# Physical meaning of KdV equation

1. Apr 20, 2015

### fian

Here is one of the KdV form

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?

2. Apr 20, 2015

### fian

Sorry, It seems like i accidentally posted it twice, it is because of the low connection.

3. Apr 20, 2015

### fian

4. Apr 23, 2015

### bigfooted

$uu_x = (\frac{1}{2}u^2)_x$

so it represents convection of kinetic energy. There is a link with the (inviscid) Burgers equation.

5. Apr 23, 2015