scholarly journals Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r = f(x)$

2008 ◽  
Vol 60 (4) ◽  
pp. 1009-1044 ◽  
Author(s):  
Shigeki MATSUTANI ◽  
Emma PREVIATO
Keyword(s):  
Jacobi Inversion

Quasi-periodic solutions to a negative-order integrable system of 2-component KdV equation

2018 ◽  
Vol 15 (03) ◽  
pp. 1850040 ◽  
Author(s):  
Jinbing Chen
Keyword(s):  
Periodic Solutions ◽  
Integrable System ◽  
Kdv Equation ◽  
Type Systems ◽  
Zero Curvature Representation ◽  
Negative Order ◽  
Symmetric Constraint ◽  
Neumann Type ◽  
Novikov Equation ◽  
Jacobi Inversion

In this paper, the backward and forward Neumann type systems are generalized to deduce the quasi-periodic solutions for a negative-order integrable system of 2-component KdV equation. The 2-component negative-order KdV (2-nKdV) equation is depicted as the zero-curvature representation of two spectral problems. It follows from a symmetric constraint that the 2-nKdV equation is reduced to a pair of backward and forward Neumann type systems, where the involutive solutions of Neumann type systems yield the finite parametric solutions of 2-nKdV equation. The negative-order Novikov equation is given to specify a finite-dimensional invariant subspace for the 2-nKdV flow. With a spectral curve given by the Lax matrix, the 2-nKdV flow is linearized on the Jacobi variety of a Riemann surface, which leads to the quasi-periodic solutions of 2-nKdV equation by using the Riemann-Jacobi inversion.

Algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations via Hamiltonian approach

2015 ◽  
Vol 12 (03) ◽  
pp. 1550038
Author(s):  
Dianlou Du ◽  
Xiao Yang
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Equations ◽  
Poisson Structure ◽  
Hamiltonian Approach ◽  
Nonlinear Schrödinger ◽  
Hamiltonian Flows ◽  
Jacobi Inversion ◽  
Liouville Integrable ◽  
Separated Variables

The algebraic-geometrical solutions of three (2 + 1)-dimensional equations (including mKP equation and coupled mKP equation) are discussed by Hamiltonian approach. First, the Poisson structure on CN × RN is introduced to give a Hamiltonian system associated with the derivative nonlinear Schrödinger (DNLS) hierarchy. The Hamiltonian system is proved to be Liouville integrable, accordingly the solutions of three (2 + 1)-dimensional nonlinear equations can be solved by three compatible Hamiltonian flows. Second, the canonical separated variables and Hamilton–Jacobi theory is used to definite action-angle variables for Hamiltonian flows. At last, by Riemann–Jacobi inversion, the algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations are obtained. Besides, the algebraic-geometrical solutions of the first two DNLS equations are also given.

Finite Genus Solutions to the Generalised Kaup-Newell Hierarchy and Two 2+1 Dimensional Modified Korteweg-de Vries Equations

2017 ◽  
Vol 72 (7) ◽  
pp. 589-594
Author(s):  
Xiao Yang ◽  
Jiayan Han
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
Algebraic Curve ◽  
Inversion Theorem ◽  
First Order ◽  
Hamiltonian Flows ◽  
De Vries ◽  
Jacobi Inversion ◽  
Finite Genus

AbstractA generalised Kaup-Newell (gKN) hierarchy is introduced, which starts with a system of first-order ordinary differential equations and includes the Gerdjikov-Ivanov equation. By introducing an appropriate generating function, its related Hamiltonian systems and algebraic curve are given. The Hamiltonian systems are proved to be integrable, then the gKN hierarchy is solved by Hamiltonian flows. The algebraic curve is provided with suitable genus, then based on the trace formula and Riemann-Jacobi inversion theorem, finite genus solutions of the gKN hierarchy are obtained. Besides, two 2+1 dimensional modified Korteweg-de Vries (mKdV) equations are also solved.

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