Abstract
The variational principle is used to construct a multi-symplectic structure of the generalized KdV-type equation. Accordingly, the local energy conservation law, the local momentum conservation law, and the Cartan form of the generalized KdV-type equation are given. An explicit multi-symplectic scheme for the generalized KdV equation based on the Fourier pseudo-spectral method and the symplectic Euler scheme is constructed. Through a numerical examination, the explicit multi-symplectic Fourier pseudo-spectral scheme for the generalized KdV equation not only preserve the discrete global energy conservation law and the global momentum conservation law with high accuracy, but show long-time numerical stability as well.
Highlights
The variational principle is used to construct a multi-symplectic structure of the generalized KdV-type equation
The generalized KdV-type equation, which can degenerate to the mKdV equation and the generalized KdV equation, is given
The variational principle is successfully used to establish a multi-symplectic structure for the KdV-type equation
Summary
Discretizing the multi-symplectic structure of the generalized KdV Eq (2.17) with the Fourier pseudo-spectral method in the space domain, the discrete form of the generalized KdV Eq (1.2) can be obtained as follows: du − Dω = 0, dt. The Fourier pseudo-spectral semi-discretization (3.3) has N semi-discrete multi-symplectic conservation laws www.nature.com/scientificreports dχi dt. Taking the wedge product with dzi on both sides of Eq (3.6) and noticing dzi ∧ Szz(zi)dzi = 0,. We show the N semi-discrete multi-symplectic conservation laws. Discretizing Eq (3.3) with respect to the time domain by the symplectic Euler scheme yields. The discrete scheme (3.13) has N full-discrete multi-symplectic conservation laws. The N full-discrete multi-symplectic conservation laws are verified[13,29]
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