Numerical Computations of Generalized Korteweg-de Vries (KdV) equations

Abstract

We consider the following generalized Korteweg-deVries (KdV) equation 𝑢𝑡+𝑎𝑢𝑥+2𝑏𝑢𝑢𝑥+𝑐𝑢𝑥𝑥𝑥−𝑑𝑢𝑥𝑥=0.

The above equation is the generalized version of the KDV equation 𝑢𝑡+𝑢𝑥+2𝑢𝑢𝑥+𝛿𝑢𝑥𝑥𝑥=0.

Here 𝑢=𝑢(𝑥,𝑡) is a scalar function of 𝑥∈𝑅and 𝑡≥0, while 𝛿>0 is a parameter. This equation is used to model the unidirectional propagation of water waves. The scalar 𝑢represents the amplitude of the wave.

In this presentation we investigate the various limits of the solutions of the generalized equation as one or more of the parameters as 𝑎,𝑏,𝑐 and 𝑑 tend to zero. This is carried out through numerical computations using the pseudo-spectral method.