This dissertation contains two completely independent parts. In Part 1, I investigate effective field theories and their applications in lattice gauge theory. Quantum chromodynamics (QCD) as a part of the standard model (SM) describes the physics of quarks and gluons. There are several numerical and analytical methods to tackle the QCD problems. Lattice QCD is the dominant numerical method. Effective field theories, on the other hand, provide analytic methods to describe the low-energy dynamics of QCD. To use the effective theories in lattice QCD, I develop chiral perturbation theory for heavy-light mesons with staggered quarks---an implementation of fermions on lattice. I use this effective chiral theory to study the pattern of taste splitting in masses of the mesons with staggered quarks. I also calculate the leptonic decay constant of the heavy-light mesons with staggered quarks to one-loop order in the chiral expansion. The resulting chiral formula provides a suitable fit form to combine and analyze a large number of decay constants of heavy-light mesons computed from different lattice ensembles with various choices of input parameters. I perform a comprehensive chiral fit to the lattice data for D mesons computed by the MILC collaboration. Consequently, I determine the physical values of the decay constants of D mesons. These precise results place narrow restrictions on the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. In Part 2, I introduce the concept of a nonlinear eigenvalue problem by investigating three nonlinear differential equations. First, equation y'(x) = cos[[pi]xy(x)] is investigated. A discrete set of initial conditions y(0) = a[subscript n], leading to unstable separatrix behavior, are identified as the eigenvalues of the problem. I calculate the asymptotic behavior of the initial conditions a[subscript n] and their corresponding solutions for large n by reducing the equation to a linear one-dimensional random-walk problem. Second, I investigate equation y''(x) = 6[y(x)]2+x, whose solutions are called the first Painlevé transcendent. I calculate different types of critical initial conditions that give rise to separatrix solutions for this equation. I work out the asymptotic behaviors of the initial conditions by reducing the problem to a linear Schrödinger equation. Finally, I investigate the second Painlevé transcendent, corresponding to equation y''(x) = 2[y(x)]3 + xy(x). I find that this equation exhibits patterns similar to the first Painlevé equation.