void kdv_exact_rational ( double x, double t, double *u, double *ut, double *ux, 
  double *uxx, double *uxxx );
void kdv_exact_sech ( double x, double t, double *u, double *ut, double *ux, 
  double *uxx, double *uxxx );
void kdv_parameters ( double *a_in, double *v_in, double *t0_in, 
  double *tstop_in, double *a_out, double *v_out, double *t0_out, 
  double *tstop_out );
double kdv_residual ( double u, double ut, double ux, double uxxx );