  This paper proposes a new class of arbitrarily high-order conservative
numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This
approach is based on the scalar auxiliary variable (SAV) method. The equation
is reformulated into an equivalent system by introducing a scalar auxiliary
variable, and the energy is reformulated into a sum of two quadratic terms.
Therefore, the quadratic preserving Runge-Kutta method will preserve all the
three invariants (momentum, mass and the reformulated energy) in the discrete
time flow (assuming the spatial variable is continuous). With the Fourier
pseudo-spectral spatial discretization, the scheme conserves the first and
third invariant quantities (momentum and energy) exactly in the space-time full
discrete sense. The discrete mass possesses the precision of the spectral
accuracy. Our numerical experiment shows the great efficiency of this scheme in
simulating the breathers for the mKdV equation.
