Abstract:
With the help of the generalized Jacobi elliptic function, an improved Jacobi elliptic function method is used to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation are investigated, and the exact solutions are derived with the aid of the homogenous balance principle.

Abstract:
In this paper, the author discusses the elliptic type gradient estimate for the solution of the time-dependent Schr\"{o}dinger equations on noncompact manifolds. As its application, the dimension-free Harnack inequality and the Liouville type theorem for the Schr\"{o}dinger equation are proved.

Abstract:
We prove the existence of positive solutions with optimal local regularity to homogeneous elliptic equations of Schr\"{o}dinger type, under only a form boundedness assumption on $\sigma \in D'(\Omega)$ and ellipticity assumption on $\mathcal{A}\in L^\infty(\Omega)^{n\times n}$, for an arbitrary open set $\Omega\subseteq \mathbf{R}^n$. We demonstrate that there is a two way correspondence between the form boundedness and the existence of positive solutions to this equation, as well as weak solutions to certain elliptic equations with quadratic nonlinearity in the gradient. As a consequence, we obtain necessary and sufficient conditions for both the form-boundedness (with a sharp upper form bound) and the positivity of the quadratic form of the Schr\"{o}dinger type operator with arbitrary distributional potential $\sigma \in D'(\Omega)$, and give examples clarifying the relationship between these two properties.

Abstract:
It is shown how the time-dependent Schr\"{o}dinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schr\"{o}dinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an {\it a priori} assumption equivalent to Feynman's dynamical postulate. De Broglie's concepts of 'matter waves' and their phase and group velocities are also critically discussed.

Abstract:
We announce three results in the theory of Jacobi matrices and Schr\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\"odinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szeg\H{o} asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.

Abstract:
We prove that Jacobi, CMV, and Schr\"odinger operators, which are reflectionless on a homogeneous set E (in the sense of Carleson), under the assumption of a Blaschke-type condition on their discrete spectra accumulating at E, have purely absolutely continuous spectrum on E.

Abstract:
In this paper, we use two integral methods, the first integral method and the direct integral method to study a higher-order nonlinear Schr dinger equation (NLSE). The application of the first integral method yield trigonometric function solutions and solitary wave solutions. Using the direct integration lead to shock wave solution and Jacobi elliptic function solutions. The direct integral method is more concise and direct than the first integral method.

Abstract:
The coupled nonlinear Schr\"{o}dinger equations (CNLSEs) of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.

Abstract:
摘要 研究弹性细杆静力学的薛定谔粒子波动比拟。类似于Kirchhoff动力学比拟, 依据弹性细杆曲率平衡微分方程与一维定态非线性薛定谔方程数学形式的相似性, 给出两者的动力学比拟关系, 称为Schr？dinger粒子波动比拟。基于比拟关系, 给出弹性细杆方程的Jacobi椭圆函数解, 并画出此解所描述的弹性细杆的空间位形。Schr？dinger粒子波动比拟建立了波函数的量子态与弹性细杆的几何构型的对应关系,给予波函数的量子态直观的几何图像, 为弹性细杆方程的求解提供了新的途径。 Abstract The Schr？dinger analogy of thin elatic rod is studied. Compared with the Kirchhoff dynamics analogy, the Schr？dinger analogy is proposed. By the new analogy, the Kirchhoff equation of elastic rod can be written as curvature equation which is similar to nonlinear Schr？dinger equation. Thus, the Jacobi elliptic function solution of Schr？dinger equation can be taken into elastic rod equation. The space configurations of the elastic rod described by the solution are given. Schr？dinger analogy reveals the relations between the quantum state of wave function and the geometry configuration of elastic rod, and gives a new way to solve the Kirchhoff equation.

Abstract:
Two "elliptic analogues'' of the nonlinear Schr\"odinger hiererchy are constructed, and their status in the Grassmannian perspective of soliton equations is elucidated. In addition to the usual fields $u,v$, these elliptic analogues have new dynamical variables called ``Tyurin parameters,'' which are connected with a family of vector bundles over the elliptic curve in consideration. The zero-curvature equations of these systems are formulated by a sequence of $2 \times 2$ matrices $A_n(z)$, $n = 1,2,...$, of elliptic functions. In addition to a fixed pole at $z = 0$, these matrices have several extra poles. Tyurin parameters consist of the coordinates of those poles and some additional parameters that describe the structure of $A_n(z)$'s. Two distinct solutions of the auxiliary linear equations are constructed, and shown to form a Riemann-Hilbert pair with degeneration points. The Riemann-Hilbert pair is used to define a mapping to an infinite dimensional Grassmann variety. The elliptic analogues of the nonlinear Schr\"odinger hierarchy are thereby mapped to a simple dynamical system on a special subset of the Grassmann variety.