Nonlinear Water Research Articles

Actionable human–water system modelling under uncertainty

Abstract. This paper develops an actionable interdisciplinary model that quantifies and assesses uncertainties in water resource allocation under climate change. To achieve this objective, we develop an innovative socio-ecological grand ensemble that combines climate, hydrological, and microeconomic ensemble experiments with a widely used decision support system for water resource planning and management. Each system is populated with multiple models (multi-model), which we use to evaluate the impacts of multiple climate scenarios and policies (multi-scenario, multi-forcing) across systems so as to identify plausible futures where water management policies meet or miss their objectives and to explore potential tipping points. The application of the methods is exemplified by a study conducted in the Douro River basin (DRB), an agricultural basin located in central Spain. Our results show how marginal climate changes can trigger non-linear water allocation changes in the decision support systems (DSSs) and/or non-linear adaptive responses of irrigators to water shortages. For example, while some irrigators barely experience economic losses (average profit and employment fall by < 0.5 %) under mild water allocation reductions of 5 % or lower, profit and employment fall by up to 12 % (∼ 24 ×) when water allocation is reduced by 10 % or less (∼ 2×). This substantiates the relevance of informing the potential natural and socio-economic impacts of adaptation strategies and related uncertainties for identifying robust decisions.

Physical modelling study on wave damping induced by an idealized floating kelp farm

A physical modelling study was carried out to investigate random wave damping promoted by an idealized floating kelp farm. The experimental conditions spanned intermediate water depths and both linear and nonlinear water waves. Unlike previous studies of wave damping promoted by vegetation, the floating kelp farm was placed close to the water surface with a ratio between vegetation height and water depth close to 0.25. The wave transmission coefficient induced by the floating kelp farm ranged between 0.56 and 0.96. This coefficient decreased for longer floating kelp farms and it was a function of the ratio between kelp farm length and incident wavelength and of the relative wave depth. Spectral analysis showed that wave damping was not frequency-dependent for wave frequencies close to the peak frequency. The wave transmission coefficients of a floating kelp farm with about 100 culture lines and with an extension of approximately 200 m were similar to those of submerged detached breakwaters with a relative crest freeboard smaller than -0.4. Furthermore, the bulk drag coefficient of near-surface idealized floating kelp farms can be modelled as a function of the Keulegan-Carpenter number. This study highlights the potential viability of nature-based solutions such as floating kelp farms for coastal protection.

Soliton solutions of nonlinear coupled Davey–Stewartson Fokas system using modified auxiliary equation method and extended (G′/G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G^{2})$$\\end{document}-expansion method

The captivating realm of the nonlinear coupled Davey–Stewartson Fokas system is explored in this research paper. As a powerful tool, the proposed system is utilized for the realistic representation of various non-linear dynamical mechanisms in different fields of sciences and engineering including non-linear optical fibers, plasma physics and water waves theory. Two distinct exact methods, namely the modified auxiliary equation method and the extended (G′/G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G^{2})$$\\end{document}-expansion method, are utilized to acquire the exact soliton solutions of the non-linear coupled Davey–Stewartson Fokas system. A plethora of novel soliton solutions containing anti-kink, kink, bright, dark, dark-bright, bright-dark and some other singular soliton solutions, have been obtained using the employed exact methods. The significance of proposed manuscript lies in the novelty of obtained solutions. Kink, dark and bright solitons have wide applications in optical fiber communications, plasma physics and water waves dynamics. The acquired nontrivial exact solutions contain exponential, trigonometric, rational and hyperbolic functions. Some obtained solutions are visually represented through graphical simulations of 3D, 2D-contour and 2D-line plots, providing a comprehensive visualization of the soliton dynamics.The modulation instability of the proposed nonlinear system has been investigated, which ensures the stability of the system.

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

The deep latent space particle filter for real-time data assimilation with uncertainty quantification

In data assimilation, observations are fused with simulations to obtain an accurate estimate of the state and parameters for a given physical system. Combining data with a model, however, while accurately estimating uncertainty, is computationally expensive and infeasible to run in real-time for complex systems. Here, we present a novel particle filter methodology, the Deep Latent Space Particle filter or D-LSPF, that uses neural network-based surrogate models to overcome this computational challenge. The D-LSPF enables filtering in the low-dimensional latent space obtained using Wasserstein AEs with modified vision transformer layers for dimensionality reduction and transformers for parameterized latent space time stepping. As we demonstrate on three test cases, including leak localization in multi-phase pipe flow and seabed identification for fully nonlinear water waves, the D-LSPF runs orders of magnitude faster than a high-fidelity particle filter and 3-5 times faster than alternative methods while being up to an order of magnitude more accurate. The D-LSPF thus enables real-time data assimilation with uncertainty quantification for the test cases demonstrated in this paper.

