Poisson structure and action-angle variables for the Hirota equation

被引：0

作者：

Zhang, Yu ^{[1
]}

Tian, Shou-Fu ^{[1
]}

机构：

[1] China Univ Min & Technol, Sch Math, Xuzhou 221116, Peoples R China

来源：

ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK | 2023年 / 74卷 / 06期

关键词：

Hirota equation; Poisson structure; Action-angle variables; Hamiltonian formalism; HAMILTONIAN STRUCTURES; VIRASORO; TRANSFORM; EVOLUTION; MODEL;

D O I：

10.1007/s00033-023-02129-z

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this work, we employ the inverse scattering transform (IST) to study the action-angle variables for the Hirota equation. Based on variational principle, we demonstrate that an Euler-Lagrange equation is equivalent to the Hirota equation. By using the IST, several properties of scattering data for the equation are discussed, and we calculate their Poisson brackets successfully with the help of tensor product. Interestingly, we reveal that the action-angle variables can be constructed by the scattering data. Furthermore, the spectral parameter expressions of conservation laws for the equation are derived, related to the Hamiltonian formulation for the equation.

引用

页数：18

共 43 条

[11] A finite genus solution of the Hirota equation via integrable symplectic maps [J].

Cao, Cewen ;

Zhang, Guangyao .

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 2012, 45 (09)

[12] Integrable nonlocal Hirota equations [J].

Cen, Julia ;

Correa, Francisco ;

Fring, Andreas .

JOURNAL OF MATHEMATICAL PHYSICS, 2019, 60 (08)

[13] N-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithm [J].

Chen, Yong ;

Peng, Wei-Qi .

PHYSICA D-NONLINEAR PHENOMENA, 2022, 435

[14] On the inverse scattering approach and action-angle variables for the Dullin-Gottwald-Holm equation [J].

Christov, O. ;

Hakkaev, S. .

PHYSICA D-NONLINEAR PHENOMENA, 2009, 238 (01) :9-19

[15] Poisson structure and action-angle variables for the Camassa-Holm equation [J].

Constantin, A ;

Ivanov, R .

LETTERS IN MATHEMATICAL PHYSICS, 2006, 76 (01) :93-108

[16] Generalized fourier transform for the Camassa-Holm hierarchy [J].

Constantin, Adrian ;

Gerdjikov, Vladimir S. ;

Ivanov, Rossen I. .

INVERSE PROBLEMS, 2007, 23 (04) :1565-1597

[17]

Dickey L.A, 1994, Soliton Equations and Hamiltonian Systems

[18] HIROTA EQUATION AS AN EXAMPLE OF AN INTEGRABLE SYMPLECTIC MAP [J].

FADDEEV, L ;

VOLKOV, AY .

LETTERS IN MATHEMATICAL PHYSICS, 1994, 32 (02) :125-135

[19]

Faddeev L. D., 1987, Hamiltonian methods in the theory of solitons, DOI DOI 10.1007/978-3-540-69969-9

[20] METHOD FOR SOLVING KORTEWEG-DEVRIES EQUATION [J].

GARDNER, CS ;

GREENE, JM ;

KRUSKAL, MD ;

MIURA, RM .

PHYSICAL REVIEW LETTERS, 1967, 19 (19) :1095-&

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