Paper Info
Title
Elliptic Soliton Solutions: $$\tau $$ τ Functions, Vertex Operators and Bilinear Identities
Abstract
We establish a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors. $$\tau $$ functions in Hirota’s form are derived and vertex operators that generate such $$\tau $$ functions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational functions. Reductions by dispersion relation are considered by employing the so-called elliptic N-th roots of the unity. $$\tau $$ functions, vertex operators and bilinear equations of the KdV hierarchy and Boussinesq equation are obtained from those of the KP. We also formulate two ways to calculate bilinear derivatives involved with the Lamé-type plane wave factors, which shows that such type of plane wave factors results in quasi-gauge property of bilinear equations.
Year
DOI
Venue
2022
10.1007/s00332-022-09835-4
Journal of Nonlinear Science
Keywords
DocType
Volume
Elliptic soliton solution, Function, Vertex operator, Bilinear identity, Weierstrass function, Lamé function
Journal
32
Issue
ISSN
Citations 
5
0938-8974
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Xing Li100.34
Da-jun Zhang232.64