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We study the focusing NLS \begin{align}\label{nls_abstract} i\partial_tu+\Delta_{x,y} u=-|u|^\alpha u\tag{NLS} \end{align} on the waveguide manifold$\mathbb{R}^d\times\mathbb{T}$ in the intercritical regime$\alpha\in(\frac{4}{d},\frac{4}{d-1})$. By assuming that the\eqref{nls_abstract} is independent of $y$, it reduces to the focusingintercritical NLS on $\mathbb{R}^d$, which is known to have standing wave andfinite time blow-up solutions. Naturally, we ask whether these specialsolutions with non-trivial $y$-dependence exist. In this paper we give anaffirmative answer to this question. To that end, we introduce the concept of\textit{semivirial} functional and consider a minimization problem $m_c$ on thesemivirial-vanishing manifold with prescribed mass $c$. We prove that for any$c\in(0,\infty)$ the variational problem $m_c$ has a ground state optimizer$u_c$ which also solves the standing wave equation $$-\Delta_{x,y}u_c+\beta_cu_c=|u|^\alpha u $$ with some $\beta_c>0$. Moreover, we prove the existence ofa critical number $c_*\in(0,\infty)$ such that \begin{itemize} \item For$c\in(0,c_*)$, any optimizer $u_c$ of $m_c$ must satisfy $\pt_y u_c\neq 0$.\item For $c\in(c_*,\infty)$, any optimizer $u_c$ of $m_c$ must satisfy $\pt_yu_c=0$. \end{itemize} Finally, we prove that the previously constructed groundstates characterize a sharp threshold for the bifurcation of scattering andfinite time blow-up solutions in dependence of the sign of the semivirial.

Normalized ground states and threshold scattering for focusing NLS on $\mathbb{R}^d\times\mathbb{T}$ via semivirial-free geometry