The Korteweg–de Vries (KdV) equation is widely known as a canonical model for weakly nonlinear and weakly dispersive waves, notably and historically for water waves in shallow depths. Being integrable, it has a rich solution set of interacting solitary and periodic waves. Recently, we extended it with several forcing/friction terms to describe the evolution of wind-driven water wave packets in shallow water. The outcome is a modified KdV–Burgers equation, whose relevant solutions are principally solitary wave trains forming a soliton gas. In this article that is extended further by allowing the water depth to be slowly spatially varying, and introducing a basic horizontal current, also slowly spatially varying. The outcome is a modified KdV–Burgers equation with spatially slowly varying coefficients. We adapt the Whitham modulation theory for a slowly varying solitary wave train, allowing for the prediction of wave amplitude growth/decay due to a combination of the slowly varying background and the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to exhibit and validate the modulation theory.

In our recent papers Maleewong and Grimshaw (2024b, 2025), we used the Korteweg-de Vries (KdV) equation and its two-dimensional extension, the Kadomtsev-Petviashvili (KP) equation to describe the evolution of wind-driven water wave packets in shallow water. Both equations were modified to include the effect of wind forcing, modelled using the Miles critical level instability theory. In this paper that is extended to a Benney-Luke (BL) equation, similarly modified for wind forcing. The motivation is that the BL equation is isotropic in the horizontal space variables, unlike the KP model, and noting that the KdV model is one-dimensional. The modified BL equation is studied using wave modulation theory as in our previous work on the forced KdV and KP equations, and with comprehensive numerical simulations. Despite the very different spatial structure the results show that under the right initial conditions and parameter settings, solitary wave trains again emerge.

In a recent paper, denoted by MG24 in this text, we used a modified Korteweg–de Vries (KdV) equation to describe the evolution of wind-driven water wave packets in shallow water. The modifications were several forcing/friction terms describing wave growth due to critical-level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave growth due to turbulent wave stress in the air near the interface, and wave decay due to a turbulent bottom boundary layer. The outcome was a KdV–Burgers type of equation that can be a stable or unstable model depending on the forcing/friction parameters. In most cases that we examined, many solitary waves are generated, suggesting the formation of a soliton gas. In this paper, we extend that model in the horizontal direction transverse to the wind forcing to produce a similarly modified Kadomtsev–Petviashvili equation (KPII for water waves in the absence of surface tension). A modulation theory is described for the cnoidal and solitary wave solutions of the unforced KP equation, focusing on the forcing/friction terms and the transverse dependence. Then, using similar initial conditions to those used in MG24, that is a sinusoidal wave with a slowly varying envelope, but supplemented here with a transverse sinusoidal term, we find through numerical simulations that the radiation field upstream is enhanced, but that a soliton gas still emerges downstream as in MG24.

The evolution of wind waves is often accompanied by the presence of a wind drift layer, a shear flow just beneath the water surface, of small depth and weak surface current, but with strong shear. The forced nonlinear Schrödinger equation is a canonical model for the evolution of wave packets, where the forcing is through a critical level instability in the air flow, and the water wave envelope is described by the soliton and breather solutions as models of a wave packet. We determine how a wind drift layer affects the forced nonlinear Schrödinger equation, and hence how the wind drift layer affects the evolution of water wave packets. Although the effect is quite small, in the absence of surface tension and in the deep water limit, we find through analytical and numerical analysis, that a wind drift layer tends to slightly increase the growth rate, enhance wave packet amplitudes and decrease the wave packet width.

The generation of water waves by wind is a fundamental and much-studied problem of scientific and operational concern. One mechanism that has obtained a wide degree of acceptance and use is the Miles air shear flow instability theory in which water waves grow due to energy transfer from the air at a critical level where the wave phase speed matches the wind speed. In this paper we revisit and extend this theory by examining how the predicted wave growth rate depends on the water depth and the wind shear parameters. At the same time we include frictional effects due to water bottom stress and at the water surface, and add an additional driving term due to wave stress in the air near the sea surface, using a turbulent parametrisation. The theory is developed using a frictional modification of the usual potential flow theory for water waves, the modified Euler equations, coupled to linearised air-flow equations. To assist our analysis we develop a long-wave approximation for the key Miles growth parameter, which explicitly shows how this depends on the water depth and the wind shear parameters. Since our focus is on the development of wave groups, we present a forced nonlinear Schrödinger equation in which the forcing term describes wave growth or decay. For very shallow water the modified Euler equations are reduced to their shallow-water counterpart, which are briefly discussed using Riemann invariants.

The critical level instability mechanism for the generation of water waves by wind is re-examined for the situation when the atmosphere is density stratified. The density stratification is confined to the middle and upper atmosphere and then two cases are investigated. In case (A) no internal gravity waves are generated in the upper atmosphere and the effect of the density stratification is very small. In case (B) vertically propagating internal gravity waves form in the upper atmosphere and travel to infinity causing an energy loss, thus inhibiting the critical level instability in the lower atmosphere. Both cases are examined quantitatively for a logarithmic wind shear profile.

