Next-basket recommendation (NBR) is a type of recommendation that aims to predict a set of items a user will purchase based on their historical transaction basket sequences. It is governed by a dynamic interplay between two distinct user intents: habitual repurchase, which involves repeating past behaviors, and exploratory interest, which involves discovering new items. However, existing NBR methods generally suffer from two limitations: (1) they often entangle these conflicting motives within a single representation, causing habits to overshadow discovery, and (2) they rely on discrete sequential modeling that ignores continuous-time intervals and item-specific periodicities. In this paper, we propose a novel solution named Time-Interval Disentangled Experts (TIDE) to address these challenges. TIDE incorporates a Hawkes-enhanced Fourier Time Encoding to capture item-specific temporal periodicities and dynamic decay. To decouple user intentions, TIDE utilizes a dual-expert architecture that integrates a Habit Expert for recurring needs and a Pattern-Guided Exploration Expert for discovery. Combined with an item-aware gating mechanism, TIDE adaptively balances repurchase and exploration. Extensive experiments on four diverse real-world datasets demonstrate that TIDE consistently outperforms representative state-of-the-art NBR methods.