Boundary value problem for PDEs and some clases of L^p bounded pseudodifferential operators

In recent years much attention has been extended in the study of differential equations of non-classical types. These articles need, on one hand, fluid mechanics, hydro-and gas dynamics and other applied disciplines, and on the other hand, the actual needs of the mathematical sciences. One of the most important classes of equations of non-classical type is the third-order equation with multiple characteristics which is a generalization of linear Korteweg-de Vries-Burgers equation, special cases which occur in the dissemination of waves in weakly dispersive media, the propagation of waves in a cold plasma, magneto-hydrodynamics, problems of nonlinear acoustics, the hydrodynamic theory of space plasma. A pioneering work in the theory of odd order partial differential equations with multiple characteristics was done by E.Del Vecchio, H.Block, in which they studied the technique of constructing fundamental solutions of these equations. Consequently, the theory of equations with multiple characteristics has been greatly developed by the Italian mathematician L.Cattabriga. In the first part of Ph.D thesis we develop and study boundary value problems for third-order equations with multiple characteristics in areas with curved boundaries, as well as some properties of the fundamental solutions of the equations, when the transition line is a curve. In addition, we construct a solution of the Cauchy problem in the classes of functions growing at infinity, depending on the behaviour of the right-hand side of the equation. Our thesis explores both linear and nonlinear boundary value problems for linear and non-linear third-order equation with multiple characteristics in the domain with curved boundaries. The main result of the first chapter is to prove the unique solvability of the general boundary value problem for the third-order equation with multiple characteristics in curved domains. The proof of the uniqueness theorem of the solution, we use the method of energy integrals. For the existence theorem, we find equivalent systems of Volterra second type integral equations. The next chapter consists of three sections and it investigates the problem with nonlinear boundary conditions for linear and non-linear equations of the third order with multiple characteristics. To prove the existence and uniqueness theorems, we will use methods of integral energy and theory of integral equations. In the last part of the thesis we analyze basic properties of pseudodifferential operators, such as the behaviour of products and adjoins of such operators, their continuity on L^2, L^p and Sobolev spaces. In the thesis we study the L^p - boundedness of vector weighted pseudodifferential operators with symbols which have derivatives with respect to x only up to order k, in the Holder continuous sense

Standard and Embedded Solitons in Nematic Optical Fibers

A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wavepackets of transverse magnetic (TM) modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations (PDEs) which describe the propagation of optical pulses along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium and derive an extended Nonlinear Schrodinger equation (eNLS) with a third order derivative nonlinearity, governing the dynamics for the amplitude of the wavepacket. In this derivation the dispersion, self-focussing and diffraction in the nematic are taken into account. Although the resulting nonlinear $PDE$ may be reduced to the modified Korteweg de Vries equation (mKdV), it also has additional complex solutions which include two-parameter families of bright and dark complex solitons. We show analytically that under certain conditions, the bright solitons are actually double embedded solitons. We explain why these solitons do not radiate at all, even though their wavenumbers are contained in the linear spectrum of the system. Finally, we close the paper by making comments on the advantages as well as the limitations of our approach, and on further generalizations of the model and method presented.Comment: "Physical Review E, in press

Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations

The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations implies the linearizability of systems of partial differential equations corresponding to those complex ordinary differential equations. The invertible complex transformations can be used to obtain invertible real transformations that map a system of nonlinear partial differential equations into a system of linear partial differential equation. Explicit invariant criteria are given that provide procedures for writing down the solutions of the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single research paper "Linearizability criteria for systems of two second-order differential equations by complex methods" which has been published in Nonlinear Dynamics. Due to citations of both parts I and II these are not replaced with the above published articl

Localized direct boundary–domain integro–differential formulations for scalar nonlinear boundary-value problems with variable coefficients

Mixed boundary-value Problems (BVPs) for a second-order quasi-linear elliptic partial differential equation with variable coefficients dependent on the unknown solution and its gradient are considered. Localized parametrices of auxiliary linear partial differential equations along with different combinations of the Green identities for the original and auxiliary equations are used to reduce the BVPs to direct or two-operator direct quasi-linear localized boundary-domain integro-differential equations (LBDIDEs). Different parametrix localizations are discussed, and the corresponding nonlinear LBDIDEs are presented. Mesh-based and mesh-less algorithms for the LBDIDE discretization are described that reduce the LBDIDEs to sparse systems of quasi-linear algebraic equations