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Krichever-Novikov Equation

The partial differential equation

(u_t)/(u_x)=1/4(u_(xxx))/(u_x)-3/8(u_(xx)^2)/(u_x^2)+3/2(p(u))/(u_x^2),

where

p(u)=1/4(4u^3-g_2u-g_3).

The special cases p(u)=(u-e_1)^2(u-e_2) and p(u)=u^3 can be reduced to the Korteweg-de Vries equation by a change of variables.

See also

Kadomtsev-Petviashvili Equation, Korteweg-de Vries Equation

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References

Krichever, I. M. and Novikov, S. P. "Holomorphic Bundles over Algebraic Curves, and Nonlinear Equations." Russ. Math. Surv. 35, 53-80, 1980. English translation of Uspekhi Mat. Nauk 35, 47-68, 1980.Mokhov, O. I. "Canonical Hamiltonian Representation of the Krichever-Novikov Equation." Math. Notes 50, 939-945, 1991. English translation of Mat. Zametki 50, 87-96, 1991.Novikov, D. P. "Algebraic-Geometric Solutions of the Krichever-Novikov Equation." Theoret. Math. Phys. 121, 1567-15773, 1999.Sokolov, V. V. "Hamiltonian Property of the Krichever-Novikov Equation." Dokl. Akad. Nauk SSSR 277, 48-50, 1984.Svinolupov, S. I.; Sokolov, V. V.; and Yamilov, R. I. "Bäcklund Transformations for Integrable Evolution Equations." Dokl. Akad. Nauk SSSR 271, 802-805, 1983. English translation of Sov. Math. Dokl. 28, 165-168, 1983.

Referenced on Wolfram|Alpha

Krichever-Novikov Equation

Cite this as:

Weisstein, Eric W. "Krichever-Novikov Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Krichever-NovikovEquation.html

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