Abstract
In this paper, we study a discretized version of the (generalized) Korteweg-de Vries equation, ∂ t u + ∂ x 3u + u4∂ x u = 0. After a number of estimates, we utilize the Contraction Mapping Principle to prove the global well-posedness of this equation in a certain discrete Banach space. Our results are analogous to those of Kenig, Ponce, and Vega in the continuous setting. However, due to the nature of the Fourier multipliers, the proofs of several of these estimates in the discrete setting require new techniques. Our results yield a numerical procedure for computing the solution. We present a numerical algorithm which is based on successive iterations to obtain a fixed point guaranteed by the Contraction Mapping Principle. This fixed point is the desired solution to the discrete equation.
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