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The Korteweg-de Vries equation and its symmetry-preserving discretization. (English) Zbl 1319.35214

Authors’ abstract: The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the Eulerian form. We also propose invariant schemes that preserve the momentum. Numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes. We find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving Galilean transformations which is demonstrated numerically as well.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35G25 Initial value problems for nonlinear higher-order PDEs
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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