Nonuniqueness for solutions of the Korteweg-de Vries equation

Abstract

Variants of the inverse scattering method give examples of nonuniqueness for the Cauchy problem for KdV {\text {KdV}} . One example gives a nontrivial C ∞ {C^\infty } solution u u in a domain { ( x , t ) : 0 > t > H ( x ) } \{ (x,t):0 > t > H(x)\} for a positive nondecreasing function H H , such that u u vanishes to all orders as t ↓ 0 t \downarrow 0 . This solution decays rapidly as x → + ∞ x \to + \infty , but cannot be well behaved as x x moves left. A different example of nonuniqueness is given in the quadrant x ≥ 0 , t ≥ 0 x \geq 0,t \geq 0 , with nonzero initial data.