构造孤子方程的Weierstrass椭圆函数解的一个新方法

利用具有Weierstrass椭圆函数解的方程，首先获得了投影Riccati方程的两组新解.由于投影Riccati方程可用于多种具孤子解的非线性演化方程的求解，因而得到了一个可以构造这些方程的Weierstrass椭圆函数解的新方法. 关键词： Weierstrass椭圆函数解 投影Riccati方程 非线性演化方程     

An action for a classical string,the equation of motion and group invariant classical solutions

A string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.      

A Series of Soliton-like and Double-like Periodic Solutions of a (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Vesselov Equation

We generalize the algebraic method presented by Fan [J. Phys. A: Math. Gen. 36 (2003) 7009)] to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations(NPDE). As an application of the method, we choose a (2 1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and successfully construct new and more general solutions including a series of nontraveling wave and coefficient functions‘ soliton-like solutions, double-like periodic and trigonometric-like function solutions.     

Measures of Simultaneous Approximation for Quasi-Periods of Abelian Varieties

We examine various extensions of a series of theorems proved by Chudnovsky in the 1980s on the algebraic independence (transcendence degree 2) of certain quantities involving integrals of the first and second kind on elliptic curves; these extensions include generalizations to abelian varieties of arbitrary dimensions, quantitative refinements in terms of measures of simultaneous approximation, as well as some attempt at unifying the aforementioned theorems. In the process we develop tools that might prove useful in other contexts, revolving around explicit “algebraic” theta functions on the one hand, and Eisenstein's theorem and G-functions on the other hand.     

Time-optimal control problem for parabolic equations with control constraints and infinite number of variables

A time optimal control problem for parabolic equations withan infinite number of variables is considered. A time optimalcontrol problem is replaced by an equivalent one with a performanceindex in the form of integral form. Constraints on controlsare assumed. To obtain the optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak (1984, Acta Universitatis LodziensisFolia Mathematica, 187–196; 1984, J. Optim. Theor. Appl.,42, 561–582) was applied.     

Some generalizations of Bessel and Flett potentials associated to the Laplace–Bessel differential operator

We introduce the notion of bi-parametric potential-type operators, which generalize the Bessel and Flett potentials associated to singular Laplace–Bessel differential operator. Explicit inversion formulas for them are obtained by using a special wavelet-like transform, generated by the so-called ‘modified beta-semigroup’.     

Weierstrass Multiplicative Points on a Compact Riemann Surface

We introduce Weierstrass multiplicative points and develop the theory of Weierstrass multiplicative points for multiplicative meromorphic functions and Prym differentials on a compact Riemann surface. We prove some analogs of the Weierstrass and Noether theorems on the gaps of multiplicative functions. We obtain two-sided estimates for the number of Weierstrass multiplicative points and q-points. We propose a method for studying the Weierstrass and Noether gaps and Weierstrass multiplicative points by means of filtrations in the Jacobi variety of a compact Riemann surface.     

Marginally Trapped Surfaces in L<Superscript>4</Superscript> and an Extended Weierstrass–Bryant Representation

Let E be the total space of a locally trivial torus bundle over a surface Σ_g of genus g > 1. Using Seiberg–Witten theory and spectral sequences, we prove that E carries a symplectic structure if and only if the homology class of the fiber [T²] is nonzero in H₂(E, R). Mathematics Subject Classifications (2000): 53C42, 53A10.     

Surfaces in Three-Dimensional Lie Groups

We derive the Weierstrass (or spinor) representation for surfaces in the three-dimensional Lie groups Nil, \(\widetilde{SL}_2\), and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. Using the spectral properties of the corresponding Dirac operators, we find analogs of the Willmore functional for surfaces in these geometries.