We develop and validate a high-order reconstruction (HOR) method for the phase-resolved reconstruction of a nonlinear wave field given a set of wave measurements. HOR optimizes the amplitude and phase of $L$ free wave components of the wave field, accounting for nonlinear wave interactions up to order $M$ in the evolution, to obtain a wave field that minimizes the reconstruction error between the reconstructed wave field and the given measurements. For a given reconstruction tolerance, $L$ and $M$ are provided in the HOR scheme itself. To demonstrate the validity and efficacy of HOR, we perform extensive tests of general two- and three-dimensional wave fields specified by theoretical Stokes waves, nonlinear simulations and physical wave fields in tank experiments which we conduct. The necessary $L$ , for general broad-banded wave fields, is shown to be substantially less than the free and locked modes needed for the nonlinear evolution. We find that, even for relatively small wave steepness, the inclusion of high-order effects in HOR is important for prediction of wave kinematics not in the measurements. For all the cases we consider, HOR converges to the underlying wave field within a nonlinear spatial-temporal predictable zone ${\mathcal{P}}_{NL}$ which depends on the measurements and wave nonlinearity. For infinitesimal waves, ${\mathcal{P}}_{NL}$ matches the linear predictable zone ${\mathcal{P}}_{L}$ , verifying the analytic solution presented in Qi et al. (Wave Motion, vol. 77, 2018, pp. 195–213). With increasing wave nonlinearity, we find that ${\mathcal{P}}_{NL}$ contains and is generally greater than ${\mathcal{P}}_{L}$ . Thus ${\mathcal{P}}_{L}$ provides a (conservative) estimate of ${\mathcal{P}}_{NL}$ when the underlying wave field is not known.

Derived equivalences and stable equivalences of Morita type, and new (candidate) invariants thereof, between symmetric algebras will be investigated, using transfer maps as a tool. Close relationships will be established between the new invariants and the validity of the Auslander–Reiten conjecture, which states the invariance of the number of non-projective simple modules under stable equivalence. More precisely, the validity of this conjecture for a given pair of algebras, which are stably equivalent of Morita type, will be characterized in terms of data refining Hochschild homology (via Külshammer ideals) being invariant and also in terms of cyclic homology being invariant. Thus, validity of the Auslander–Reiten conjecture implies a whole set of ring theoretic and cohomological data to be invariant under stable equivalence of Morita type, and hence also under derived equivalence. We shall also prove that the Batalin–Vilkovisky algebra structure of Hochschild cohomology for symmetric algebras is preserved by derived equivalence. The main tools to be developed and used are transfer maps and their properties, in particular a crucial compatibility condition between transfer maps in Hochschild homology and Hochschild cohomology via the duality between them.

Simple-minded systems (s.m.s.'s) in stable module categories are defined by orthogonality and generating properties so that the images of the simple modules under a stable equivalence form such a system. Under stable equivalences s.m.s.'s are shown to be invariant; thus, the set of all s.m.s.'s is an invariant of a stable module category. The s.m.s.'s of several classes of algebras are described and connections to the Auslander–Reiten conjecture are pointed out.

When moderately steep waves travel over a periodic rippled bed, class III Bragg resonance may occur due to the third-order quartet wave-bottom interaction among one bottom and three surface wave components. The theory, however, has not been experimentally confirmed. To verify the existence of class III Bragg resonance, we consider the simplest possible case involving a single incident wave and conduct a series of physical experiments. The experiments show that as the theory predicts, class III Bragg resonance could generate not only the reflected waves (from subharmonic resonance) but also the transmitted waves (from superharmonic resonance), and the reflection and transmission coefficients vary linearly along the rippled bottom patch. Furthermore, the experimental data of the reflected and transmitted waves (due to class III resonance) agree quantitatively well with the prediction by high-order spectral (HOS) method computations. To further understand the characteristics of class III resonance, we apply HOS simulations to study the more general cases including the presence of two incident waves of different frequencies as well as the presence of an incident wave group with Gaussian envelope. The results show that in addition to class I and class II Bragg resonances, class III Bragg resonance can significantly influence the evolution of surface wave fields passing over a rippled bottom, especially in the case of shallow water.