Dispersion-manageability conjecture for non-scattering data

Let $u_0\in H^1({\mathbb R}^d)$ be non-scattering data, meaning that the solution to $(NLS)_-$ with $u(0,x)=u_0(x)$ does not scatter. Let $t_0,t_1$ be the management parameters and let $\gamma_+,\gamma_-$ be the corresponding managed coefficients. A solution of the dispersion-managed NLS model is dispersion-manageable when it exists globally and satisfies

$$c_0\leq \sup_{t\geq 0}\left\|u(t,\cdot)\right\|_{L_x^\infty({\mathbb R}^d)}\leq C_0$$

for some $c_0,C_0\in(0,\infty)$.

Non-scattering-data dispersion-manageability conjecture. Given such $u_0$, both of the following assertions hold: (i) for fixed $t_0,t_1$, there exist $\gamma_+=\gamma_+(u_0)$ and $\gamma_-=\gamma_-(u_0)$; and (ii) for fixed $\gamma_+,\gamma_-$, there exist $t_0=t_0(u_0)$ and $t_1=t_1(u_0)$; in either case, the solution of the dispersion-managed NLS model with $u(0)=u_0$ is dispersion-manageable.

This conjecture proposes a parameter-selection mechanism producing global, non-scattering managed solutions from non-scattering initial data. Its resolution is not given in the supplied text.