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Nonuniqueness for solutions of the Korteweg-de Vries equation


Authors: Amy Cohen and Thomas Kappeler
Journal: Trans. Amer. Math. Soc. 312 (1989), 819-840
MSC: Primary 35Q20
MathSciNet review: 988885
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Abstract: Variants of the inverse scattering method give examples of nonuniqueness for the Cauchy problem for $ {\text{KdV}}$. One example gives a nontrivial $ {C^\infty }$ solution $ u$ in a domain $ \{ (x,t):0 < t < H(x)\} $ for a positive nondecreasing function $ H$ , such that $ u$ vanishes to all orders as $ t \downarrow 0$ . This solution decays rapidly as $ x \to + \infty $ , but cannot be well behaved as $ x$ moves left. A different example of nonuniqueness is given in the quadrant $ x \geq 0,t \geq 0$, with nonzero initial data.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0988885-1
Article copyright: © Copyright 1989 American Mathematical Society