Here is one of the KdV form(adsbygoogle = window.adsbygoogle || []).push({});

u_t + u_x + uu_x + u_{xxx} = 0

Where u is elevation, x is spatial variable, and t is time variable. The first two terms describe the linear water wave, the third term represent the nonlinear effect, and the last term is the dispersion.

From what i understand, the nonlinear term explain the energy focusing that keeps the shape of the wave packet. But, how is u multiplied by u_x represents the energy focusing? For example, like in predator-prey model, the nonlinear term xy explain the interaction between the two species, where x and y are the number of predators and prey respectively.

Also, how does the last term, the third derivative of u with respect to x, explain the dispersion which is the deformation of the waves?

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# Physical meaning of KdV equation

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