Abstract
In order to find higher dimensional integrable models, we study differential equations of hyperelliptic functions up to genus four. For genus two, differential equations of hyperelliptic functions can be written in the Hirota form. If the genus is more than one, we have KdV equation. If the genus is more than two, we have KdV and another KdV equations. If the genus becomes more than three, there appear differential equations which cannot be written in the Hirota form, which means that the Hirota form is not enough to characterize the integrable differential equations. We have shown that some differential equations are satisfied for general genus. We can obtain differential equations for general genus step by step.
Highlights
Through studies of soliton system, we have solved non-linear problems of very interesting phenomena
By using the Lie group structure as our guiding principle, we have revealed that the two dimensional integrable models such as KdV/mKdV/sinh-Gordon are the consequence of the SO(2,1)– SL(2,R) Lie group structure [21,22,23,24,25] 1 Here we would like to to study higher-dimensional integrable models
Eq(2.19), we reduce the power of xi in the range xij1, pj “ 1, 2, ̈ ̈ ̈, gq and comparing the coefficients of left- and right-hand side of Eq(4.6), we have following differential equations for general genus λ2g1℘g1,jλ2g ℘gj
Summary
Through studies of soliton system, we have solved non-linear problems of very interesting phenomena. Starting with the fermionic bilinear identity of glp, Rq, they have obtained KP hierarchy and finite higherdimensional Hirota forms by the reduction of KP hierarchy Another systematic approach to high-dimensional integrable models is to find differential equations for higher genus hyperelliptic functions by using the analogy of differential equation of Weierstrass ℘ function. We would like to examine the connections between i) higher-dimensional integrable differential equations, ii) higher-rank Lie group structure and iii) higher genus hyperelliptic functions
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