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We show that the one dimensional discrete nonlinear Schr\"odinger chain(DNLS) at finite temperature has three different dynamical regimes (ultra-low,low and high temperature regimes). This has been established via (i) one pointmacroscopic thermodynamic observables (temperature $T$ , energy density$\epsilon$ and the relationship between them), (ii) emergence and disappearanceof an additional almost conserved quantity (total phase difference) and (iii)classical out-of-time-ordered correlators (OTOC) and related quantities(butterfly speed and Lyapunov exponents). The crossover temperatures$T_{\textit{l-ul}}$ (between low and ultra-low temperature regimes) and$T_{\textit{h-l}}$ (between high and low temperature regimes) extracted fromthese three different approaches are consistent with each other. The analysispresented here is an important step forward towards the understanding of DNLSwhich is ubiquitous in many fields and has a non-separable Hamiltonian form.Our work also shows that the different methods used here can serve as importanttools to identify dynamical regimes in other interacting many body systems.

Dynamical regimes of finite temperature discrete nonlinear Schr\"odinger chain