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
∂ u ∂ t − ( 6 ε 2 u 2 + 6 u ) ∂ u ∂ x + ∂ 3 u ∂ x 3 = 0 , {\displaystyle {\frac {\partial u}{\partial t}}-(6\varepsilon ^{2}u^{2}+6u){\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0,}
where ε {\displaystyle \varepsilon } is an arbitrary real parameter.
See also Korteweg–de Vries equation
Notes
References Shingareva, Inna; Lizárraga-Celaya, Carlos (2011). Solving Nonlinear Partial Differential Equations with Maple and Mathematica. Wien; New York: Springer Science & Business Media. ISBN 978-3-7091-0517-7. OCLC 751824407.