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We study a two-dimensional Fermi-Pasta-Ulam lattice in the long-amplitude, small-wavelength limit. The one-dimensional lattice has been thoroughly studied in this limit, where it has been established that the dynamics of the lattice is well-approximated by the Korteweg–De Vries (KdV) equation for timescales of the order ε^−3. Further it has been shown that solitary wave solutions of the FPU lattice in the one dimensional case are well approximated by solitary wave solutions of the KdV equation. A two-dimensional analogue of the KdV equation, the Kadomtsev–Petviashvili (KP-II) equation, is known to be a good approximation of certain two-dimensional FPU lattices for similar timescales, although no proof exists. In this thesis we present a rigorous justification that the KP-II equation is the long-amplitude, small-wavelength limit of a two-dimensional FPU model we introduce, analogous to the one-dimensional FPU system with quadratic nonlinearity. We also prove that the cubic KP-II equation is the limit of a model analogous to a one-dimensional FPU system with cubic nonlinearity. Further we study whether stability of line solitons in the KP-II equation extends to stability of one-dimensional FPU solitary waves in the two-dimensional lattices. / Thesis / Doctor of Philosophy (PhD)

Global dynamic optimization arises in many engineering applications such as parameter estimation, global optimal control, and optimization-based worst-case uncertainty analysis. In branch-and-bound deterministic global optimization algorithms, a major computational bottleneck is generating appropriate lower bounds for the globally optimal objective value. These bounds are typically constructed using convex relaxations for the solutions of dynamic systems with respect to decision variables. Tighter convex relaxations thus translate into tighter lower bounds, which will typically reduce the number of iterations required by branch-and-bound. Subgradients, as useful local sensitivities of convex relaxations, are typically required by nonsmooth optimization solvers to effectively minimize these relaxations. This thesis develops novel techniques for efficiently computing tight convex relaxations with the corresponding subgradients for the solutions of ordinary differential equations (ODEs), to ultimately improve efficiency of deterministic global dynamic optimization. Firstly, new bounding and comparison results for dynamic process models are developed, which are more broadly applicable to engineering models than previous results. These new results show for the first time that in a state-of-the-art ODE relaxation framework, tighter enclosures of the original ODE system's right-hand side will necessarily translate into enclosures for the state variables that are at least as tight, which paves the way towards new advances for bounding in global dynamic optimization. Secondly, new convex relaxations are proposed for the solutions of ODE systems. These new relaxations are guaranteed to be at least as tight as state-of-the-art ODE relaxations. Unlike established ODE relaxation approaches, the new ODE relaxation approach can employ any valid convex and concave relaxations for the original right-hand side, and tighter such relaxations will necessarily yield ODE relaxations that are at least as tight. In a numerical case study, such tightness does indeed improve computational efficiency in deterministic global dynamic optimization. This new ODE relaxation approach is then extended in various ways to further tighten ODE relaxations. Thirdly, new subgradient evaluation approaches are proposed for ODE relaxations. Unlike established approaches that compute valid subgradients for nonsmooth dynamic systems, the new approaches are compatible with reverse automatic differentiation (AD). It is shown for the first time that subgradients of dynamic convex relaxations can be computed via a modified adjoint ODE sensitivity system, which could speed up lower bounding in global dynamic optimization. Lastly, in the situation where convex relaxations are known to be correct but subgradients are unavailable (such as for certain ODE relaxations), a new approach is proposed for tractably constructing useful correct affine underestimators and lower bounds of the convex relaxations just by black-box sampling. No additional assumptions are required, and no subgradients must be computed at any point. Under mild conditions, these new bounds are shown to converge rapidly to an original nonconvex function as the domain of interest shrinks. Variants of the new approach are presented to account for numerical error or noise in the sampling procedure. / Thesis / Doctor of Philosophy (PhD)

Nonlinear partial differential equations are difficult to solve, with many of the approximate solutions in the literature being numerical in nature. In this work, we apply the Homotopy Analysis Method to give approximate analytical solutions to nonlinear ordinary and partial differential equations. The main goal is to apply different linear operators, which can be chosen, to solve nonlinear problems. In the first three chapters, we study ordinary differential equations (ODEs) with one or two linear operators. As we progress, we apply the method to partial differential equations (PDEs) and use several linear operators. The results are all purely analytical, meaning these are approximate solutions that we can evaluate at points and take their derivatives. Another main focus is error analysis, where we test how good our approximations are. The method will always produce approximations, but we use residual errors on the domain of the problem to find a measure of error. In the last two chapters, we apply similarity transforms to PDEs to transform them into ODEs. We then use the Homotopy Analysis Method on one, but are able to find exact solutions to both equations.

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(Page numbers are given in brackets after sub-headings.) Chapter 1 :-Application of operational methods of solution to ordinary differential equations with constant coefficients. Motion of helical springs --- Free vibration ( p.2 ) --- Forced vibration ( p.7 ) --- Resonance ( p.10 ) ----- Linked springs ( p.12 ) --- Analogous electrical problems ( P. 15 ). Appendix A :-Laplace Transformations. ( p.18 ) Appendix B :- Brief explanation of the operational methods of solution used in the chapter. ( p.19 ) Appendix C :- Example of partial fraction working. ( p.21a) Chapter 2 :- Application of Fourier Series and of operational methods of solution to partial differential equations. Small transverse vibrations of a light elastic string ( p.22 ) Vibration of a string initially plucked at its middle point ( p.28 ) --- Alternative method of solution of the preceeding problem by operational methods ( p.31 ) --- Discussion of the shape assumed by the string at any time ( p.36 ) Chapter 3 :- Application of Bessel Functions to the solution of partial differential equations of the second order. Small vibrations of a light circular membrane ( p.39 ) Appendix A :- Transformation of the differential equation of motion from Cartesian to cylindrical coordinates. ( p.46 ). Appendix B :- Note on roots of Bessel Functions. ( p.47 ) Appendix C :- Bessel Functions - Some important results applied in the chapter. p.48 ). Chapter 4 :- Application of Legendre Polynomials to the solution of partial differential equations of the second order. Conduction of heat in a solid body ( p.50 ) ---------- Flow of heat in a solid sphere ( p.53 ). Appendix :-Statement of the Divergence Theorem used in the chapter. ( p.61 ). Chapter 5 :- Example of Simultaneous Partial Differential Equations. The flow of electricity in a long imperfectly insulated cable ( p.62 ) ---- Analogous problem of the flow of heat in rod of small uniform cross section ( p.69 ). / Thesis / Master of Arts (MA)

This dissertation is concerned with analysis of orbital stability of solitary waves and well-posedness of the Cauchy problem in the integrable evolution equations. The analysis is developed by using tools from integrable systems, such as higher-order conserved quantities, B\"{a}cklund transformation, and inverse scattering transform. The main results are obtained for the massive Thirring model, which is an integrable nonlinear Dirac equation, and for the derivative NLS equation. Both equations are related with the same Kaup-Newell spectral problem. Our studies rely on the spectral properties of the Kaup-Newell spectral problem, which convey key information about solution behavior of the nonlinear evolution equations. / Dissertation / Doctor of Philosophy (PhD)