kdv_exact

kdv_exact, a MATLAB code which evaluates an exact solution u(x,t) of the Korteweg-deVries (KdV) partial differential equation (PDE), representing the motion of a soliton. The equation has the form u' - 6 u ux + uxxx = 0.

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

        ut - 6 u ux + uxxx = 0

        u(x,t) = - 1/2 v ( sech ( 1/2 * sqrt ( v ) * ( x - v * t - a ) )^2

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

kdv_exact is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and codes:

kdv_exact_test

kdv_etdrk4, a MATLAB 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, a MATLAB 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.

matlab_exact, a MATLAB code which evaluates exact solutions to a few selected examples of ordinary differential equations (ODE) and partial differential equations (PDE).

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:

• kdv_exact_rational.m, returns an exact solution of the PDE using a rational function;
• kdv_exact_sech.m, returns an exact solution of the PDE using sech();
• kdv_parameters.m, returns parameters of the PDE.
• kdv_residual.m, returns the residual of the PDE at (x,t) for given values of u, ut, ux, uxxx.