The modified Korteweg–de Vries (KdV) equation is an integrable nonlinear partial differential equation:^[1]

${\displaystyle u_{t}+u_{xxx}+\alpha u^{2}u_{x}=0\,}$

where ${\displaystyle \alpha }$ is an arbitrary (nonzero) constant.

This is a special case of the Gardner equation.

• Korteweg–de Vries equation

• Griffiths, Graham W.; Schiesser, W. E. (2011). Traveling Wave Analysis of Partial Differential Equations. Amsterdam; Boston: Academic Press. ISBN 978-0-12-384652-5. OCLC 657600287.
• Polyanin, Andrei D.; Zaitsev, Valentin F. (2003-10-29). "9.1.2. Cylindrical, Spherical, and Modified Korteweg-de Vries Equations". Handbook of Nonlinear Partial Differential Equations. Boca Raton, Fla: Chapman and Hall/CRC. ISBN 978-1-58488-355-5.

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