Solving negative and mixed Toda hierarchies by Jacobi inversion problems

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.

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Serre duality, Abel’s theorem, and Jacobi inversion for supercurves over a thick superpoint

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2015.01.013 ◽

2015 ◽

Vol 90 ◽

pp. 95-103 ◽

Cited By ~ 2

Author(s):

Mitchell J. Rothstein ◽

Jeffrey M. Rabin

Keyword(s):

Serre Duality ◽

Jacobi Inversion

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BTZ black hole and Jacobi inversion for fundamental domains of infinite volume

Contemporary Mathematics - Algebraic Structures and Their Representations ◽

10.1090/conm/376/06973 ◽

2005 ◽

pp. 385-391

Author(s):

Floyd L. Williams

Keyword(s):

Black Hole ◽

Infinite Volume ◽

Btz Black Hole ◽

Fundamental Domains ◽

Jacobi Inversion

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887815500383 ◽

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.

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