The Gardner equation is an integrable nonlinear partial differential equation introduced by the mathematician Clifford Gardner in 1968 to generalize KdV equation and modified KdV equation. The Gardner equation has applications in hydrodynamics, plasma physics and quantum field theory^[1]

${\displaystyle {\frac {\partial u}{\partial t}}+(2au-3bu^{2}){\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0,}$

where ${\displaystyle a}$ and ${\displaystyle b}$ are constants.

References

1. Graham W. Griffiths William E. Shiesser, Traveling Wave Analysis of Partial Differential Equations, Academy Press
2. Richard H. Enns George C. McCGuire, Nonlinear Physics, Birkhauser,1997
3. Inna Shingareva, Carlos Lizárraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple, Springer.
4. Eryk Infeld and George Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge 2000
5. Saber Elaydi, An Introduction to Difference Equations, Springer 2000
6. Dongming Wang, Elimination Practice, Imperial College Press 2004
7. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis. Springer, 1998 ISBN 9780387983004
8. George Articolo, Partial Differential Equations & Boundary Value Problems with Maple V, Academic Press 1998 ISBN 9780120644759