Generalized Korteweg–De Vries equation

In mathematics, the generalized Korteweg–De Vries (gKdV) equation is a nonlinear partial differential equation that extends the classic Korteweg–De Vries equation (KdV equation). The KdV equation is a mathematical model for waves on shallow water surfaces; the generalized form allows for different types of nonlinearity, making it applicable to a wider range of physical phenomena.^[1]

The equation is written as:^[2]

${\displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}f(u)=0}$

Here, ${\displaystyle u(x,t)}$ represents the wave's amplitude as a function of position ${\displaystyle x}$ and time ${\displaystyle t}$. The function ${\displaystyle f(u)}$ describes the nonlinear effects. The original Korteweg–De Vries equation is the specific case where ${\displaystyle f(u)=3u^{2}}$. A commonly studied form of the gKdV equation uses ${\displaystyle f(u)={\frac {u^{k+1}}{k+1}}}$ for some positive integer ${\displaystyle k}$.