Optical soliton solutions of nonlinear differential Boussinesq water wave equation via two analytical techniques

In this article, we examine the optical soliton solutions of the 4th-order nonlinear Boussinesq water wave equation using the modified (G′/G2)− expansion and F-expansion methods. We utilize the similarity transformation to transform the partial differential form of the equation into an ordinary differential form. These techniques present a variety of solutions, including bright, dark, singular, dark-periodic, and singular-periodic soliton solutions. We plot the derived solutions in several profiles, such as 2D, 3D, and contour, to illustrate their physical appearance. The determined findings might be useful and have an enormous effect on the research of nonlinear phenomena in a variety of physical areas of science, like shallow water waves, acoustics, fluid dynamics, laser optics, communication systems, and heat transfer. The achieved results demonstrate the validity, applicability, and effectiveness of the presented method. We plot and validate the found solutions using the computational program MATHEMATICA.

Laboratory investigations of wave-induced transport of plastic debris over a rippled bottom

Laboratory experiments are conducted in a wave flume to investigate the effect of water waves on the transport of plastic pellets over a rippled bottom. The horizontal velocities of plastic debris are analyzed over the rippled bottom for different wave conditions and plastic elements with different properties. Laboratory investigations determined the characteristic transport patterns of wave-induced plastic debris with a density of ∼2.0g/cm3 moving along the rippled bottom. In the first, swing-type motion, the grains move only in the ripple trough with velocities lower than 0.10 m/s. For sliding-type movement, the grains move along the entire rippled surface with velocities in the range of 0.10–0.13 m/s. For higher velocities in the range of 0.15–0.20 m/s, a saltation-type motion becomes dominant. The results show that plastic grains may move up to 2–3 cm above the ripple crest depending on hydrodynamic conditions. The analysis shows that for velocity-skewed flows, sliding-type motion and onshore transport dominate. For acceleration-skewed flows, saltation-type motion and offshore transport dominate, which is attributed to higher boundary layer thickness and phase lag effects. The analysis of the relationship between the particle Reynolds number and the thickness of the turbulent boundary layer reveals that for values of Rep≥1000 and a boundary layer thicknessδ≥50 mm saltation-type motion becomes dominant. The direction of transport is affected not only by the density of the sediment and the wave skewness coefficients but also by the dimensions of the bottom ripples. The laboratory investigations also provide insight into the hydrodynamic conditions affecting the transport of plastic debris along the bottom covered with ripples in oscillating nonlinear water flows.

A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

Physics-informed neural networks for fully non-linear free surface wave propagation

This study proposed fully nonlinear free surface physics-informed neural networks (FNFS-PINNs), an advancement within the framework of PINNs, to tackle wave propagation in fully nonlinear potential flows with the free surface. Utilizing the nonlinear fitting capabilities of neural networks, FNFS-PINNs offer an approach to addressing the complexities of modeling nonlinear free surface flows, broadening the scope for applying PINNs to various wave propagation scenarios. The improved quasi-σ coordinate transformation and dimensionless formulation of the basic equations are adopted to transform the time-dependent computational domain into the stationary one and align variable scale changes across different dimensions, respectively. These innovations, alongside a specialized network structure and a two-stage optimization process, enhance the mathematical formulation of nonlinear water waves and solvability of the model. FNFS-PINNs are evaluated through three scenarios: solitary wave propagation featuring nonlinearity, regular wave propagation under high dispersion, and an inverse problem of nonlinear free surface flow focusing on the back-calculation of an initial state from its later state. These tests demonstrate the capability of FNFS-PINNs to compute the propagation of solitary and regular waves in the vertical two-dimensional scenarios. While focusing on two-dimensional wave propagation, this study lays the groundwork for extending FNFS-PINNs to other free surface flows and highlights their potential in solving inverse problems.

