We　employ　the　heat　perturbation　correlation　function　to　study　thermal　transport　in　the　one-dimensional　Fermi-Pasta-Ulam-β　lattice　with　both　nearest-neighbor　and　next-nearest-neighbor　couplings.　We　find　that　such　a　system　bears　a　peculiar　phonon　dispersion　relation,　and　thus　there　exists　a　competition　between　phonon　dispersion　and　nonlinearity　that　can　strongly　affect　the　heat　correlation　function＇s　shape　and　scaling　property.　Specifically,　for　small　and　large　anharmoncities,　the　scaling　laws　are　ballistic　and　superdiffusive　types,　respectively,　which　are　in　good　agreement　with　the　recent　theoretical　predictions;　whereas　in　the　intermediate　range　of　the　nonlinearity,　we　observe　an　unusual　multiscaling　property　characterized　by　a　nonmonotonic　delocalization　process　of　the　central　peak　of　the　heat　correlation　function.　To　understand　these　multiscaling　laws,　we　also　examine　the　momentum　perturbation　correlation　function　and　find　a　transition　process　with　the　same　turning　point　of　the　anharmonicity　as　that　shown　in　the　heat　correlation　function.　This　suggests　coupling　between　the　momentum　transport　and　the　heat　transport,　in　agreement　with　the　theoretical　arguments　of　mode　cascade　theory.　©　2017　American　Physical　Society.