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. The equation is written as:
∂ t u + ∂ x 3 u + ∂ x f ( u ) = 0 {\displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}f(u)=0}
Here, u ( x , t ) {\displaystyle u(x,t)} represents the wave's amplitude as a function of position x {\displaystyle x} and time t {\displaystyle t} . The function f ( u ) {\displaystyle f(u)} describes the nonlinear effects. The original Korteweg–De Vries equation is the specific case where f ( u ) = 3 u 2 {\displaystyle f(u)=3u^{2}} . A commonly studied form of the gKdV equation uses f ( u ) = u k + 1 k + 1 {\displaystyle f(u)={\frac {u^{k+1}}{k+1}}} for some positive integer k {\displaystyle k} .
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