Approximations for KdV equation as a Hamiltonian system
Several Fourier spectral approximations and collocation approximations for the Korteweg-de Vries (KdV) equation are considered paying special attention to the preservation of its Hamiltonian character. In the first part some basic facts on (infinite dimensional) Hamiltonian systems are briefly reviewed. Also the authors observe that the KdV equation \(u_t + u_{xxx}+6 u u_x=0\) can be written in Hamiltonian form by using two different Hamiltonian functions, and therefore there are two conserved quantities. Now Fourier spectral methods are applied to each of these formulations in order to get semidiscrete approximations for the standard basis \(\{ 1/2, \cos x, \sin x, \ldots, \cos (Nx), \sin (Nx) \}\), showing that for \(N=1\) both formulations give the same finite dimensional approximation. However for \( N>1\), only one of these discretizations retains its Hamiltonian nature. Next, the authors consider collocation methods applied to both Hamiltonians for the periodic KdV equation. Once more they find that for \(N=1\) both spatial discretizations are Hamiltonian but for \(N>1\) only one of them retains this character.
scientific article; zbMATH DE number 1471574