AN INTEGRABLE HIERARCHY AND ITS EXPANDING LAX INTEGRABLE MODEL

In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy possessing bi-Hamiltonian structure is obtained by choosing V with derivatives in x and spectral potentials. Then integrable coupling, i.e. expanding Lax integrable model of the hierarchy obtained is presented by constructing a subalgebra of loop algebra A2.     

一族可积Hamilton方程

本文利用屠规彰格式,导出了一族新的可积系,包含４个未知函数,具有双Ｈａｍｉｌｔｏｎ结构,且以ＴＣ族为特例。     

WEAK CONVERGENCE OF HENSTOCK INTEGRABLE SEQUENCES

Some relationships between pointwise and weak convergence of a sequence of Henstock integrable functions are studied, In particular it is provided an example of a sequence of Henstock integrable functions whose pointwise limit is different from the weak one. By introducing an asymptotic version of the Henstock equiintegrability notion it is given a necessary and sufficient condition in order that a pointwisely convergent sequence of Henstock integrable functions is weakly convergent to its pointwise limit.     

（2＋1）-DIMENSIONAL Tu HIERARCHY AND ITS INTEGRABLE COUPLINGS AS WELL AS THE MULTI-COMPONENT INTEGRABLE HIERARCHY

Under the frame of the(2 1)-dimensional zero curvature equation and Tu model,(2 1)-dimensional Tu hierarchy is obtained.Again by employing a subalgebra of the loop algebra ■_2 the integrable coupling system of the above hierarchy is presented.Finally,A multi-component integrable hierarchy is ob- tained by employing a multi-component loop algebra ■_M.     

关于一个可积的广义Hamilton方程族

本文利用ｒ－矩阵生成了一个广义的Ｈａｍｉｌｔｏｎ方程族，并证明了它是广义可积的，然后讨论了它和（４）中Ｌｉｏｖｖｉｌｌｅ可积的新的广义Ｈａｍｉｌｔｏｎ方程族之间的关系。     

一类可积的广义Duffing方程

本文应用李群方法,通过求解Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程,得出了一类广义Ｄｕｆｆｉｎｇ方程的通解。     

DIAGONAL BOUNDARY K-MATRICES FOR TWO INTEGRABLE NINETEEN-VERTEX MODELS

1991MRSubjectClassification82DRecentlythere11asbeenaconsiderableinterestinthestudyofthelatticeintegrable1llodelswithopenBC[1--4].Inarecelltpapel['],acompletelistofintegrablenineteen-vertexluodelswaspresellted.Thelistcolilprisesseveralwell--knowllmodels[6]andfournewoiles.The1'0firnewonesweresolvedbyKlulllpcr[7].alldtheBCforthenewoileswerestudio(lillRef.[8].Illthepresentpaper,wewillstlldytheBCforanothertwomodelsillthelist,whichillcludethe0liesinRef.Ig-10]respectively.BysolvingtheRE,three…     

AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS

Within the zero curvature formulation, a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type. The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity. When the involved two potential vectors are scalar, all the resulting integrable lattice equations are reduced to the standard AblowitzLadik hierarchy.     

AN INTEGRABLE COUPLING OF THE GENERALIZED AKNS HIERARCHY

A direct method for establishing integrable couplings is proposed in this paper by constructing a new loop algebra G. As an illustration by example, an integrable coupling of the generalized AKNS hierarchy is given. Furthermore, as a reduction of the generalized AKNS hierarchy, an integrable coupling of the well-known G J hierarchy is presented. Again a simple example for the integrable coupling of the MKdV equation is also given. This method can be used generally.     

INTEGRABLE COUPLING OF THE TC HIERARCHY OF EQUATIONS

A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the integrable coupling of the famous KdV equation is presented.     

TWO CLASSES OF INTEGRABLE COUPLING FOR AKNS HIERARCHY

A set of new matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A 2M . Then we use the idea of enlarging spectral problems to make an enlarged spectral problems. It follows that the multi-component AKNS hierarchy is presented. Further, two classes of integrable coupling of the AKNS hierarchy are obtained by enlarging spectral problems.     

BIFURCATION OF CUBIC INTEGRABLE SYSTEM UNDER CUBIC PERTURBATION

In this paper, we consider the bifurcation for a class of cubic integrable system under cubic perturbation. Using bifurcation theory and qualitative analysis, we obtain a complete bifurcation diagram of the system in a neighbourhood of the origin for parameter plane.     

THE COMPLETENESS PROBLEM IN SPACES OF PETTIS INTEGRABLE FUNCTIONS

Abstract

Two subspaces of the space of Banach space valued Pettis integrable functions are considered: the space P(μ, X, var) of Pettis integrable functions with integrals of finite variation in a Banach space X and LLN(μ,X,var), the space of functions satisfying the law of large numbers. It is proved that LLN(μ,X*,var) is always complete and P(μ, X*,var) is complete if Martin's axiom and the perfectness of μ are assumed. Moreover, a non-trivial example of a non-conjugate Banach space X with non-complete P(μ, X, var) is presented.     

一类非线性演化方程族的可积耦合

Starting from a Tu Guizhang‘s isospectral‘problem, a Lax pair is obtained by means of Tu scheme ( we call it Tu Lax pair ). By applying a gauge transformation between matrices, the Tu Lax pair is changed to its equivalent Lax pair with the traces of spectral matrices being zero, whose compatibility gives rise to a type of Tu hierarchy of equations. By making use of a high order loop algebra constructed by us, an integrable coupling system of the Tu hierarchy of equations are presented. Especially, as reduction cases, the integrable couplings of the celebrated AKNS hierarchy, TD hierarchy and Levi hierarchy are given at the same time.     

EFFECTIVE STABILITY FOR NEARLY INTEGRABLE MAPPINGS WITH INTERSECTION PROPERTY

We consider a nearly integrable mapping with a fixed twist frequency. Under the intersection property and Diophantine condition we prove that the mapping possesses effective stability. In particular, the mapping considered may have different dimensions of action-angle variables in our case.     

一个新的无限维Liouville可积系及其变换

本文讨论了一个新的等谱特征问题，按屠规彰格式导出了相应的Ｌａｘ可积的非线性发展方程族。证明了它是Ｌｉｏｕｖｉｌｌｅ可积系，并且讨论了谱参数改变符号时相应的可积系的变化情况。     

EMBEDDING C0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract

We show that the normed space of μ-measurable Pettis integrable functions on a probability space with values in a Banach space X contains a copy of the sequence space c₀ if and only if X contains a copy of c₀. In this case, if the probability μ has infinite range, a copy of c₀ consisting of μ-measurable functions can be found, such that it is complemented in the bigger space of all weakly μ-measurable Pettis integrable functions.