Our results suggest a new method to control and manipulate the population of states in two level atom radiation interaction,which is very essential for quantum information processing.We have also studied the variation of atomic population inversion with time, when a two level atom interacts with light field, where the light field has a sinusoidal frequency variation with a constant phase. In both coherent field and squeezed field cases, the population inversion variation is completely different from the phase zero frequency modulation case. It is observed that in the presence of a non zero phase φ, the population inversion oscillates sinusoidally.Also the collapses and revivals gradually disappears when φ increases from 0 to π/2. When φ = π/2 the evolution of population inversion is identical to the case when a two level atom interacts with a Fock state. Thus, by applying a phase shifted frequency modulation one can induce sinusoidal oscillations of atomic inversion in linear medium, those normally observed in Kerr medium. We noticed that the entanglement between the atom and field can be controlled by varying the period of the field frequency fluctuations. The system has been solved numerically and the behaviour of it for different initial conditions and different susceptibility values are analysed. It is observed that, for weak cavity coupling the effect of susceptibility is minimal. In cases of strong cavity coupling, susceptibility factor modifies the nature in which the probability oscillates with time. Effect of susceptibility on probability of states is closely related to the initial state of the system.