additional problems for integrable systems

In 2002, P. Zhou analyzed the behavior of solutions of the Cauchy problem for perturbations of the defocusing NLS equation i u_t + u_{xx}- 2|u|^2 u - \epsilon |u|^\ell u =0, u(x,t=0) = f(x) \to 0 \qquad as\quad |x|\to \infty. An open problem of great and basic interest is to analyze solutions of the Cauchy problem for perturbations of the focusing NLS equation i\tilde{u}t+ \tilde{u}{xx} +2|\tilde{u}|^2 u +\epsilon |\tilde{u}|^{\ell} \tilde{u} =0, \tilde{u}(x,t=0) = \tilde{f}(x) \to 0\qquad as\quad |x|\to \infty. The principal new difficulty in analyzing (eq14) over (eq13), is that for solutions $u(x,t)$ of (eq13) we have decay in the sup norm \sup_x |u(x,t) | \le c/t^{1/2} \to 0 \qquad as\quad t\to\infty. There are two open problems here. The first is to use Riemann--Hilbert/steepest-descent methods to prove the results in Venakides et al.\ rigorously. The second problem concerns the fact that if we replace the exponential forces $e^x$ by a general force $F(x)$, \ddx_k =F(x_{k-1}- x_k) - F(x_k - x_{k+1}), \qquad k\ge 1 then numerical simulations show that the solution of the shock problem for this system behaves in the same way as $t\to\infty$ as the Toda system, provided (eq16) has a 2-periodic solution x_k (t+T) = x_k(t), \quad T>0, \qquad x_{k+2}(t) = x_k(t) for $k\in \mathbb{Z}$. Second open problem: Analyze the general shock problem (eq16) for suitably small perturbations $F(x)$ of $e^x$. One would like, rather, a definition which is intrinsic to probability