Gisser-Sánchez revisited: A model of optimal groundwater withdrawal under irrigation including surface–groundwater interaction

We revisit the classic problem of determining economically optimal groundwater withdrawal rates for irrigation. The novelty compared to previous mathematical analyses is the inclusion of non-linear groundwater-surface water interaction that allows for incorporating the impact of capture, i.e. the fact that all or part of the pumped groundwater comes out of reduced surface water flow or increased recharge. We additionally included the option to internalize environmental externalities (e.g. streamflow depletion) and maximize social welfare rather than farmer’s profit. This analysis results in a fixed optimal groundwater withdrawal rate qopt when withdrawal q remains smaller than some critical withdrawal rate (maximum capture) qcrit and provides depletion trajectories, either under competition or optimal control, if q is larger than qcrit. Based on the relative value of q, qcrit and qopt it also yields four quadrants of distinct withdrawal strategies. Using global hydrogeological and hydroeconomic datasets we map the global occurrence of these four quadrants and provide global estimates of optimal groundwater withdrawal rates and depletion trajectories. For the quadrants with groundwater depletion (q > qcrit) we derive and compare depletion trajectories under competition, optimal control and optimal control including environmental externalities, and assessed globally where the differences between these depletion modes are small, which is known as the Gisser-Sánchez effect. We find that the Gisser-Sánchez effect is globally ubiquitous, but only if environmental externalities are ignored. The inclusion of environmental externalities in optimal control withdrawal result in notably reduced groundwater decline and larger values of social welfare in many of the major depletion areas.

Hydrological time series prediction based on IWOA-ALSTM

The prediction of hydrological time series is of great significance for developing flood and drought prevention approaches and is an important component in research on smart water resources. The nonlinear characteristics of hydrological time series are important factors affecting the accuracy of predictions. To enhance the prediction of the nonlinear component in hydrological time series, we employed an improved whale optimisation algorithm (IWOA) to optimise an attention-based long short-term memory (ALSTM) network. The proposed model is termed IWOA-ALSTM. Specifically, we introduced an attention mechanism between two LSTM layers, enabling adaptive focus on distinct features within each time unit to gather information pertaining to a hydrological time series. Furthermore, given the critical impact of the model hyperparameter configuration on the prediction accuracy and operational efficiency, the proposed improved whale optimisation algorithm facilitates the discovery of optimal hyperparameters for the ALSTM model. In this work, we used nonlinear water level information obtained from Hankou station as experimental data. The results of this model were compared with those of genetic algorithms, particle swarm optimisation algorithms and whale optimisation algorithms. The experiments were conducted using five evaluation metrics, namely, the RMSE, MAE, NSE, SI and DR. The results show that the IWOA is effective at optimising the ALSTM and significantly improves the prediction accuracy of nonlinear hydrological time series.

Nonisospectral water wave field: Fast and adaptive modal identification and prediction via reduced-order nonlinear solutions.

Real-world water wave fields exhibit significant nonlinear and nonisospectral characteristics, making it challenging to predict their evolution by relying solely on numerical simulation or exact solutions using integrable system theory. Hence, this paper introduces a fast and adaptive method of modal identification and prediction in nonisospectral water wave fields using the reduced-order nonlinear solution (RONS) scheme. Specifically, we discuss the coarse graining and mode extraction of wave field snapshots from the data-driven and physics-driven perspectives and utilize the RONS method for principle modal prediction of nonisospectral water wave fields. This is achieved by investigating the standard and nonisospectral Gardner system describing nonlinear water waves as a demonstration. Through detailed comparison and analysis, the fundamental solitary behaviors and dispersive effects in the Gardner system are discussed. Subsequently, a neighbor approximation is developed that combines the essences of symbolic precomputation and numerical computation in the RONS procedure, which exploits the locality of nonlinear interactions in water wave fields.

