This thesis contains two parts. In the first part, we discuss certain class of KP solitons in connections with singular projective curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\ldots, lm+k\rangle$ where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions, then we check that quasi-periodic solutions of $l$-th generalized KdV hierarchy indeed degenerate to soliton solutions we begin with when we degenerate the underlying algebraic curve and the line bundle over it properly. For these KP solitons, we also construct the space curves from commutative rings of differential operators in the sense of the well-known Burchnall-Chaundy theory. This part is mainly based on a published paper \cite{Kodama-Xie2021KP}. In the second part, we discuss solutions of the full Kostant-Toda (f-KT) lattice and their connections with the flag varieties. Firstly, we carry out Kowalevski-Painlev\'e analysis for f-KT equation. In particular, we associate each solution of the indicial equations with a Weyl group element, provide explicit formulas for eigenvalues of Kowalevski matrix and at last parameterize all the Laurent series solutions by $\mathcal{G} \slash \mathcal{B} \times \mathbb{C}^n$ where $\mathcal{G} \slash \mathcal{B}$ is the flag variety and $\mathbb{C}^n$ represents the spectral parameters. Secondly, we use iso-spectral deformation theory to study f-KT in the Hessenberg form, and give explicit form of the wave functions and entries in the Lax matrix expressed by $\tau$-functions with which we study $\ell$-banded Kostant-Toda hierarchy. W (open full item for complete abstract)

In this dissertation we studied nonlinear partial differential equations in two different directions. We apply the bifurcation theory to investigate a number of positive solutions of the semilinear Dirichlet boundary value problem on a n-dimensional ball for the second order elliptic equation with periodic nonlinearity containing a positive parameter. Our approach appeals to the well known results of B. Gidas, W.-M. Ni, L. Nirenberg, the bifurcation theorems of M. G. Crandall and P. H. Rabinowitz, and the stationary phase method. Further, we investigate the issue of global existence of the solutions of the Cauchy problem for the semilinear Tricomi-type equations, appearing in the boundary value problems problems of gas dynamics. We study Cauchy problem trough integral equation and give some sufficient conditions for the existence of the global weak solutions. We prove necessity of these conditions. We obtain necessary condition for the existence of the self-similar solutions for the semilinear Tricomi-type equation. In our approach we employ the fundamental solution and the Lp-Lq estimates for the linear Tricomi-type equations.

Drilled shafts socketed into rock are widely used as foundations for bridges and other important structures. Rock-socketed drilled shafts are also used to stabilize a landslide. The main loads applied on the drilled shafts are axial compressive or uplift loads as well as lateral loads with accompanying moments. Although there exist several analysis and design methods especially for rock-socketed drilled shafts under lateral loading, these methods were developed with assumptions without actual validations with field load test results. Some of the methods have been found to provide unsafe designs when compared to recently available field test data. Therefore, there is a need to develop a more rational design approach for laterally loaded drilled shafts socketed in rock. A hyperbolic non-linear p-y criterion for rock is developed in this study that can be used in conjunction with existing computer programs, such as COM624P, LPILE, and FBPIER, to predict the deflection, moment, and shear responses of a shaft under the applied lateral loads. Considerations for the effects of joints and discontinuities on the rock mass modulus and strength are included in the p-y criterion. Evaluations based on comparisons between the predicted and measured responses of full-scale lateral load tests on fully instrumented drilled shafts have shown the applicability of the proposed p-y criterion and the associated methods for determining the required input of rock parameters. In addition to the development of a hyperbolic p-y criterion for rock, a method for predicting lateral capacities of drilled shafts in rock and/or soils is developed for assessing the safety margin of the designed shafts against the design loads. A computer program LCPILE is developed using VC++ to facilitate computations. An elastic solution based on a variational approach is also developed for determining drilled shaft elastic deflection due to applied lateral loads in a two-layer soil layer system. The computational (open full item for complete abstract)