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Hyperelliptic solutions of modified Korteweg-de Vries equation of genus \(g\): Essentials of the Miura transformation. (English) Zbl 1040.37063

Summary: Explicit hyperelliptic solutions of the modified Korteweg-de Vries equations without any ambiguous parameters are constructed in terms of only the hyperelliptic al-functions over the nondegenerate hyperelliptic curve \(y^2= f(x)\) of arbitrary genus \(g\). In the derivation, any \(\theta\)-functions or Baker-Akhiezer functions are not essentially used. Then the Miura transformation naturally appears as a connection between the hyperelliptic \(\wp\)-functions and hyperelliptic al-functions.

MSC:

37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14H42 Theta functions and curves; Schottky problem
14H70 Relationships between algebraic curves and integrable systems
33E05 Elliptic functions and integrals
35Q53 KdV equations (Korteweg-de Vries equations)
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