Canonical explicit B\"{a}cklund transformations with spectrality for constrained flows of soliton hierarchies

It is shown that explicit B\"{a}cklund transformations (BTs) for the high-order constrained flows of soliton hierarchy can be constructed via their Darboux transformations and Lax representation, and these BTs are canonical transformations including B\"{a}cklund parameter $\eta$ and possess a spectrality property with respect to $\eta$ and the 'conjugated' variable $\mu$ for which the pair $(\eta, \mu)$ lies on the spectral curve. As model we present the canonical explicit BTs with the spectrality for high-order constrained flows of the Kaup-Newell hierarchy and the KdV hierarchy.Comment: 21 pages, Latex, to be published in "PHYSICA A

Deriving N-soliton solutions via constrained flows

The soliton equations can be factorized by two commuting x- and t-constrained flows. We propose a method to derive N-soliton solutions of soliton equations directly from the x- and t-constrained flows.Comment: 8 pages, AmsTex, no figures, to be published in Journal of Physics

Constructing N-soliton solution for the mKdV equation through constrained flows

Based on the factorization of soliton equations into two commuting integrable x- and t-constrained flows, we derive N-soliton solutions for mKdV equation via its x- and t-constrained flows. It shows that soliton solution for soliton equations can be constructed directly from the constrained flows.Comment: 10 pages, Latex, to be published in "J. Phys. A: Math. Gen.

The relation between the Toda hierarchy and the KdV hierarchy

Under three relations connecting the field variables of Toda flows and that of KdV flows, we present three new sequences of combination of the equations in the Toda hierarchy which have the KdV hierarchy as a continuous limit. The relation between the Poisson structures of the KdV hierarchy and the Toda hierarchy in continuous limit is also studied.Comment: 11 pages, Tex, no figures, to be published in Physics Letters

Contrastive Learning with Prompt-derived Virtual Semantic Prototypes for Unsupervised Sentence Embedding

Contrastive learning has become a new paradigm for unsupervised sentence embeddings. Previous studies focus on instance-wise contrastive learning, attempting to construct positive pairs with textual data augmentation. In this paper, we propose a novel Contrastive learning method with Prompt-derived Virtual semantic Prototypes (ConPVP). Specifically, with the help of prompts, we construct virtual semantic prototypes to each instance, and derive negative prototypes by using the negative form of the prompts. Using a prototypical contrastive loss, we enforce the anchor sentence embedding to be close to its corresponding semantic prototypes, and far apart from the negative prototypes as well as the prototypes of other sentences. Extensive experimental results on semantic textual similarity, transfer, and clustering tasks demonstrate the effectiveness of our proposed model compared to strong baselines. Code is available at https://github.com/lemon0830/promptCSE.Comment: Findings of EMNLP 202