In this thesis, we investigate the Strominger system on non-Kähler manifolds. We will present a natural generalization of the Strominger system for non-Kähler hermitian manifolds M with c₁(M) = 0. These manifolds are more general than balanced hermitian manifolds with holomorphically trivial canonical bundles. We will then consider explicit examples when M can be realized as a principal torus fibration over a Kähler surface S. We will solve the Strominger system on such construction which also includes manifolds of topology (k−1)(S²×S⁴)#k(S³×S³). We will investigate the anomaly cancellation condition on the principal torus fibration M. The anomaly cancellation condition reduces to a complex Monge-Ampère-type PDE, and we will prove existence of solution following Yau’s proof of the Calabi-conjecture [Yau78], and Fu and Yau’s analysis [FY08]. Finally, we will discuss the physical aspects of our work. We will discuss the Strominger system using α'-expansion and present a solution up to (α')¹-order. In the α'-expansion approach on a principal torus fibration, we will show that solving the anomaly cancellation condition in topology is necessary and sufficient to solving it analytically. We will discuss the potential problems with α'-expansion approach and consider the full Strominger system with the Hull connection. We will show that the α'-expansion does not correctly capture the behaviour of the solution even up to (α')¹-order and should be used with caution.

In Chapter 2 we study the uniqueness problem of sign-changing solutions for a nonlinear scalar equation. It is well-known that positive solution is radially symmetric and unique up to a translation. Recently, there are many works on the existence and multiplicity of sign-changing solutions. However much less is known for uniqueness, even in the radially symmetric class. In Chapter 2, we solve this problem for nearly critical nonlinearity by Lyaponov-Schmidt reduction. Moreover, we can also prove the non-degeneracy. / In Chapter 3 we are concerned with the uniqueness problem for coupled nonlinear Schrodinger equations. The problem is to classify all positive solutions. In Chapter 3, some sufficient conditions are given. In particular, we have a sufficient and necessary condition in one dimension. The proof is elementary because only the implicit function theorem, integration by parts, and the uniqueness for scalar equation are needed. / In Chapter 4 we go back to the nonlinear scalar equation and consider the traveling wave solutions. Using an infinite dimensional Lyaponov-Schmidt reduction, new examples of traveling wave solutions are constructed. Our approach explains the difference between two dimension and higher dimensions, and also explores a connection between moving fronts and the mean curvature flow. This is the first such traveling waves connecting the same states. / This thesis is devoted to the study of nonlinear elliptic equations and systems. It is divided into two parts. In the first part, we study the uniqueness problem, and in the second part, we are concerns with traveling wave solutions. / Yao, Wei. / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 132-142). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Wong, Chun Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 133-135). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

In this master thesis, it is proposed to solve in the large thedifferential equationZ2(d2y/dz2) + z (dy/dz) (b0+b1zm) + (c0+c1zm)y = 0Here, m is an arbitrary positive integer, the variable z is complex as are the constants bi, ci (i = 0, 1). It is also assumed that the roots of the indicial equation about the regular singular point z=0 are such that their difference is incongruent to zero modulo m.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1927. / "Sonderabdruck aus der "Mathematischen Zeitschrift", Band 27, Heft 1"--T.p. verso. Vita. Includes bibliographical references.

Thesis (Ph. D.)--Kent State University, 2008. / Title from PDF t.p. (viewed Nov. 10, 2009). Advisor: Hassan Allouba. Keywords: stochastic partial differential equations; compact support property. Includes bibliographical references (p. 30).

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Thesis (Ph. D.)--West Virginia University, 2009. / Title from document title page. Document formatted into pages; contains v, 58 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 56-58).