We use the Korteweg-de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air-water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV-Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

For weakly nonlinear waves in one space dimension, the nonlinear Schrödinger Equation is widely accepted as a canonical model for the evolution of wave groups described by modulation instability and its soliton and breather solutions. When there is forcing such as that due to wind blowing over the water surface, this can be supplemented with a linear growth term representing linear instability leading to the forced nonlinear Schrödinger Equation. For water waves in two horizontal space dimensions, this is replaced by a forced Benney–Roskes system. This is a two-dimensional nonlinear Schrödinger Equation with a nonlocal nonlinear term. In deep water, this becomes a local nonlinear term, and it reduces to a two-dimensional nonlinear Schrödinger Equation. In this paper, we numerically explore the evolution of wave groups in the forced Benney–Roskes system using four cases of initial conditions. In the one-dimensional unforced nonlinear Schrödinger equa tion, the first case would lead to a Peregrine breather and the second case to a line soliton; the third case is a long-wave perturbation, and the fourth case is designed to stimulate modulation instability. In deep water and for finite depth, when there is modulation instability in the one-dimensional nonlinear Schdrödinger Equation, the two-dimensional simulations show a similar pattern. However, in shallow water where there is no one-dimensional modulation instability, the extra horizontal dimension is significant in producing wave growth through modulation instability.

In many physical contexts, notably including deep-water waves, modulation instability in one space dimension is often studied by using the nonlinear Schrödinger equation. The principal solutions of interest are solitons and breathers which are adopted as models of wave packets. The Peregrine breather in particular is often invoked as a model of a rogue wave. In this paper, we add a linear growth term to the nonlinear Schrödinger equation to model the amplification of propagating wave groups. This is motivated by an application to wind-generated water waves, but this forced nonlinear Schrödinger equation potentially has much wider applicability. We describe a series of numerical simulations which in the absence of the forcing term would generate solitons and/or breathers. We find that overall the effect of the forcing term is to favour the generation of solitons with amplitudes growing at twice the linear growth rate over the generation of breathers.

A modified fully nonlinear model of an air–water system in deep water is presented in which the effect of wind in the air is simply represented by a direct link between the air–water interface pressure and the interface slope. The water system is a fully nonlinear Euler model of incompressible and irrotational fluid flow. Our main aim is to establish and compare with a reduced model represented by a forced nonlinear Schrödinger (FNLS) equation that describes wave groups in a weakly nonlinear asymptotic limit. Wave groups are described by the soliton and breather solutions generated by four cases of initial conditions in the relevant parameter regime. Numerical simulations of wave group formation in both models are compared, both with and without wind forcing. The FNLS model gives a good prediction to the modified fully nonlinear model when the wavenumber and wave frequency of the initial carrier waves are close to unity in dimensionless units based on typical carrier wavenumber and wave frequency. Wind forcing induces an exponential growth rate in the maximum amplitude wave. When the wave steepness becomes high in the fully nonlinear model some wave breaking is observed, but the FNLS model continues to predict large waves without breaking and there is then agreement only in the initial stage for the relevant initial conditions and parameter value ranges.

The rainfall data set usually contains missing values due to easily broken sensors. In Thailand, many public agencies collect rainfall values, including National Hydro Informatics (HII), Thai Meteorological Department, etc., since the data are valuable in terms of rainfall prediction, which is important for an agricultural country like Thailand. The rainfall data is normally collected hourly, and because there are many sensor locations, it is hard to maintain these sensors. The sensor data can be lost transiently and/or may yield anomaly values. Since there is a lot of data flowing to the server every day, it is hard to inspect manually or even semi-manually. This project collaborates with HII to develop a system that automates the rainfall data quality improvement process. The machine learning algorithms are used as tools for data cleansing. The derived data can be exposed as an open data set for many developers to explore new innovations. We explore data set characteristics and adopt both statistical and machine learning methods. The results show that the approach used both statistical and machine learning resulting in higher accuracy than using only statistical or machine learning approaches. We also develop a web application to visualize rainfall data results after cleansing and be connected to the models for the automatic cleansing pipelines.

Water level data obtained from telemetry stations typically contains large number of outliers. Anomaly detection and a data imputation are necessary steps in a data monitoring system. Anomaly data can be detected if its values lie outside of a normal pattern distribution. We developed a median-based statistical outlier detection approach using a sliding window technique. In order to fill anomalies, various interpolation techniques were considered. Our proposed framework exhibited promising results after evaluating with F1-score and root mean square error (RMSE) based on our artificially induced data points. The present system can also be easily applied to various patterns of hydrological time series with diverse choices of internal methods and fine-tuned parameters. Specifically, the Spline interpolation method yielded a superior performance on non-cyclical data while the long short-term memory (LSTM) outperformed other interpolation methods on a distinct tidal data pattern.

