Matsutani, Shigeki Hyperelliptic solutions of modified Korteweg-de Vries equation of genus \(g\): Essentials of the Miura transformation. (English) Zbl 1040.37063 J. Phys. A, Math. Gen. 35, No. 19, 4321-4333 (2002). 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. Cited in 3 Documents 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) PDFBibTeX XMLCite \textit{S. Matsutani}, J. Phys. A, Math. Gen. 35, No. 19, 4321--4333 (2002; Zbl 1040.37063) Full Text: DOI arXiv