Giant exoplanets on 1--3 day orbits, known as ultra-hot Jupiters, induce detectable tides in their host stars. The energy of those tides dissipates at a rate related to the properties of the stellar interior. At the same time, a planet loses its orbital angular momentum and spirals into the host star. The decrease in the orbital period is empirically accessible with precise transit timing and can be used to probe planet-star tidal interactions. Statistical studies show that stars of GK spectral types, with masses below 1.1 $M_ odot $, are depleted in hot Jupiters. This finding is evidence of tidal orbital decay during the main-sequence lifetime. Theoretical considerations show that in some configurations the tidal energy dissipation can be boosted by non-linear effects in dynamical tides, which are wave-like responses to tidal forcing. To probe the regime of these dynamical tides in GK stars, we searched for orbital period shortening for six selected hot Jupiters in systems with 0.8--1 $M_ odot $ host stars: HATS-18, HIP 65A, TrES-3, WASP-19, WASP-43, and WASP-173A. For the hot Jupiters in our sample, we analysed transit timing datasets based on mid-transit points homogeneously determined from observations performed with the Transiting Exoplanet Survey Satellite and high-quality data available in the literature. For the TrES-3 system, we also used new transit light curves we acquired with ground-based telescopes. We searched mid-transit times for shortening of orbital periods by statistically testing quadratic transit ephemerides. Theoretical predictions on the dissipation rate for dynamical tides were calculated under the regimes of internal gravity waves (IGWs) undergoing wave breaking (WB) in stellar centres and weak non-linear (WNL) wave-wave interactions in radiative layers. Stellar parameters of the host stars, such as mass and age, which were used in those computations, were homogeneously redetermined using evolutionary models with the Bayesian inference. We found that transit times follow the refined linear ephemerides for all ultra-hot Jupiters of our sample. The non-detection of orbital decay allowed us to place lower constraints on the tidal dissipation rates in those planet-star systems. In three systems, HATS-18, WASP-19, and WASP-43, we reject a scenario with total dissipation of IGWs. We conclude that their giant planets are not massive enough to induce WB. Our observational constraints for HIP 65A, TrES-3, and WASP-173A are too weak to probe the WB regime. Calculations show that WB is not expected in the former two, leaving the WASP-173A system as a promising target for further transit timing observations. The WNL dissipation was tested in the WASP-19 and WASP-43 systems, showing that the theoretical dissipation rates are overestimated by at least one order of magnitude. For the remaining systems, decades or even centuries of transit timing measurements are needed to probe the WNL regime entirely. Among them, TrES-3 and WASP-173A have the predicted WNL dissipation rates that coincide with the values obtained from gyrochronology. Tidal dissipation in the GK stars of our sample is not boosted by WB in their radiative cores, preventing their giant planets from rapid orbital decay. Weakly non-linear tidal dissipation could drive orbital shrinkage and stellar spin-up on gigayear timescales. Although our first results suggest that theory might overestimate the dissipation rate and some fine-tuning would be needed for at least a fraction of planet-star configurations, some predictions coincide intriguingly with the gyrochronological estimates. We identify the WASP-173A system as a promising candidate for exploring this problem in the shortest possible time of the coming decades.
