Abstract
The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UεC3(ℝ), that U is piecewise of class C 4, and that U (j) decays at an algebraic rate for j≦4. The faster the decay of U (j) the smoother the solution will be for t≠0. If U and its first four derivatives decay faster than ¦x¦−n for all n, then the solution will be infinitely differentiable for t≠0. For t>0, the decay rate of u(x, t) as x→ + ∞ increases with the decay rate of U; but the decay rate as x→ -∞ depends on the regularity of U. A solution u 1 of the Korteweg-de Vries equation such that u 1(·, 0)εC∞(ℝ) may fail to remain in class C ∞ for all time if u 1(x, 0) does not decay fast enough as ¦x¦→∞.
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Communicated by J. B. McLeod
This research was performed in part as a Visiting Member of the Courant Institute of Mathematical Science.
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Cohen, A. Existence and regularity for solutions of the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 71, 143–175 (1979). https://doi.org/10.1007/BF00248725
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DOI: https://doi.org/10.1007/BF00248725