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We employ the heat perturbations correlation function to study thermaltransport in the one-dimensional (1D) Fermi-Pasta-Ulam-$\beta$ lattice withboth nearest-neighbor and next-nearest-neighbor couplings. We find that such asystem bears a peculiar phonon dispersion relation, and thus there exists acompetition between phonon dispersion and nonlinearity that can strongly affectthe heat correlation function' shape and scaling property. Specifically, forsmall and large anharmoncity, the scaling laws are ballistic and superdiffusivetypes, respectively, which are in good agreement with the recent theoreticalpredictions; whereas in the intermediate range of the nonlinearity, we observean unusual multiscaling property characterized by a \emph{nonmonotonic}delocalization process of the central peak of the heat correlation function. Tounderstand this multiscaling laws, we also examine the momentum perturbationscorrelation function and find a transition process with the same turning pointof the anharmonicity as that shown in the heat correlation function. Thissuggests coupling between the momentum transport and the heat transport, inagreement with the theoretical arguments of mode cascade theory.

physical review. e

Heat perturbations spreading in the Fermi-Pasta-Ulam-$\beta$ system with  next-nearest-neighbor coupling: Competition between phonon dispersion and  nonlinearity