For the NLS in 1D, quintic is $L^2$ critical. So you are quite comfortably in the subcritical regime. Indeed, you have the result of

Tsutsumi, Yoshio; "$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups"

which implies for $H^s$ (and hence $L^2$) initial data, you have global existence of solutions in the space $C(\mathbb{R}; L^2(\mathbb{R}))$ intersected with some Strichartz space.

For local solutions in $H^s$, in the sense of having solutions in the space $C((T_{min}, T_{max}); H^s(\mathbb{R}))$, you can look at

Kato, Tosio; "On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness"

(Remark, the nonlinearities considered in Kato's paper allow more general forms than that of Tsutsumi's.)

The main problem here is that we have no direct conservation law on the level of $H^s$, and so global well-posedness cannot follow, as in the $L^2$ case, from local well-posedness. On the other hand, the Strichartz type estimates used in Kato's work is sufficient to allow a direct proof of global well posedness provided that the initial data in $H^s$ is sufficiently small.

(Remark: for general results concerning NLS, a good reference book is Cazenave's *Semilinear Schroedinger equations*,)

However, global wellposedness for the cubic NLS is true in 1 spatial dimension. Here we can use the fact that we are looking at an algebraic nonlinearity: that the right hand side is $\pm \phi\bar{\phi} \phi$. In particular, for this equation we can prove persistence of regularity. For this equation the statement is

**Proposition** (Prop. 3.11, p132, Tao, *Nonlinear dispersive equations*)

Let $u\in C((t_1,t_2); H^s(\mathbb{R}))$ be a (local) solution for $s\geq 0$, with $t_1 < 0 < t_2$. Then there exists a constant $C$ depending on $s$ such that
$$ \|u\|_{L^\infty((t_1,t_2); H^s(\mathbb{R}))} \leq \|u(0)\|_{H^s} \exp (C \|u\|^3_{L^2((t_1,t_2); L^\infty(\mathbb{R}))}) $$
provided that the right hand side of the estimate is finite.

It turns out that for the 1D Schrodinger equation you can control $L^4_t L^\infty_x$ using Strichartz estimates from $L^2$ of initial data, and so restricting to finite intervals you also pick up $L^2_t L^\infty_x$ which then allows you to use the $L^2$ global well-posedness result above to obtain global wellposedness in any $H^s$ for $s > 0$; however, we lose control over the $H^s$ norm over time. If you run the argument using the above proposition, you get that $\|u(t)\|_{H^s(\mathbb{R})}$ is allowed to grow exponentially in $t$. For more details see Section 3.4 in Tao, *op. cit.*