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Kadomtsev-Petviashvili Equation

The partial differential equation

3/4U_y+W_x=0,
(1)

where

W_y+U_t-1/4U_(xxx)+3/2UU_x=0
(2)

(Krichever and Novikov 1980; Novikov 1999). Zwillinger (1997, p. 131) and Calogero and Degasperis (1982, p. 54) give the equation as

partial/(partialx)(u_t+u_(xxx)-6uu_x)+/-u_(yy)=0.
(3)

The modified Kadomtsev-Petviashvili equation is given by

u_(xt)=u_(xxx)+3u_(yy)-6u_x^2u_(xx)-6u_yu_(xx)
(4)

(Clarkson 1986; Zwillinger 1997, p. 133).

See also

Kadomtsev-Petviashvili-Burgers Equation, Korteweg-de Vries Equation, Krichever-Novikov Equation

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References

Baker, H. F. Abelian Functions: Abel's Theorem and the Allied Theory, Including the Theory of the Theta Functions. New York: Cambridge University Press, p. xix, 1995.Calogero, F. and Degasperis, A. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations. New York: North-Holland, 1982.Clarkson, P. A. "The Painlevé Property, a Modified Boussinesq Equation and a Modified Kadomtsev-Petviashvili Equation." Physica D 19, 447-450, 1986.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.Novikov, D. P. "Algebraic-Geometric Solutions of the Krichever-Novikov Equation." Theoret. Math. Phys. 121, 1567-15773, 1999.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 131, 1997.

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Kadomtsev-Petviashvili Equation

Cite this as:

Weisstein, Eric W. "Kadomtsev-Petviashvili Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Kadomtsev-PetviashviliEquation.html

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