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 ${\text {KdV}}$. 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.