Narrow Results

The historical and geometrical origin of Gauge Transformation and Yang's phase loop of gauge theory are discussed. In the present talk, we present the following points: 1. Parallel transport of a vector; 2. Weyl 1918 gauge transformation 3. Concept of non-integrable phase factor; 4. Berry's quantum geometrical phase; 5. Parallel transport of quantum state vector produces the phase physics.

We examine in terms of exact solutions of the time-dependent Schrödinger equation, the quantum tunnelling process in Bose–Einstein condensates of two interacting species trapped in a double well configuration. Based on the two series of time-dependent SU(2) gauge transformations, we diagonalize the Hamilton operator and obtain analytic time-evolution formulas of the population imbalance and the berry phase. The particle population imbalance of species A between the two wells is studied analytically.

In the shallow-water studies, the Whitham–Broer–Kaup (WBK) system can be used to describe the propagation of the long waves. In this paper, based on the Lax pair of the WBK system, we derive the gauge transformation from the WBK system to the Ablowitz–Kaup–Newell–Segur (AKNS) system with the help of symbolic computation. Applying the Darboux transformation of the AKNS system, we obtain some soliton solutions of the WBK system. Those results might be useful in the investigations on the propagation of solitons in such situation as shallow water.

In this paper, the gauge transformation of the constrained semi-discrete KP(cdKP) hierarchy is constructed explicitly by the suitable choice of the generating functions. Under the m-step successive gauge transformation T_m, we give the transformed (adjoint) eigenfunctions and the τ-function of the transformed Lax operator of the cdKP hierarchy.

We point out that the Darboux transformation in the paper “Analytic solutions and Darboux transformation to a new Hamiltonian lattice hierarchy” [Mod. Phys. Lett. B30 (2016) 1650100] is not right, and present a correct Darboux transformation.

In this paper, we give the definition and multi-folds gauge transformations of the multi-component B-type Kakomtsev–Petviashvili (MBKP) hierarchy and multi-component CKP (MCKP) hierarchy. Besides, we derive new solutions after the one-fold gauge transformation of the MBKP hierarchy. The remarkable differences between the BKP (CKP) hierarchy and the MBKP(MCKP) hierarchy are given in this paper. What is more, we introduce some facts about the multi-component constrained BKP and multi-component constrained CKP hierarchies including some obstacles and some new multi-component equations.

In this paper, we mainly study the gauge transformations of the constrained q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy. Different from the usual case, we have to consider the additional constraints on the Lax operator of the constrained q-deformed KP hierarchy, since the form of the Lax operator must be kept when constructing the gauge transformations. For this reason, the selections of generating functions in elementary gauge transformation operators $T_D$ and $T_I$ must be very special, which are from the constraints in the Lax operator. At last, we consider the successive applications of n-step of $T_D$ and k-step of $T_I$ gauge transformations.

We extend the Falceto–Zambon version of Marsden–Ratiu Poisson reduction to Poisson quasi-Nijenhuis structures with background on manifolds. We define gauge transformations of Poisson quasi-Nijenhuis structures with background, study some of their properties and show that they are compatible with reduction procedure. We use gauge transformations to construct Poisson quasi-Nijenhuis structures with background.

In this paper, we firstly recall some basic facts about the discrete KP(d-KP) and discrete modified KP(d-mKP) hierarchies, and then we find that d-KP hierarchy and d-mKP hierarchy are linked by a gauge transformation. What’s more, we give three gauge transformation operators of the d-mKP hierarchy and give their successive applications. We further construct the ghost symmetry and use this symmetry to give the definition the d-mKP hierarchy with self-consistent sources. We also give gauge transformations of a newly defined constrained d-mKP(cd-mKP) hierarchy and the constrained d-mKP hierarchy with self-consistent sources(cd-mKPHSCSs).

The Heisenberg supermagnet model is an important supersymmetric integrable system which is the super extension of the Heisenberg ferromagnet model. By virtue of introducing the general auxiliary matrix variables, we construct a new $(2+1)$-dimensional generalized integrable Heisenberg supermagnet models under two constraints. Meanwhile, we establish their corresponding gauge equivalent counterparts. Moreover, we derive new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

The determinant representation of the gauge transformation operators is establised. In this process, the generalized Wronskian determinant is introduced. As a simple application, the authors present a construction of the special τ-function obtained firstly by Chau et al. (Commun. Math. Phys., 149(1992), 263), which involves the generalized Wronskian determinant. Also, some properties of this determinant are given.

By using gauge transformations, we manage to obtain new solutions of (2 + 1)-dimensional Kadomtsev–Petviashvili (KP), Kaup–Kuperschmidt (KK) and Sawada–Kotera (SK) equations from nonzero seeds. For each of the preceding equations, a Galilean type transformation between these solutions u₂ and the previously known solutions u′₂ generated from zero seed is given. We present several explicit formulas of the single-soliton solutions for u₂ and u′₂, and further point out the two main differences of them under the same value of parameters, i.e., height and location of peak line, which are demonstrated visibly in three figures.