Focusing Singularity in a Derivative Nonlinear Schr\"odinger Equation

We present a numerical study of a derivative nonlinear Schr\"odinger equation with a general power nonlinearity, $|\psi|^{2\sigma}\psi_x$. In the $L^2$-supercritical regime, $\sigma>1$, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schr\"odinger equation with supercritical power law nonlinearity.Comment: 24 pages, 17 figure

Embedded Eigenvalues and the Nonlinear Schrodinger Equation

A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances. The proof is computer assisted as it depends on the sign of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.Comment: 29 pages, 27 figures: fixed a typo in an equation from the previous version, and added two equations to clarif

A multiregional endogenous growth model with forward looking agents

The paper presents a multiregional endogenous growth model designed for calibration with real world data and for numerical policy evaluation. It integrates four strands of research: (1) the Ramsey model of consumer behaviour, (2) Tobin's q-theory of investment, (3) Romer's theory of endogenous growth through horizontal product innovation, and (4) the Dixit-Stiglitz-Ethier theory of intra-industry trade. Integrating the latter into a multiregional model is also an essential ingredient of the New Economic Geography. Thus, the paper is related to this literature as well, but lacks another essential feature of this tradition; the model to be presented does not exhibit catastrophic agglomeration. A symmetric first nature will always generate a “flat earth†steady state equilibrium. The model has an arbitrary (possibly large) number of regions with a representative household and a production sector in each of them. There are three types of goods, non-tradable local goods, horizontally diversified tradables, and designs of tradable products that are the exclusive property of their producers (called “blueprintsâ€, for short). Goods are produced combining - in identical proportions for all three of them - four inputs, labour, capital local and tradable goods. Blueprint production benefits from a technological positive externality. Technologies and preferences are uniform across space. Beyond goods markets and real factor markets there is a frictionless global bond market. All markets are perfectly competitive except the tradables market, which is monopolistic in the familiar Dixit-Stiglitz-Ethier style. The engine of sustainable long run growth is the accumulation of blueprints. Agents act under perfect foresight. The paper explains the formal structure, the solution and calibration techniques and illustrates the application by a small example.

Green's Function for Discrete Second-Order Problems with Nonlocal Boundary Conditions

We investigate a second-order discrete problem with two additional conditions which are described by a pair of linearly independent linear functionals. We have found the solution to this problem and presented a formula and the existence condition of Green's function if the general solution of a homogeneous equation is known. We have obtained the relation between two Green's functions of two nonhomogeneous problems. It allows us to find Green's function for the same equation but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.</p

Spectral Analysis for Matrix Hamiltonian Operators

In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schr\"odinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though we focus on a proof of the 3d cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability. Source code for verification of our comptuations, and for further experimentation, are available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.Comment: 57 pages, 22 figures, typos fixe

Existence Results for Singular Boundary Value Problem of Nonlinear Fractional Differential Equation

By applying a fixed point theorem for mappings that are decreasing with respect to a cone, this paper investigates the existence of positive solutions for the nonlinear fractional boundary value problem: 0+()+(,())=0, 0<<1, (0)=′(0)=′(1)=0, where 2<<3, 0+ is the Riemann-Liouville fractional derivative

On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods

Continuum and crystal strain gradient plasticity with energetic and dissipative length scales

This work, standing as an attempt to understand and mathematically model the small scale materials thermal and mechanical responses by the aid of Materials Science fundamentals, Continuum Solid Mechanics, Misro-scale experimental observations, and Numerical methods. Since conventional continuum plasticity and heat transfer theories, based on the local thermodynamic equilibrium, do not account for the microstructural characteristics of materials, they cannot be used to adequately address the observed mechanical and thermal response of the micro-scale metallic structures. Some of these cases, which are considered in this dissertation, include the dependency of thin films strength on the width of the sample and diffusive-ballistic response of temperature in the course of heat transfer. A thermodynamic-based higher order gradient framework is developed in order to characterize the mechanical and thermal behavior of metals in small volume and on the fast transient time. The concept of the thermal activation energy, the dislocations interaction mechanisms, nonlocal energy exchange between energy carriers and phonon-electrons interactions are taken into consideration in proposing the thermodynamic potentials such as Helmholtz free energy and rate of dissipation. The same approach is also adopted to incorporate the effect of the material microstructural interface between two materials (e.g. grain boundary in crystals) into the formulation. The developed grain boundary flow rule accounts for the energy storage at the grain boundary due to the dislocation pile up as well as energy dissipation caused by the dislocation transfer through the grain boundary. Some of the abovementioned responses of small scale metallic compounds are addressed by means of the numerical implementation of the developed framework within the finite element context. In this regard, both displacement and plastic strain fields are independently discretized and the numerical implementation is performed in the finite element program ABAQUS/standard via the user element subroutine UEL. Using this numerical capability, an extensive study is conducted on the major characteristics of the proposed theories for bulk and interface such as size effect on yield and kinematic hardening, features of boundary layer formation, thermal softening and grain boundary weakening, and the effect of soft and stiff interfaces

Numerical study of convective fluid flow in porous and non-porous media.

Ph. D. University of KwaZulu-Natal, Pietermaritzburg 2015.Abstract available in PDF file