A Weakly Nonlinear System for Waves and Sheared Currents over Variable Bathymetry

The wave–current–seabed interaction problem is studied by using a coupled-mode system developed for modeling wave scattering by non-homogeneous, sheared currents in variable bathymetry regions. The model is based on a modal series expansion of wave velocity based on vertical eigenfunctions, dependent on local depth and flow parameters, including propagating and evanescent modes. The latter representation is able to accurately satisfy the wave flow continuity condition and the no-entrance boundary condition on the sloping parts of the seabed. A new derivation of a simplified nonlinear system is introduced using decomposition to a mean flow and a perturbative wave field. To force the system to consider incoming waves at the inlet, boundary knowledge of periodic, travelling nonlinear water waves over a flat bottom is required. For this purpose, specific solutions are derived using the semi-analytical method based on the stream function formulation, for cases of water waves propagating above linearly and exponentially sheared currents. Results obtained by the application of the CMS concerning the propagation of waves and currents—in particular, examples characterized by depth inhomogeneities—are presented and discussed, illustrating the applicability and performance of the method.

Exploring solitary wave solutions to the simplified modified camassa-holm equation through a couple sophisticated analytical approaches

The simplified modified Camassa-Holm equation, a developed Korteweg-de Vries equation, is an integrable nonlinear dispersive water wave equation with potential applications in diverse fields such as shallow water waves, liquid drop patterning, water surface waves in channels, tsunamis in coastal regions, ocean engineering, etc. In this study, we derived solitary wave solutions to the described equation, employing two reliable approaches: the enhanced modified simple equation and the extended Kudryashov methods. The enhanced modified simple equation technique yields solutions with variable coefficients through a generalized wave transformation. On the other hand, the extended Kudryashov method exploits the solutions of the Riccati and Bernoulli differential equations. The determined results include diverse soliton solutions expressed in the form of trigonometric, hyperbolic, and rational functions. Notably, for specific parameter values, these solutions exhibit wave of distinct shapes, including kink, smooth kink, flat kink, periodic, singular periodic, one-sided kink, etc. To apparently explain the physical behavior and the impact of the fractional parameter in the solutions, we plotted three- and two-dimensional graphs. The obtained solutions have potential applications in oceanography, spanning offshore rigs, ocean gravity waves, and wave energy.

WaveNets: physics-informed neural networks for full-field recovery of rotational flow beneath large-amplitude periodic water waves

AbstractWe formulate physics-informed neural networks (PINNs) for full-field reconstruction of rotational flow beneath nonlinear periodic water waves using a small amount of measurement data, coined WaveNets. The WaveNets have two NNs to, respectively, predict the water surface, and velocity/pressure fields. The Euler equation and other prior knowledge of the wave problem are included in WaveNets loss function. We also propose a novel method to dynamically update the sampling points in residual evaluation as the free surface is gradually formed during model training. High-fidelity data sets are obtained using the numerical continuation method which is able to solve nonlinear waves close to the largest height. Model training and validation results in cases of both one-layer and two-layer rotational flows show that WaveNets can reconstruct wave surface and flow field with few data either on the surface or in the flow. Accuracy in vorticity estimate can be improved by adding a redundant physical constraint according to the prior information on the vorticity distribution.

Numerical Investigation of Fractional Kawahara Equation via Haar Scale Wavelet Method

The Kawahara equation is a fifth-order dispersive equation that plays a significant role in explaining the creation of non-linear water waves in the long-wavelength region. In this research, the Kawahara equation is solved numerically using the novel Haar scale-3 wavelet method in conjunction with the collocation method. The quasilinearisation approach and the Caputo derivative are used to characterise the non-linearity and fractional behaviour of the equation, respectively. To verify that the findings obtained are legitimate, residual and error estimates are generated. A thorough comparison is made between the present solutions and the numerical findings that have already been published in the literature, which demonstrates the advantages and effectiveness of the suggested technique. The Haar wavelet method reveals a dynamic system of alternative solutions for a wide variety of physical parameters.

Study on the Optimization of Wujiang’s Water Resources by Combining the Quota Method and NSGA-II Algorithm

Recently, the Chinese government has implemented stringent water requirements based on the concept of ‘Basing four aspects on water resources’. However, existing research has inadequately addressed the constraints of water resources on population, city boundaries, land, and production, failing to adequately analyze the interplay between water resource limitations and urban development. Recognizing the interconnectedness between urban water use and economic development, a multi-objective model becomes crucial for optimizing urban water resources. This study establishes a nonlinear multi-objective water resources joint optimization model, aligning with the “Basing four aspects on water resources” requirement to maximize urban GDP and minimize total water use. A genetic algorithm (NSGA-II Algorithm) is applied to solve this complex nonlinear multi-objective model and obtain the Pareto solution set, addressing information loss inherent in the traditional water quota method. The model was tested in Wujiang District, an area located in China’s Jiangsu Province that has been rapidly urbanizing over the past few decades, and yielded 50 non-inferior water resource optimization schemes. The results reveal that the Pareto solution set visually illustrates the competition among objectives and comprehensively displays the interplay between water and urban development. The model takes a holistic approach to consider the relationships between water resources and urban population, land use, and industries, clearly presenting their intricate interdependencies. This study serves as a valuable reference for the rational optimization of water resources in urban development.