A steady forced Korteweg–de Vries (fKdV) model which includes gravity, capillary, and pressure distributions is solved numerically using the wavelet Galerkin method. The anti-derivatives of Daubechies wavelets are developed as the basis of the solution subspaces for the mixed boundary condition type. Accuracy of numerical solutions can be improved by increasing the number of wavelet levels in the multi-resolution analysis. The theoretical result of convergence rate is also shown. The problem can be viewed as gravity-capillary wave flows over an applied pressure distribution. The flow regime can be characterized by subcritical, supercritical, and critical flows depending on the value of the Froude number. Trapped depression and elevation waves are found over the pressure distribution. For a near-critical flow regime, a generalized solitary wave with ripples is presented. This shows a capillary effect in balance to gravity and the pressure force on the free surface.

Compute Unified Device Architecture (CUDA) implementations are presented of a well-balanced finite volume method for solving a shallow water model. The CUDA platform allows programs to run parallel on GPU. Four versions of the CUDA algorithm are presented in addition to a CPU implementation. Each version is improved from the previous one. We present the following techniques for optimizing a CUDA program: limiting register usage, changing the global memory access pattern, and using loop unroll. The accuracy of all programs is investigated in 3 test cases: a circular dam break on a dry bed, a circular dam break on a wet bed, and a dam break flow over three humps. The last parallel version shows 3.84x speedup over the first CUDA implementation. We use our program to simulate a real-world problem based on an assumed partial breakage of the Srinakarin Dam located in Kanchanaburi province, Thailand. The simulation shows that the strong interaction between massive water flows and bottom elevations under wet and dry conditions is well captured by the well-balanced scheme, while the optimized parallel program produces a 57.32x speedup over the serial version.

A time delay epidemic model is presented for the spread of the Coronavirus 2019 (COVID-19) in China. The time delay effects affect infected individuals. Monte Carlo simulation is performed to estimate the transmission and recovery rates. The basic reproduction number is estimated in terms of the average infected ratio. This ratio can be used to monitor the policy performance of disease control during the spread of the disease.

This paper extends a previous study of free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation, to an analogous study of flow over two localised holes, or a combination of an obstacle and a hole. Importantly the terminology obstacle or hole can be reversed for a stratified fluid and refers more precisely to the relative polarity of the forcing and the solitary wave solution of the unforced Korteweg–de Vries equation. As in the previous study, our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. In the transcritical regime at early times, undular bores are produced upstream and downstream of each forcing site. We then describe the interaction of these undular bores between the forcing sites, and the outcome at very large times.

This paper developed the anti-derivative wavelet bases to handle the more general types of boundary conditions: Dirichlet, mixed and Neumann boundary conditions. The boundary value problem can be formulated by the variational approach, resulting in a system involving unknown wavelet coefficients. The wavelet bases are constructed to solve the unknown solutions corresponding to the types of solution spaces. The augmentation method is presented to reduce the dimension of the original system, while the convergence rate is in the same order as the multiresolution method. Some numerical examples have been shown to confirm the rate of convergence. The examples of the singularly perturbed problem with Neumann boundary conditions are also demonstrated, including highly oscillating cases.

Flood simulation of a region in southern Thailand during January 2017 is presented in this work. The study area covers the Tapi river, the longest river in southern Thailand. The simulation is performed by applying the two-dimensional shallow water model in the presence of strong source terms to the local bottom topography. The model is solved numerically by our finite volume method with well-balanced property and linear reconstruction technique. This technique is accurate and efficient at solving for complex flows in the wet/dry interface problem. Measurements of flows are collected from two gauging stations in the area. The initial conditions are prepared to match the simulated flow to the measurements recorded at the gauging stations. The accuracy of the numerical simulations is demonstrated by comparing the simulated flood area to satellite images from the same period. The results are in good agreement, indicating the suitability of the shallow water model and the presented numerical method for simulating floodplain inundation.

The numerical method of data-driven modeling is presented in this work. It can be used to find the corresponding governing equations in terms of differential equations from some provided spatial and time series data. The problem can be reduced to solve by sparse linear regression technique that can be referred to one of the powerful technique in machine learning algorithm. Here we propose to apply the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) and then study the capability of its predictions. The environmental problem involves advection and diffusion effects are demonstrated to show the accuracy and the efficiency for finding all of the coefficients in the partial differential equation corresponding to the input data.

Finite volume method with reconstruction and bottom modification techniques for simulating open channel flows in arbitrary cross-sectional areas is presented. These techniques are introduced to handle the difficulty in approximating water depth at wet/dry areas. Various numerical experiments with source terms are demonstrated to confirm the accuracy of numerical scheme. Further, we have applied the present method to simulate open channel flow in the Yom River, Phrae Province, Thailand. The simulation results are compared with measured data.