# kdv_exact

kdv_exact, an Octave code which evaluates an exact solution of the Korteweg-deVries (KdV) partial differential equation (PDE) that represents the motion of a soliton.

The equation for u(x,t), the height of the wave, has the form

```        ut - 6 u ux + uxxx = 0
```
for which an exact solution is
```        u(x,t) = - 1/2 v ( sech ( 1/2 * sqrt ( v ) * ( x - v * t - a ) )^2
```
where parameter "a" is an arbitrary phase, and "v" represents the wave velocity.

### Languages:

kdv_exact is available in a MATLAB version and an Octave version and a Python version.

### Related Data and codes:

kdv_etdrk4, an Octave code which uses the exponential time differencing (ETD) RK4 method to solve the Korteweg-deVries (KdV) partial differential equation (PDE), by Aly-Khan Kassam, Lloyd Trefethen.

kdv_ift, an Octave code which uses the Inverse Fourier Transform (IFT) method to solve the Korteweg-deVries (KdV) partial differential equation (PDE), by Aly-Khan Kassam, Lloyd Trefethen.

pendulum_nonlinear_exact, an Octave code which evaluates an exact formula for the solution of the the ordinary differential equations (ODE) that represent the behavior of a nonlinear pendulum of length L under a gravitational force of strength G.

navier_stokes_2d_exact, an Octave code which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 2D.

navier_stokes_3d_exact, an Octave code which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations (NSE) over an arbitrary domain in 3D.

stokes_2d_exact, an Octave code which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.

### Reference:

• John D Cook,
Solitons and the KdV equation,
03 November 2023,
https://www.johndcook.com/blog/2023/11/03/solitons-and-the-kdv-equation/
• John D Cook,
Rational solution to the Korteweg-De Vries equation,
13 November 2023,
https://www.johndcook.com/blog/2023/11/13/rational-kdv/

### Source Code:

Last revised on 15 November 2023.