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Boussinesq Equation

The linear Boussinesq equation is the partial differential equation

u_(tt)-alpha^2u_(xx)=beta^2u_(xxtt)
(1)

(Whitham 1974, p. 9; Zwillinger 1997, p. 129). The nonlinear Boussinesq equation is

u_(tt)-u_(xx)-u_(xxxx)+3(u^2)_(xx)=0
(2)

(Calogero and Degasperis 1982; Zwillinger 1997, p. 130). The modified Boussinesq equation is

1/3u_(tt)-u_tu_(xx)-3/2u_x^2u_(xx)+u_(xxxx)=0
(3)

(Clarkson 1986; Zwillinger 1997, p. 132).

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References

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.Whitham, G. B. Linear and Nonlinear Waves. New York: Wiley, 1974.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 129-130, 1997.

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Boussinesq Equation

Cite this as:

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

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