基于Bell多项式求解非线性发展方程Hirota双线性形式的新算法

基于Bell多项式,构造了获得非线性发展方程双线性形式的一个新算法,并且开发了相应的程序包.非线性发展方程与其双线性形式之间的变换可由该程序包自动推导,同时给出了一些具有代表性的实例验证了该程序包的有效性与可靠性.     

用Hirota双线性导数变换求MNLS方程的Rogue波解

修正的非线性薛定谔方程(MNLS方程)与导数非线性薛定谔方程(DNLS方程)是两个紧密相关且完全可积的非线性偏微分方程.该文通过Hirota双线性导数变换方法,首先求得MNLS方程在平面简谐波背景下的空间周期解,即Akhmediev型呼吸子解,再通过长波极限得其Rogue波解.根据简单的参数归零法使之自然地约化为DNLS方程的Rogue波解,并借助于一个积分变换将其变换为Chen-Lee-Liu方程的Rogue波解.文章还简要讨论了MNLS方程和DNLS方程在非局域情形整体解的存在性问题.     

N=2 超对称KdV 方程的双线性形研究

利用双线性方法研究$N=2$超对称KdV方程. 通过适当的相关变换, 将$N=2, a=4$和$N=2, a=1$超对称KdV方程转化成双线性形式, 由此构造了相应方程的解. 对于$N=2, a=1$ 超对称KdV方程, 还得到了它的双线性B\"acklund变换和Lax 表示.     

与二阶多项式谱问题相联系的非等谱流,对称和李代数

与二阶多项式谱问题相联系的非等谱流，对称和李代数陈登远，曾云波（中国科学技术大学数学系，合肥２３００２７）ＮＯＮ－ＩＳＯＳＰＥＣＴＲＡＬＦＬＯＷＳ，ＳＹＭＭＥＴＲＩＥＳＡＮＤＬＩＥＡＬＧＥＢＲＡＡＳＳＯＣＩＡＴＥＤＷＩＴＨＴＨＥＳＥＣＯＮＤ－ＯＲＤＥ...     

Zakharov方程组与Hirota方程的同宿轨道

利用Hirota方法得到Zakharov方程组和Hirota方程同宿轨道的解析表达式.     

Hirota型非线性发展方程的整体光滑解

本文研究如下Hirota型非线性发展方程 i?_tφ+α?_x²φ+iβ?_x²φ+iγ?_x(|φ|²φ)+δ|φ|²φ=0,Cauchy问题整体光滑解的存在性,唯一性及解的收敛性与衰减估计.     

（2＋1）维非线性Schroedinger型方程的同宿轨道

研究了几类（2＋1）维非线性Schroedinger型方程同宿轨道的问题．利用Hirota双线性算子方法,通过给出的相关变换,得到了包括（2＋1）维的长短波相互作用方程,广义Zakharov方程,Mel’nikov方程和g-Schroedinger方程的同宿轨道解的显式解析表达式,从而讨论了这些方程的同宿轨道．     

用Hirota双线性方法构造一种(3+1)维高维孤子方程的多孤子解

通过两种方法构造了一种(3+1)维高维孤子方程的孤子解.第一种方法是利用对数函数变换,将其化成双线性形式的方程,在用级数扰动法求解双线性方程的单孤子解、双孤子解和N-孤子解.第二种方法是用广义有理多项式与试探法相结合,构造了(3+1)维高维孤子方程的怪波解.     

双线性型图与交错型图不是完美图

本文证明了双线性型图与交错型图都不是完美图,从而解决了双线性型图与交错型图的完美图判别问题.     

Construction of Symmetric and Symplectic Bilinear Forms

Let V be a finite-dimensional vector-space. A linear mapping on V is called simple if V( - 1) is 1-dimensional. Let S be a set of simple bijections on V. We discuss conditions entraining that each element of S is orthogonal (respectively symplectic) under an appropriate symmetric (respectively symplectic) bilinear form on V.     

Exact Estimates for Moments of Random Bilinear Forms

The present paper concentrates on the analogues of Rosenthal's inequalities for ordinary and decoupled bilinear forms in symmetric random variables. More specifically, we prove the exact moment inequalities for these objects in terms of moments of their individual components. As a corollary of these results we obtain the explicit expressions for the best constant in the analogues of Rosenthal's inequality for ordinary and decoupled bilinear forms in identically distributed symmetric random variables in the case of the fixed number of random variables.     

紧 n -Lie 代数与不变双线性型

该文研究 n -Lie代数的非退化不变双线性型. 给出了实数域上紧 n -Lie 代数的概念, 并对其结构进行了研究.     

第三类Painlevé方程(δ=0或γ=0)解的渐近性态

对带有一般实参数第三类Painlevé方程,已有γ<0,δ>0时,解的有界性以及振荡渐近解的表达形式的结论.在本文中,我们给出当δ=0或γ=0时其振荡渐近解的表达形式.     

Bell多项式在经典Boussinesq方程中的应用

本文借助于Bell多项式研究经典Boussinesq方程,将其转换成Hirota双线性形式,构造了带参数的B(a)cklund变换,进而重新导出了其Lax表示.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).     

Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

In ([11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]), we have studied quadratic Leibniz algebras that are Leibniz algebras endowed with symmetric, nondegenerate, and associative (or invariant) bilinear forms. The nonanticommutativity of the Leibniz product gives rise to other types of invariance for a bilinear form defined on a Leibniz algebra: the left invariance, the right invariance. In this article, we study the structure of Leibniz algebras endowed with nondegenerate, symmetric, and left (resp. right) invariant bilinear forms. In particular, the existence of such a bilinear form on a Leibniz algebra 𝔏 gives rise to a new algebra structure ☆ on the underlying vector space 𝔏. In this article, we study this new algebra, and we give information on the structure of this type of algebras by using some extensions introduced in [11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]. In particular, we improve the results obtained in [22 Lin, J., Chen, Z. (2010). Leibniz algebras with pseudo-Riemannian bilinear forms. Front. Math. China 5(1):103–115.[Crossref], [Web of Science ®] , [Google Scholar]].     

一类变系数KdV方程的Painlevé分析和自Bcklund变换

利用符号计算对一类系数函数是x和t的函数的变系数K dV方程进行了Pa in levé分析,得到了该方程具有Pa in levé性质时系数函数必须满足的约束条件.利用Pa in levé截断法给出了该方程的一个自B ck lund变换,作为例子根据得到的自B ck lund变换给出了两组精确解.     

Quadratic and Symmetric Bilinear Compositions of Quadratic Forms Over Commutative Rings

Over commutative rings in which 2 is a zero-divisor, to compose a quadratic form with symmetric bilinear forms or with quadratic forms is not quite the same. In this article, the relation between the two classes of compositions is clarified and the results applied to find the ranks of minimal compositions.