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Quasi-periodic solutions of the $2+1$ dimensional modified Korteweg-de Vries equation. (English) Zbl 0937.35155
Summary: A new $2 + 1$ dimensional modified Korteweg-de Vries equation is proposed and decomposed into the first two members of the well-known Kaup-Newell hierarchy, which are reduced further into integrable ordinary differential equations in the invariant set produced by the stationary Kaup-Newell equation. The Abel-Jacobi coordinates are introduced to straighten out the flows, from which quasi-periodic solutions of the $2 + 1$ dimensional modified Korteweg-de Vries equation are obtained in terms of the Riemann theta functions.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35B10 Periodic solutions of PDE 37K20 Relations of infinite-dimensional systems with algebraic geometry, etc.
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##### References:
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