Summability in of a vector measure

We show a picture of the relations among different types of summability of series in the space  of integrable functions with respect to a vector measure m of relatively norm compact range. In order to do that, we study the class of the so‐called m‐1‐summing operators. We give several applications regarding the existence of copies of c ₀ in , as well as on m‐1‐summing operators which are weakly compact, Asplund or weakly precompact.     

Pseudo–almost‐periodic solutions for delayed differential equations with integrable dichotomies and bi–almost‐periodic Green functions

Using the existence of integrable bi–almost‐periodic Green functions of linear homogeneous differential equations and the contraction fixed point, we are able to prove the existence of almost and pseudo–almost‐periodic mild solutions under quite general hypotheses for the differential equation with constant delay in a Banach space X, where τ>0 is a fixed constant. The results extend the corresponding ones in the case of exponential dichotomy. Some examples illustrate the importance of the concepts.     

Limit circle invariance for two differential systems on time scales

In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter λ. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value λ₀, then this property is preserved for all complex values λ. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non‐hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors.     

Quantum integrable models of interacting bosons and classical r-matrices with spectral parameters

Using the technique of classical $r$-matrices with spectral parameters we construct a general form of quantum Lax operators of interacting boson systems corresponding to an arbitrary simple (or reductive) Lie algebra. We prove quantum integrability of these models in the physically important case of $\mathfrak{g}=gl\left(n\right)$ and “diagonal” in the root basis classical $r$-matrices. We consider in detail two classes of non-skew-symmetric classical $r$-matrices with spectral parameters and obtain the corresponding quantum Lax operators and quantum integrable many-boson hamiltonians that generalize Bose–Hubbard dimer hamiltonians.     

STUDIES OF MELNIKOV METHOD AND TRANSVERSAL HOMOCLINIC ORBITS IN THE CIRCULAR PLANAR RESTRICTED THREE－BODY PROBLEM

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Integrable boundary conditions for the B_2 model

New integrable B_2 model with off-diagonal boundary reflections is proposed. The general solutions of the reflection matrix for the B_2 model are obtained by using the fusion technique. Wefind that the reflection matrix has 7 free boundary parameters, which are used to describe the degree of freedom of boundary couplings, without breaking the integrability of the system. The new quantization conditions will induce the novel structure of the energy spectrum and the boundary states. The corresponding boundary effects can be studied based on the results in this paper. Meanwhile, the reflection matrix of high rank models associated with Bnalgebra can also be obtained by using the method suggested in this paper.     

Lump solutions of the 2D Toda equation

In this research, the lump solution, which is rationally localized and decays along the directions of space variables, of a 2D Toda equation is studied. The effective method of constructing the lump solutions of this 2D Toda equation is derived, and the constraint conditions that make the lump solutions analytical and positive are obtained as well. Finally, three classes of lump solutions are constructed, 3D plots, density plots, and contour plots are given to illustrate this proposed method.     

玻色化方法在超对称可积系统的应用

利用玻色化方法可以避免超对称可积系统中反对易费米场带来的计算困难. 本文以N=1超对称mKdVB系统为例, 利用玻色化方法, 将其转化为只有玻色场的耦合系统. 应用标准的WTC方法, 证明了该耦合系统具有Painlevé性质. 运用Painlevé截断方法, 可以得到玻色化后超对称mKdVB系统的非局域对称. 为了求解与非局域对称相关的Lie第一性原理, 引入新的场将玻色化后系统拓展为更大的系统. 通过引入新的场, 该非局域对称局域化为Lie点对称. 因此, 可以利用Lie点对称约化方法研究拓展后的系统, 得到超对称mKdVB系统的孤子与其他孤波相互作用解.     

通过Painlevé分析从低维模型中寻找高维可积模型

将Painlevé方法推广到更一般的形式, 可以从给定的低维可积模型中得到无穷多个新的可积模型. 新的可积模型与原模型相比都是较高维的, 它们保持保角不变性和Painlevé性质. 本文主要以KdV、NLS和KP方程为例, 运用WTC法、截断展开、领头项分析等方法, 给出了(3+1)维可积模型的具体形式.