Breathers of Gerdjikov-Ivanov equation under background of elliptic function

As one specific type of local solutions of nonlinear evolution equation, the breathers have the characteristic of envelope oscillation structure. This kind of oscillation is periodic. According to the periodicity of the distribution and evolution directions, there are three kinds of breathers, namely, the Kuznetsov-Ma breather (KMB), the Akhmediev breather (AB), and the general breather (GB). In recent years, the propagation of envelope breathers under the periodic background has been observed in many nonlinear physical fields, including nonlinear optical fibers and hydrodynamics. It is believed that the breathers can arise due to the modulational instability of the periodic waves, and they demonstrate many rich physical properties and dynamic behaviors of interactions. Therefore, recently great attention has been paid to the breathers under the periodic background in nonlinear science. As an important integrable model, the Gerdjikov-Ivanov (GI) equation can be used to describe various nonlinear phenomena in many physical fields such as in the quantum field theory, weak nonlinear dispersive water wave, and nonlinear optics. It is very meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the GI equation have not been given under the elliptic function background. In this study, firstly, elliptic function solutions of the GI equation are solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtain the basic solution of the Lax pair corresponding to the Jacobi elliptic function seed solution. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution can be expressed in terms of the Weierstrass elliptic function. Secondly, by the once iterated Darboux transformation, three types of breather solutions under the elliptic function background are constructed including the GB, the KMB and the AB. In addition, we analyze the dynamic behaviors of these three kinds of breathers, and present their three-dimensional space-time structures. By the twice iterated Darboux transformation, under the dn-periodic background we exhibit three types of interactions between two breathers, i.e. a GB and a KMB, an AB and a KMB, and a GB and an AB. Finally, we also present three types of interactions between two breathers under the general periodic background.

A unified deep learning framework for water quality prediction based on time-frequency feature extraction and data feature enhancement

Deep learning methods exhibited significant advantages in mapping highly nonlinear relationships with acceptable computational speed, and have been widely used to predict water quality. However, various model selection and construction methods resulted in differences in prediction accuracy and performance. Hence, a unified deep learning framework for water quality prediction was established in the paper, including data processing module, feature enhancement module, and data prediction module. In the established model, the data processing module based on wavelet transform method was applied to decomposing complex nonlinear meteorology, hydrology, and water quality data into multiple frequency domain signals for extracting self characteristics of data cyclic and fluctuations. The feature enhancement module based on Informer Encoder was used to enhance feature encoding of time series data in different frequency domains to discover global time dependent features of variables. Finally, the data prediction module based on the stacked bidirectional long and short term memory network (SBiLSTM) method was employed to strengthen the local correlation of feature sequences and predict the water quality. The established model framework was applied in Lijiang River in Guilin, China. The maximum relative errors between the predicted and observed values for dissolved oxygen (DO), chemical oxygen demand (CODMn) were 12.4% and 20.7%, suggesting a satisfactory prediction performance of the established model. The validation results showed that the established model was superior to all other models in terms of prediction accuracy with RMSE values 0.329, 0.121, MAE values 0.217, 0.057, SMAPE values 0.022, 0.063 for DO and CODMn, respectively. Ablation tests confirmed the necessity and rationality of each module for the established model framework. The established method provided a unified deep learning framework for water quality prediction.

Stability of constrained solitary waves for the Ostrovsky–Vakhnenko model in the coastal zone

In this paper, we consider a nonlinear water wave model describing the evolution of the internal tides in the coastal zone. First, we prove the existence of minimizers for the corresponding energy functional on the L2-norm constrained manifold by the concentrated compactness principle, and thus we obtain a ground state traveling wave solution of the model. Moreover, we show that the solution has exponential decay in the far field. Finally, we show that the traveling wave solution is non-degenerate, and thus the spectral stability of the solution can be obtained by using the instability index theory.